1
|
Narayanamoorthy S, Sathiyapriya SP. A pertinent approach to solve nonlinear fuzzy integro-differential equations. SPRINGERPLUS 2016; 5:449. [PMID: 27119053 PMCID: PMC4830805 DOI: 10.1186/s40064-016-2045-4] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 12/03/2015] [Accepted: 03/23/2016] [Indexed: 11/10/2022]
Abstract
Fuzzy integro-differential equations is one of the important parts of fuzzy analysis theory that holds theoretical as well as applicable values in analytical dynamics and so an appropriate computational algorithm to solve them is in essence. In this article, we use parametric forms of fuzzy numbers and suggest an applicable approach for solving nonlinear fuzzy integro-differential equations using homotopy perturbation method. A clear and detailed description of the proposed method is provided. Our main objective is to illustrate that the construction of appropriate convex homotopy in a proper way leads to highly accurate solutions with less computational work. The efficiency of the approximation technique is expressed via stability and convergence analysis so as to guarantee the efficiency and performance of the methodology. Numerical examples are demonstrated to verify the convergence and it reveals the validity of the presented numerical technique. Numerical results are tabulated and examined by comparing the obtained approximate solutions with the known exact solutions. Graphical representations of the exact and acquired approximate fuzzy solutions clarify the accuracy of the approach.
Collapse
Affiliation(s)
- S Narayanamoorthy
- Department of Mathematics, Bharathiar University, Coimbatore, TamilNadu 641046 India
| | - S P Sathiyapriya
- Department of Mathematics, Bharathiar University, Coimbatore, TamilNadu 641046 India
| |
Collapse
|
2
|
Narayanamoorthy S, Sathiyapriya SP. Homotopy perturbation method: a versatile tool to evaluate linear and nonlinear fuzzy Volterra integral equations of the second kind. SPRINGERPLUS 2016; 5:387. [PMID: 27047713 PMCID: PMC4816960 DOI: 10.1186/s40064-016-2038-3] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 08/25/2015] [Accepted: 03/20/2016] [Indexed: 12/26/2022]
Abstract
In this article, we focus on linear and nonlinear fuzzy Volterra integral equations of the second kind and we propose a numerical scheme using homotopy perturbation method (HPM) to obtain fuzzy approximate solutions to them. To facilitate the benefits of this proposal, an algorithmic form of the HPM is also designed to handle the same. In order to illustrate the potentiality of the approach, two test problems are offered and the obtