Adapting Laplace residual power series approach to the Caudrey Dodd Gibbon equation

In real-life applications, nonlinear differential equations play an essential role in representing many phenomena. One well-known nonlinear differential equation that helps describe and explain many chemicals, physical, and biological processes is the Caudrey Dodd Gibbon equation (CDGE). In this paper, we propose the Laplace residual power series method to solve fractional CDGE. The use of terms that involve fractional derivatives leads to a higher degree of freedom, making them more realistic than those equations that involve the derivation of an integer order. The proposed method gives an easy and faster solution in the form of fast convergence. Using the limit theorem of evaluation, the experimental part presents the results and graphs obtained at several values of the fractional derivative order.

Fractional analysis of non-linear fuzzy partial differential equations by using a direct procedure

In this study, an accurate analytical solution is presented for fuzzy FPDEs. It is done by using a novel method called the Laplace-residual power series (LRPSM) to build a series solution to the given problems. The fundamental instruments of the employed method are the Laplace transform, fractional Laurent, and fractional power series. Using the idea of a limit at infinity, we provide a series solution to a fuzzy FPDE with quick convergence and simple coefficient finding. We analyze three cases to obtain approximate and exact solutions to show the effectiveness and reliability of the Laplace- residual power series approach. To demonstrate the accuracy of the suggested procedure, we compare the findings to the real data.

The fractional analysis of thermo-elasticity coupled systems with non-linear and singular nature

It is mentioned that understanding linear and non-linear thermo-elasticity systems is important for understanding temperature, elasticity, stresses, and thermal conductivity. One of the most crucial aspects of the current research is the solution to these systems. The fractional form of several thermo-elastic systems is explored, and elegant solutions are provided. The solutions of fractional and integer thermo-elastic systems are further discussed using tables and diagrams. The closed contact between the LRPSM and exact solutions is displayed in the graphs. Plotting fractional problem solutions demonstrates their convergence towards integer-order problem solutions for suitable modeling. The tables confirm that greater precision is rapidly attained as the terms of the derived series solution increase. The faster convergence and stability of the suggested method support its modification for other fractional non-linear complex systems in nature.

Theoretical estimation of the thermal damages of living tissues caused by laser irradiation in tumor thermal therapy

This article aims to provide theoretical predictions for the thermal reactions of human tissues during tumor thermotherapy when exposed to laser irradiation and an external heat source. For the construction of a theoretical study of bioheat transfer, the selection of a suitable thermal model capable of accurately predicting the required thermal responses is essential. The effect of heat production by heat treatment on a spherical multilayer tumor tissue is evaluated using this approach. Analytical solution for the non-homogenous differential equations is derived in the Laplace domain. The study examines the impact of thermal relaxation time on tissue temperature and the subsequent thermal damage. The numerical findings of thermal damage and temperatures are depicted in a graphical representation. This model explains laser treatment, physical events, metabolic support, and blood perfusion. The numerical outcomes of the recommended model are validated by comparing them to the literatures.

Phase-type distributions in mathematical population genetics: An emerging framework

A phase-type distribution is the time to absorption in a continuous- or discrete-time Markov chain. Phase-type distributions can be used as a general framework to calculate key properties of the standard coalescent model and many of its extensions. Here, the 'phases' in the phase-type distribution correspond to states in the ancestral process. For example, the time to the most recent common ancestor and the total branch length are phase-type distributed. Furthermore, the site frequency spectrum follows a multivariate discrete phase-type distribution and the joint distribution of total branch lengths in the two-locus coalescent-with-recombination model is multivariate phase-type distributed. In general, phase-type distributions provide a powerful mathematical framework for coalescent theory because they are analytically tractable using matrix manipulations. The purpose of this review is to explain the phase-type theory and demonstrate how the theory can be applied to derive basic properties of coalescent models. These properties can then be used to obtain insight into the ancestral process, or they can be applied for statistical inference. In particular, we show the relation between classical first-step analysis of coalescent models and phase-type calculations. We also show how reward transformations in phase-type theory lead to easy calculation of covariances and correlation coefficients between e.g. tree height, tree length, external branch length, and internal branch length. Furthermore, we discuss how these quantities can be used for statistical inference based on estimating equations. Providing an alternative to previous work based on the Laplace transform, we derive likelihoods for small-size coalescent trees based on phase-type theory. Overall, our main aim is to demonstrate that phase-type distributions provide a convenient general set of tools to understand aspects of coalescent models that are otherwise difficult to derive. Throughout the review, we emphasize the versatility of the phase-type framework, which is also illustrated by our accompanying R-code. All our analyses and figures can be reproduced from code available on GitHub.

Modelling simultaneous transport of natural and anthropogenic radionuclides in fractured media - Diffusion into the heterogeneous layered rock matrix for an arbitrary length decay chain

Transport models used for assessing the safety of radioactive waste repositories hosted in fractured bedrock typically do not consider fluxes of naturally occurring radionuclides in the rock and their further migration in flow-bearing fractures. A consistent model that simultaneously describes the transport of radionuclides from both natural and anthropogenic sources has been developed, where decay chains and rock heterogeneity are accounted for. The model accounts for advective flow in the fracture, a decay chain of arbitrary length, and diffusion into and out of the adjacent rock matrix composed of different geological layers. The proposed solution has been verified against a previously published steady state case which considers a homogeneous rock matrix of infinite extent where porewater ingrowth is not accounted for. The model is also applied to some different calculation examples for both transient and limiting steady state cases to represent typical applications of the model as well as to illustrate the effect of different parameters and processes on the transport of natural radionuclides in fractured rocks. This study presents a novel and powerful tool to simulate migration of both anthropogenic and natural radionuclides in and from crystalline rocks to the biosphere. The presented modelling is essential in safety and performance assessment of deep geological disposal of radioactive waste in fractured rocks. The obtained analytical solution can be used to compare relative fluxes of natural and anthropogenic radionuclides, which is useful for validation of the radionuclide transport parameters obtained from field and laboratory experiments.

Modified conformable double Laplace-Sumudu approach with applications

In this study, we combine two novel methods, the conformable double Laplace-Sumudu transform (CDLST) and the modified decomposition technique. We use the new approach called conformable double Laplace-Sumudu modified decomposition (CDLSMD) method, to solve some nonlinear fractional partial differential equations. We present the essential properties of the CDLST and produce new results. Furthermore, five interesting examples are discussed and analyzed to show the efficiency and applicability of the presented method. The results obtained show the strength of the proposed method in solving different types of problems.

SDIQR mathematical modelling for COVID-19 of Odisha associated with influx of migrants based on Laplace Adomian decomposition technique

In this study, the Laplace Adomian decomposition technique (LADT) is employed to analyse a numerical study with the SDIQR mathematical model of COVID-19 for infected migrants in Odisha. The analytical power series and LADT are applied to the Covid-19 model to estimate the solution profiles of the dynamical variables. We proposed a mathematical model that incorporates both the resistive class and the quarantine class of COVID-19. We also introduce a procedure to evaluate and control the infectious disease of COVID-19 through the SDIQR pandemic model. Five compartments like susceptible (S ), diagnosed (D ), infected (I ), quarantined (Q ) and recovered (R ) population are found in our model. The model can only be solved approximately rather than analytically as it contains a system of nonlinear differential equations with reaction rates. To demonstrate and validate our model, the numerical simulations for infected migrants are plotted with suitable parameters.

Analytical solution of Nye-Tinker-Barber model by Laplace transform

The Nye-Tinker-Barber model is a classical convection-diffusion model for nutrient uptake by plant roots in cylindrical coordinates and has one nonlinear left Robin boundary condition with Michaelis-Menten function of concentration. First the Michaelis-Menten function is fitted into a function of time by numerical concentration at root surface from difference scheme, and then the Laplace and numerical inverse Laplace transforms - Zakian inversion method are taken to obtain the approximate analytical solution. Compared with other solutions made by difference scheme, Stehfest inversion method and previous analytical methods, it is found that the analytical solution obtained by Laplace and Zakian inversion transforms has higher accuracy and computation efficiency. This analytical method can be extended to other nutrient uptake models with Michaelis-Menten function.

Heat transfer analysis for tissue with surface heat flux based on the non-linearized form of the three-phase-lag model

The three-phase-lag model of heat conduction has been proposed for considering thermoelastic effect in medium. The bioheat transfer equations based on Taylor's series approximation of the three-phase-lag model were derived in conjunction with a modified energy conservation equation. For exploring the effect of non-linear expansion in the phase lag times, the Taylor's series of second-order expansion was applied. The resulting equation involves mixed derivative terms and higher-order derivatives of temperature with respect to time. The hybrid application of the Laplace transform method and a modified discretization technique was extended to solve the equations and explore the effect of thermoelasticity on the thermal behavior in living tissue with surface heat flux. The influence of thermoelastic parameters and phase lags on heat transfer in tissue has been investigated. The present results illustrate the thermal response oscillation is excited in medium for the thermoelastic effect, the phase lag times significantly affect the amplitude and frequency of the oscillation, and the expansion order of TPL model evidently affects the predicted temperature.

Analytical and numerical analysis of the dual-pulse lag heat transfer in a three-dimensional tissue subjected to a moving multi-point laser beam

An extensive algorithm based on both analytical and numerical solution methodologies is proposed to obtain transient temperature distributions in a three-dimensional living tissue subjected to a moving single-point and multi-point laser beam by considering metabolic heat generation and blood perfusion rate. Here, the dual-phase lag/Pennes equation is analytically solved by using the method of Fourier series and the Laplace transform. The ability to model single-point or multi-point laser beams as an arbitrary function of place and time is a significant advantage of the proposed analytical approach, which can be used to solve similar heat transfer problems in other living tissues. Besides, the related heat conduction problem is numerically solved based on the finite element method. The effects of laser beam transitional speed, laser power, and the number of laser points on the temperature distribution within the skin tissue are investigated. Moreover, the temperature distribution predicted by the dual-phase lag model is compared with that of the Pennes model under different working conditions. For the studied cases, it is observed that the maximum tissue temperature decreased about 63% by an increase of 6mm/s in the speed of the laser beam. An increase in the laser power from 0.8W/cm³ to 1.2W/cm³ results in a 28 °C increase in the maximum temperature of the skin tissue. It is observed that the maximum temperature predicted by the dual-phase lag model is always lower than that of the Pennes model and the temperature variations over time are sharper, while their results are entirely consistent over the simulation time. The obtained numerical results indicated that the dual-phase lag model is preferred in heating processes occurring at short intervals. Among the investigated parameters, the laser beam speed has the most considerable effect on the difference between the results of the Pennes and the dual-phase lag models.

Heat distribution and the condition of hypothermia in the multi-layered human head: A mathematical model

The conduction, perfusion and metabolic heat generation based partial differential equation has been used to study the heat transfer in human head. The main objective of this study is to predict the temperature distribution at the multi-layered human head that results in hypothermic condition. The temperature profiles have been estimated at the interface points of brain, skull and scalp with respect to various parameters including atmospheric temperature, arterial temperature and metabolic heat generation. The variational finite element method and analytical method based on Laplace transform has been employed to establish the solution of the formulated model, and the resulting outcomes are illustrated graphically. Under cold exposure, the blood capillaries around scalp exchange core heat with the external cold environment and experience lowering in the tissue temperature of the blood in the scalp. It is reflected in the graphical view of the model that the prolonged exposure to cold transmits its effect into the deep brain capillaries, wherein the temperature gradually lowers down below the normal body temperature that results hypothermia and hence abnormal body homoeostasis.

Theoretical investigation on resonance characteristics of a vapor bubble based on Laplace transform method

In the present paper, resonance characteristics of the vapor bubble oscillating in an acoustic field are investigated analytically. The analytical solution of the non-dimensional perturbation of the instantaneous bubble radius during the transient process in the initial oscillation stage is explicitly obtained and physically analyzed at the resonance situation based on the Laplace transform method. And the typical oscillation behaviors obtained from the analytical solution are thoroughly exhibited and analyzed in the time and frequency domains. In addition, the corresponding oscillation behaviors at the non-resonance situation are also investigated for the purpose of comparisons. Through our investigation, several essential conclusions can be drawn as follows: (1) The analytical solution of the non-dimensional perturbation of the instantaneous bubble radius can be divided into four terms according to the physical meaning. Among them, it is the term related to the acoustic field that causes the progressively violent bubble oscillation. (2) The vapor bubble with a smaller equilibrium radius could respond faster and more significantly to the acoustic field during the oscillation. (3) The bubble oscillation characteristics always exhibit significant differences at the resonance and non-resonance situations in both the time and frequency domains, even if the difference between the natural frequency of the oscillating vapor bubble and the angular frequency of the acoustic field is greatly small.

Fractional Order Dual-Phase-Lag Model of Heat Conduction in a Composite Spherical Medium

In the paper, a solution of the fractional dual-phase-lag heat conduction problem is presented. The considerations are related to the heat conduction in a multi-layered spherical medium with azimuthal symmetry. The final form of the analytical solution is given in a form of the double series of spherical Bessel functions and Legendre functions. Numerical calculations concern the study of the effect of the order of the Caputo derivative on the temperature distribution in a composite solid sphere, hemisphere and spherical cone.

A novel grey model based on Susceptible Infected Recovered Model: A case study of COVD-19

The COVID-19 pandemic has lasted for nearly two years, and the global epidemic situation is still grim and growing. Therefore, it is necessary to make correct predictions about the epidemic to implement appropriate and effective epidemic prevention measures. This paper analyzes the classic Susceptible Infected Recovered Model (SIR) to understand the significance of model characteristics and parameters, and uses the differential and difference information of the grey system to put forward a grey prediction model based on SIR infectious disease model. The Laplace transform is used to calculate the model reduction formula, and finally obtain the modeling steps of the model. It is applied to large and small numerical cases to verify the validity of different orders of magnitude data. Meanwhile, data of different lengths are modeled and predicted to verify the robustness of model. Finally, the new model is compared with three classical grey prediction models. The results show that the model is significantly superior to the comparison model, indicating that the model can effectively predict the COVID-19 epidemic, and is applicable to countries with different population magnitude, can carry out stable and effective simulation and prediction for data of different lengths.

Fractional telegraph equation under moving time-harmonic impact

The time-fractional telegraph equation with moving time-harmonic source is considered on a real line. We investigate two characteristic versions of this equation: the "wave-type" with the second and Caputo fractional time-derivatives as well as the "heat-type" with the first and Caputo fractional time-derivatives. In both cases the order of fractional derivative 1 < α < 2. For the time-fractional telegraph equation it is impossible to consider the quasi-steady-state corresponding to the solution being a product of a function of the spatial coordinate and the time-harmonic term. The considered problem is solved using the integral transforms technique. The solution to the "wave-type" equation contains wave fronts and describes the Doppler effect contrary to the solution for the "heat-type" equation. Numerical results are illustrated graphically for different values of nondimensional parameters.

Self-heating phenomenon of piezoelectric elements excited by a tone-burst electric field

Tone-burst excitation is often used for ultrasonic transducers of specific operation modes or for overcoming transducer overheating problems associated with continuous wave (CW) excitation. In this study, a theoretical model for the self-heating phenomenon of a piezoelectric disc element is established to estimate the temperature rise induced by a tone-burst electric field. An analytical solution for the temperature rise of the piezoelectric element is obtained by using Laplace transform method. Numerical simulations and experimental measurements are performed to investigate the influence of different excitation parameters on the temperature rise. By comparing the experimental results with the simulation results, the temperature-rise difference between tone-burst and CW excitations is quantified, and the validity of the theoretical model is verified. Furthermore, a multiparameter estimation method is proposed for the heat convection coefficient and dielectric properties under high-field operating conditions. These results are useful in both optimization of heat dissipation performance and characterization of high-power ultrasonic transducers.

One Dimensional Reduction of a Renewal Equation for a Measure-Valued Function of Time Describing Population Dynamics

Despite their relevance in mathematical biology, there are, as yet, few general results about the asymptotic behaviour of measure valued solutions of renewal equations on the basis of assumptions concerning the kernel. We characterise, via their kernels, a class of renewal equations whose measure-valued solution can be expressed in terms of the solution of a scalar renewal equation. The asymptotic behaviour of the solution of the scalar renewal equation, is studied via Feller's classical renewal theorem and, from it, the large time behaviour of the solution of the original renewal equation is derived.

A FRACTIONAL ORDER HIV/AIDS MODEL USING CAPUTO-FABRIZIO OPERATOR

Background:

HIV is a virus that is directed at destroying the human immune system thereby exposing the human body to the risk of been affected by other common illnesses and if it is not treated, it generates a more chronic illness called AIDS.

Materials and Methods:

In this paper, we employed the fixed-point theory in developing the uniqueness and existence of a solution of fractional order HIV/AIDS model having Caputo-Fabrizio operator. This approach adopted in this work is not conventional when solving biological models by fractional derivatives.

Results:

The results showed that the model has two equilibrium points namely, disease-free, and endemic equilibrium points, respectively. We showed conditions necessitating the existence of the endemic equilibrium point and showed that the disease-free equilibrium point is locally asymptotically stable. We also tested the stability of our solution using the iterative Laplace transform method on our model which was also shown stable agreeing with the disease-free equilibrium.

Conclusions:

Numerical simulations of our model showed clear comparison with our analytical results. The numerical solutions show that given fractional operator like the Caputo-Fabrizio operator, it is less noisy and hence plays a major role in making a precise decision and gives room or opportunity (‘freedom’) to use data of specific patients as the model can be easily adjusted to accommodate this, as it a better fit for the patients’ data and provide meaningful predictions. Finally, the result showed the advantage of using fractional order derivative in the analysis of the dynamics of HIV/AIDS over the classical case.

A new general integral transform for solving integral equations

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A new general integral transform which is covered all class of integral transform in the class of Laplace transform.

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We investigated the application of this new transform for solving ODE with constant and variable coefficient.

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This new transform can handle easily for fractional order integral equations and fractional order differential equations.

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We have discussed the advantage and disadvantage of other integral transformed which is defined during last 2 decades.

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We proved the related theorems for this new transform.

Introduction

Integral transforms are important to solve real problems. Appropriate choice of integral transforms helps to convert differential equations as well as integral equations into terms of an algebraic equation that can be solved easily.

During last two decades many integral transforms in the class of Laplace transform are introduced such as Sumudu, Elzaki, Natural, Aboodh, Pourreza, Mohand, G_transform, Sawi and Kamal transforms.

Objectives

In this paper, we introduce a general integral transform in the class of Laplace transform. We study the properties of this transform. Then we compare it with few exiting integral transforms in the Laplace family such as Laplace, Sumudu, Elzaki and G\_transforms, Pourreza, Aboodh and etc.

Methods

A new integral transform is introduced. Then some properties of this integral transform are discussed. This integral transform is used to solve this new transform is used for solving higher order initial value problems, integral equations and fractional order integral equation.

Results

It is proved that those new transforms in the class of Laplace transform which are introduced during last few decades are a special case of this general transform. It is shown that there is no advantage between theses transforms unless for special problems.

Conclusion

It has shown that this new integral transform covers those exiting transforms such as Laplace, Elzaki and Sumudu transforms for different value of p(s) and q(s). We used this new transform for solving ODE, integral equations and fractional integral equations. Also, we can introduce new integral transforms by using this new general integral transform.

Modelling and Prediction of Cutting Temperature in the Machining of H13 Hard Steel of Transient Heat Conduction

Cutting heat conduction undergoes three stages that include intensity transient-state, transient-state, and steady-states. Especially during machining with coated cutting tools, in the conduction process, cutting heat needs to pass through a few micron thick coatings and then flow into the tool body. This heat conduction presents typical non-Fourier heat conduction characteristics. This paper focuses on the cutting temperature in transient heat conduction with a coated tool. A new analytical model to characterize the thermal shock based on the non-Fourier heat conduction was proposed. The distribution of cutting temperature in mono-layer coated tools during the machining was then illustrated. The cutting temperature distribution predicted by the Fourier heat conduction model was employed to compare with that by non-Fourier heat conduction in order to reveal the non-Fourier heat conduction effect in transient heat conduction. The results show that the transient heat conduction analytical model is more suitable for the intensity transient-state and transient-state in the process of cutting heat conduction.

Theoretical and numerical analysis for transmission dynamics of COVID-19 mathematical model involving Caputo-Fabrizio derivative

This manuscript is devoted to a study of the existence and uniqueness of solutions to a mathematical model addressing the transmission dynamics of the coronavirus-19 infectious disease (COVID-19). The mentioned model is considered with a nonsingular kernel type derivative given by Caputo-Fabrizo with fractional order. For the required results of the existence and uniqueness of solution to the proposed model, Picard's iterative method is applied. Furthermore, to investigate approximate solutions to the proposed model, we utilize the Laplace transform and Adomian's decomposition (LADM). Some graphical presentations are given for different fractional orders for various compartments of the model under consideration.

Doppler effect described by the solutions of the Cattaneo telegraph equation

The Cattaneo telegraph equation for temperature with moving time-harmonic source is studied on the line and the half-line domain. The Laplace and Fourier transforms are used. Expressions which show the wave fronts and elucidate the Doppler effect are obtained. Several particular cases of the considered problem including the heat conduction equation and the wave equation are investigated. The quasi-steady-state solutions are also examined for the case of non-moving time-harmonic source and time-harmonic boundary condition for temperature.

Non-commensurate fractional linear systems: New results

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It presents a partial fraction decomposition of non commensurate systems.

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Suitable inversion of each fraction is done in two ways: series and integer/fractional decomposition.

A study of non-commensurate fractional linear system is done in a parallel way to the commensurate case. A partial fraction decomposition is accomplished using a recursive procedure. Each partial fraction is inverted in two different ways. The decomposition integer/fractional is done also. Some examples are presented.

Variational finite element approach to study heat transfer in the biological tissues of premature infants

The body temperature of newborn preterm infants depends on the heat transfer between the infant and the external environment. Factors that influence the heat exchange include the temperature and humidity of the air and the temperature of surfaces in contact with and around the infant. Neonatal thermoregulation has a different pattern as they have an immature thermoregulatory system. For this purpose, mathematical models can provide detailed insights for the heat transfer processes and its applications for clinical purposes. A new multi-compartment mathematical model of the neonatal thermoregulatory system is presented. The formulation of the model is based on the Pennes' bio-heat equation with suitable boundary and initial conditions. The variational finite element method has been employed to determine heat transfer and exchange in the biological tissues of premature infants. The results obtained in this paper have shown that premature infants are unable to maintain a constant core temperature and resemble the empirically obtained results, proving the validity and feasibility of our model. AMS (2010): SUBJECT CLASSIFICATION: 92BXX, 92CXX, 92C35, 92C50, 46N60.

A temporal record of the past with a spectrum of time constants in the monkey entorhinal cortex

Episodic memory is believed to be intimately related to our experience of the passage of time. Indeed, neurons in the hippocampus and other brain regions critical to episodic memory code for the passage of time at a range of timescales. The origin of this temporal signal, however, remains unclear. Here, we examined temporal responses in the entorhinal cortex of macaque monkeys as they viewed complex images. Many neurons in the entorhinal cortex were responsive to image onset, showing large deviations from baseline firing shortly after image onset but relaxing back to baseline at different rates. This range of relaxation rates allowed for the time since image onset to be decoded on the scale of seconds. Further, these neurons carried information about image content, suggesting that neurons in the entorhinal cortex carry information about not only when an event took place but also, the identity of that event. Taken together, these findings suggest that the primate entorhinal cortex uses a spectrum of time constants to construct a temporal record of the past in support of episodic memory.

On the space of Laplace transformable distributions

We show that the spaceS ' ( Γ ) of Laplace transformable distributions, where Γ ⊆ R d is a non-empty convex open set, is an ultrabornological (PLS)-space. Moreover, we determine an explicit topological predual ofS ' ( Γ ) .

The effects of thermal and mechanical material properties on tumorous tissue during hyperthermia treatment

Although there have been numerous reports in several articles about the viscoelastic properties of biological tissues, no effort has been made to investigate the combined thermal and mechanical behavior of the viscoelastic tissue. At present, the model of thermo-viscoelasticity theory with variable thermal conductivity and rheological properties of the volume is considered to investigate bio-thermo-mechanics behavior in living tissue within the context of the Lord-Shulman theory. The model is applied to a limited thickness, cancerous layer problem. The problem was solved analytically in the transformed domain using Laplace transform as a tool. The exact solution is obtained in the context of transformation Laplace. Numerical results are given and illustrated graphically for the distributions of temperature, displacement, and stress. Some correlations are produced with the results obtained for the absence of the thermal relaxation parameter. The effects of variable thermal and volume materials properties, blood perfusion rate on the behavior of various fields are examined.

Numerical analysis of local non-equilibrium heat transfer in layered spherical tissue during magnetic hyperthermia

A solid multi-layered concentric sphere with Gaussian space source is considered as the tissue model for magnetic hyperthermia treatment. The generalized dual-phase-lag model of bioheat transfer is used to describe the behavior of heat transport in tissue in the hyperthermia treatment process for accounting the local non-equilibrium effect. The effects of blood perfusion with the transient temperature are included in the tissue model. The hybrid numerical scheme based on Laplace transform, change of variables, and the modified discretization technique is extended to solve the present problem. The analytical solution for constant heat generation in the inner sphere is presented and evidences the accuracy and rationality of the present numerical results. In an ideal hyperthermia treatment, all the diseased tissues should be selectively heated without affecting any healthy tissue. Attempting to achieve the ideal temperature distribution, the thermal dose is estimated at the specified condition. The corresponding thermal efficacy of tumor damage has also been assessed based on the Arrenius equation.

Heat Transfer Enhancement in Unsteady MHD Natural Convective Flow of CNTs Oldroyd-B Nanofluid under Ramped Wall Velocity and Ramped Wall Temperature

This article analyzes heat transfer enhancement in incompressible time dependent magnetohydrodynamic (MHD) convective flow of Oldroyd-B nanofluid with carbon nanotubes (CNTs). Single wall carbon nanotubes (SWCNTs) and multi-wall carbon nanotubes (MWCNTs) are immersed in a base fluid named Sodium alginate. The flow is restricted to an infinite vertical plate saturated in a porous material incorporating the generalized Darcy’s law and heat suction/injection. The governing equations for momentum, shear stress and energy are modelled in the form of partial differential equations along with ramped wall temperature and ramped wall velocity boundary conditions. Laplace transformation is applied to convert principal partial differential equations to ordinary differential equations first and, later, complex multivalued functions of Laplace parameter are handled with numerical inversion to obtain the solutions in real time domain. Expression for Nusselt number is also obtained to clearly examine the difference in rate of heat transfer. A comparison for isothermal wall condition and ramped wall condition is also made to analyze the difference in both profiles. A graphical study is conducted to analyze how the fluid profiles are significantly affected by several pertinent parameters. Rate of heat transfer increases with increasing volume fraction of nanoparticle while shear stress reduces with elevation in retardation time. Moreover, flow gets accelerated with increase in Grashof number and Porosity parameter. For every parameter, a comparison between solutions of SWCNTs and MWCNTs is also presented.

Cubic-quartic bright optical solitons with improved Adomian decomposition method

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Optical soliton solutions for the perturbed nonlinear Schrodinger’s equation are revealed.

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Quadratic-cubic nonlinearity is considered.

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Bright optical solitons are retrieved by the help of the IADM.

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The numerical results together with high level accuracy plots are exhibited.

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The method proposed herein works with high degree of accuracy.

This paper numerically retrieves cubic-quartic solitons having power law of nonlinearity refractive index. An improvement of the Adomian decomposition scheme is the adopted algorithm of this work. The results are displayed along with the established error analysis.

Unsteady Radiative Natural Convective MHD Nanofluid Flow Past a Porous Moving Vertical Plate with Heat Source/Sink

In this research article, we investigated a comprehensive analysis of time-dependent free convection electrically and thermally conducted water-based nanofluid flow containing Copper and Titanium oxide (Cu and TiO 2 ) past a moving porous vertical plate. A uniform transverse magnetic field is imposed perpendicular to the flow direction. Thermal radiation and heat sink terms are included in the energy equation. The governing equations of this flow consist of partial differential equations along with some initial and boundary conditions. The solution method of these flow interpreting equations comprised of two parts. Firstly, principal equations of flow are symmetrically transformed to a set of nonlinear coupled dimensionless partial differential equations using convenient dimensionless parameters. Secondly, the Laplace transformation technique is applied to those non-dimensional equations to get the close form exact solutions. The control of momentum and heat profile with respect to different associated parameters is analyzed thoroughly with the help of graphs. Fluid accelerates with increasing Grashof number (Gr) and porosity parameter (K), while increasing values of heat sink parameter (Q) and Prandtl number (Pr) drop the thermal profile. Moreover, velocity and thermal profile comparison for Cu and TiO 2 -based nanofluids is graphed.

Low-Finesse Fabry-Pérot Interferometers Applied in the Study of the Relation between the Optical Path Difference and Poles Location

Interferometry sensors are frequently analyzed by applying the Fourier transform because the transformation separates all frequency components of its signal, making its study on a complex plane feasible. In this work, we study the relation between the optical path difference (OPD) and poles location theoretically and experimentally, using the Laplace transform and a pole-zero map. Theory and experiments are in concordance. For our study, only the cosine function was considered, which is filtered from the interference pattern. In experimental work, two unperturbed low-finesse Fabry-Pérot interferometers were used. First, a Fabry-Pérot interferometer that has a cavity length of ~1.6 mm was used. Its optical path difference was 2.33 mm and the poles were localized at points ±i12. rad/nm. Secondly, a Fabry-Pérot interferometer with a cavity length of ~5.2 mm was used, and its optical path difference was 7.59 mm and the poles were localized at points ±i40.4 rad/nm. Experimental results confirmed the theoretical analysis. Our proposal finds practical application for interferometer analysis, signal processing of optical fiber sensors, communication system analysis, and multiplexing systems based on interferometers.

Linear signal combination T2 spectroscopy

A technique is presented for performing T2 spectroscopy in magnetic resonance imaging (MRI). It is based on a weighted linear combination of T2 decay data. The data is combined in a manner that acts like a filter on the T2 spectrum. The choice of weighting coefficients determines the filter specifications (e.g. passband/stopband locations, stopband suppression factors). To perform spectroscopy, a series of filters are designed with narrow passbands centered about consecutive regions of the T2 spectrum. This provides an estimate of every region of the spectrum. Taken together, an initial estimate of the full T2 spectrum is thus obtained. However, the filtering process causes a distortion of the estimate relative to the true spectrum. To reduce this distortion, deconvolution is performed. The characteristics of the technique are first evaluated through simulation. The technique is then applied to experimental MRI data to demonstrate practical feasibility. T2 spectroscopy falls into a class of problems requiring inverse transformation with a set of exponential basis functions (i.e. the Laplace Transform). It is demonstrated how the present technique may be applied to problems involving non-exponential basis functions as well.

An asymptotic solution of the integral equation for the second moment function in geometric processes

In this study, we derive an asymptotic solution of the integral equation satisfied by the second moment function M 2 t , a . We first find the Laplace transform M 2 L s , a and then obtain M 2 t , a asymptotically by inversion. Further, we have derived the asymptotic expressions of M 2 t , a for some special lifetime distributions such as exponential, gamma, Weibull, lognormal and truncated normal. Finally, the asymptotic solution is compared with the numerical solution to evaluate its performance.

Three novel approaches to structural identifiability analysis in mixed-effects models

BACKGROUND AND OBJECTIVE

Structural identifiability is a concept that considers whether the structure of a model together with a set of input-output relations uniquely determines the model parameters. In the mathematical modelling of biological systems, structural identifiability is an important concept since biological interpretations are typically made from the parameter estimates. For a system defined by ordinary differential equations, several methods have been developed to analyse whether the model is structurally identifiable or otherwise. Another well-used modelling framework, which is particularly useful when the experimental data are sparsely sampled and the population variance is of interest, is mixed-effects modelling. However, established identifiability analysis techniques for ordinary differential equations are not directly applicable to such models.

METHODS

In this paper, we present and apply three different methods that can be used to study structural identifiability in mixed-effects models. The first method, called the repeated measurement approach, is based on applying a set of previously established statistical theorems. The second method, called the augmented system approach, is based on augmenting the mixed-effects model to an extended state-space form. The third method, called the Laplace transform mixed-effects extension, is based on considering the moment invariants of the systems transfer function as functions of random variables.

RESULTS

To illustrate, compare and contrast the application of the three methods, they are applied to a set of mixed-effects models.

CONCLUSIONS

Three structural identifiability analysis methods applicable to mixed-effects models have been presented in this paper. As method development of structural identifiability techniques for mixed-effects models has been given very little attention, despite mixed-effects models being widely used, the methods presented in this paper provides a way of handling structural identifiability in mixed-effects models previously not possible.

A neural microcircuit model for a scalable scale-invariant representation of time

Scale-invariant timing has been observed in a wide range of behavioral experiments. The firing properties of recently described time cells provide a possible neural substrate for scale-invariant behavior. Earlier neural circuit models do not produce scale-invariant neural sequences. In this article, we present a biologically detailed network model based on an earlier mathematical algorithm. The simulations incorporate exponentially decaying persistent firing maintained by the calcium-activated nonspecific (CAN) cationic current and a network structure given by the inverse Laplace transform to generate time cells with scale-invariant firing rates. This model provides the first biologically detailed neural circuit for generating scale-invariant time cells. The circuit that implements the inverse Laplace transform merely consists of off-center/on-surround receptive fields. Critically, rescaling temporal sequences can be accomplished simply via cortical gain control (changing the slope of the f-I curve).

Time-Fractional Diffusion with Mass Absorption in a Half-Line Domain due to Boundary Value of Concentration Varying Harmonically in Time

The time-fractional diffusion equation with mass absorption is studied in a half-line domain under the Dirichlet boundary condition varying harmonically in time. The Caputo derivative is employed. The solution is obtained using the Laplace transform with respect to time and the sin-Fourier transform with respect to the spatial coordinate. The results of numerical calculations are illustrated graphically.

Modelling mass diffusion for a multi-layer sphere immersed in a semi-infinite medium: application to drug delivery

We present a general mechanistic model of mass diffusion for a composite sphere placed in a large ambient medium. The multi-layer problem is described by a system of diffusion equations coupled via interlayer boundary conditions such as those imposing a finite mass resistance at the external surface of the sphere. While the work is applicable to the generic problem of heat or mass transfer in a multi-layer sphere, the analysis and results are presented in the context of drug kinetics for desorbing and absorbing spherical microcapsules. We derive an analytical solution for the concentration in the sphere and in the surrounding medium that avoids any artificial truncation at a finite distance. The closed-form solution in each concentric layer is expressed in terms of a suitably-defined inverse Laplace transform that can be evaluated numerically. Concentration profiles and drug mass curves in the spherical layers and in the external environment are presented and the dependency of the solution on the mass transfer coefficient at the surface of the sphere analyzed.

Anomalous Advection-Dispersion Equations within General Fractional-Order Derivatives: Models and Series Solutions

In this paper, an anomalous advection-dispersion model involving a new general Liouville–Caputo fractional-order derivative is addressed for the first time. The series solutions of the general fractional advection-dispersion equations are obtained with the aid of the Laplace transform. The results are given to demonstrate the efficiency of the proposed formulations to describe the anomalous advection dispersion processes.

Analytical Solution of Electro-Osmotic Peristalsis of Fractional Jeffreys Fluid in a Micro-Channel

The electro-osmotic peristaltic flow of a viscoelastic fluid through a cylindrical micro-channel is studied in this paper. The fractional Jeffreys constitutive model, including the relaxation time and retardation time, is utilized to describe the viscoelasticity of the fluid. Under the assumptions of long wavelength, low Reynolds number, and Debye-Hückel linearization, the analytical solutions of pressure gradient, stream function and axial velocity are explored in terms of Mittag-Leffler function by Laplace transform method. The corresponding solutions of fractional Maxwell fluid and generalized second grade fluid are also obtained as special cases. The numerical analysis of the results are depicted graphically, and the effects of electro-osmotic parameter, external electric field, fractional parameters and viscoelastic parameters on the peristaltic flow are discussed.

One-dimensional analytical solution for hydraulic head and numerical solution for solute transport through a horizontal fracture for submarine groundwater discharge

Submarine groundwater discharge (SGD) has been recognized as a major pathway of groundwater flow to coastal oceanic environments. It could affect water quality and marine ecosystems due to pollutants and trace elements transported through groundwater. Relations between different characteristics of aquifers and SGD have been investigated extensively before, but the role of fractures in SGD still remains unknown. In order to better understand the mechanism of groundwater flow and solute transport through fractures in SGD, one-dimensional analytical solutions of groundwater hydraulic head and velocity through a synthetic horizontal fracture with periodic boundary conditions were derived using a Laplace transform technique. Then, numerical solutions of solute transport associated with the given groundwater velocity were developed using a finite-difference method. The results indicated that SGD associated with groundwater flow and solute transport was mainly controlled by sea level periodic fluctuations, which altered the hydraulic head and the hydraulic head gradient in the fracture. As a result, the velocity of groundwater flow associated with SGD also fluctuated periodically. We found that the pollutant concentration associated with SGD oscillated around a constant value, and could not reach a steady state. This was particularly true at locations close to the seashore. This finding of the role of fracture in SGD will assist pollution remediation and marine conservation in coastal regions.

An effective time-constant algorithm for drug transport to capillaries and surrounding tissues

Expressions for a single time constant were developed in Maple (Waterloo Maple, Inc.) to calculate the rate at which a drug reaches steady-state levels in the blood capillaries and neighboring tissues. The solute concentration in the capillary region was represented by a one-dimensional convection-diffusion model. In a first case study, the plasma and the tissue reached equilibrium very quickly. Within the dynamic regime, the amount of drugs collected in both compartments increased with the Peclet number while the relaxation time to a steady-state value decreased. A similar conclusion was drawn, in a second case study, when axial and radial diffusive transports were considered important in the lungs or the skin. Also, as the mass transfer Biot number decreased, a larger amount of medication was delivered to the tissue at a given time during the transient period. Additional applications of the approach included the analysis of oxygen transport in peripheral nerves and the design of hollow fibre bioreactors.

The pulsating orb: solving the wave equation outside a ball

Transient acoustic waves are generated by the oscillations of an object or are scattered by the object. This leads to initial-boundary value problems (IBVPs) for the wave equation. Basic properties of this equation are reviewed, with emphasis on characteristics, wavefronts and compatibility conditions. IBVPs are formulated and their properties reviewed, with emphasis on weak solutions and the constraints imposed by the underlying continuum mechanics. The use of the Laplace transform to treat the IBVPs is also reviewed, with emphasis on situations where the solution is discontinuous across wavefronts. All these notions are made explicit by solving simple IBVPs for a sphere in some detail.

Laplace transform homotopy perturbation method for the approximation of variational problems

This article proposes the application of Laplace Transform-Homotopy Perturbation Method and some of its modifications in order to find analytical approximate solutions for the linear and nonlinear differential equations which arise from some variational problems. As case study we will solve four ordinary differential equations, and we will show that the proposed solutions have good accuracy, even we will obtain an exact solution. In the sequel, we will see that the square residual error for the approximate solutions, belongs to the interval [0.001918936920, 0.06334882582], which confirms the accuracy of the proposed methods, taking into account the complexity and difficulty of variational problems.

Progressive Myopia and Lid Suture Myopia are Explained by the Same Feedback Process: a Mathematical Model of Myopia

OBJECTIVE

Progressive myopia in humans and lid-sutured myopia in primates have been considered to be different processes. This report seeks to establish the connection between progressive myopia in humans and lid suture myopia in macaque monkeys.

METHODS

We followed the axial length of 4 lid-sutured macaque monkeys over an 18 month period. Their axial length is directly related to myopia. We also studied the myopia progression in corrected human subjects. Macaques and humans exhibit a linear time course of myopia progression when lid-sutured or corrected with lenses, respectively.

RESULTS

A linear progression is observed in lid-sutured eyes of four macaques, r = 0.94, p < 0.05. Human progressive myopia and lid-suture myopia can be modeled by the same feedback process. In both cases the functional equivalent is the opening of the feedback loop.

CONCLUSIONS

The open loop feedback process predicts a linear progression of myopia. This prediction was confirmed in human subjects and it is now confirmed in our macaque subjects. This process also explains the very rapid rate of myopia progression of lid sutured eyes.

Nonlinearities distribution Laplace transform-homotopy perturbation method

This article proposes non-linearities distribution Laplace transform-homotopy perturbation method (NDLT-HPM) to find approximate solutions for linear and nonlinear differential equations with finite boundary conditions. We will see that the method is particularly relevant in case of equations with nonhomogeneous non-polynomial terms. Comparing figures between approximate and exact solutions we show the effectiveness of the proposed method.

Measuring diffusion-relaxation correlation maps using non-uniform field gradients of single-sided NMR devices

Single-sided NMR systems are becoming a relevant tool in industry and laboratory environments due to their low cost, low maintenance and capacity to evaluate quantity and quality of hydrogen based materials. The performance of such devices has improved significantly over the last decade, providing increased field homogeneity, field strength and even controlled static field gradients. For a class of these devices, the configuration of the permanent magnets provides a linear variation of the magnetic field and can be used in diffusion measurements. However, magnet design depends directly on its application and, according to the purpose, the field homogeneity may significantly be compromised. This may prevent the determination of diffusion properties of fluids based on the natural inhomogeneity of the field using known techniques. This work introduces a new approach that extends the applicability of diffusion-editing CPMG experiments to NMR devices with highly inhomogeneous magnetic fields, which do not vary linearly in space. Herein, we propose a method to determine a custom diffusion kernel based on the gradient distribution, which can be seen as a signature of each NMR device. This new diffusion kernel is then utilised in the 2D inverse Laplace transform (2D ILT) in order to determine diffusion-relaxation correlation maps of homogeneous multi-phasic fluids. The experiments were performed using NMR MObile Lateral Explore (MOLE), which is a single-sided NMR device designed to maximise the volume at the sweet spot with enhanced depth penetration.

Solute transport in a single fracture involving an arbitrary length decay chain with rock matrix comprising different geological layers

A model is developed to describe solute transport and retention in fractured rocks. It accounts for advection along the fracture, molecular diffusion from the fracture to the rock matrix composed of several geological layers, adsorption on the fracture surface, adsorption in the rock matrix layers and radioactive decay-chains. The analytical solution, obtained for the Laplace-transformed concentration at the outlet of the flowing channel, can conveniently be transformed back to the time domain by the use of the de Hoog algorithm. This allows one to readily include it into a fracture network model or a channel network model to predict nuclide transport through channels in heterogeneous fractured media consisting of an arbitrary number of rock units with piecewise constant properties. More importantly, the simulations made in this study recommend that it is necessary to account for decay-chains and also rock matrix comprising at least two different geological layers, if justified, in safety and performance assessment of the repositories for spent nuclear fuel.

Analytical solutions for systems of partial differential-algebraic equations

This work presents the application of the power series method (PSM) to find solutions of partial differential-algebraic equations (PDAEs). Two systems of index-one and index-three are solved to show that PSM can provide analytical solutions of PDAEs in convergent series form. What is more, we present the post-treatment of the power series solutions with the Laplace-Padé (LP) resummation method as a useful strategy to find exact solutions. The main advantage of the proposed methodology is that the procedure is based on a few straightforward steps and it does not generate secular terms or depends of a perturbation parameter.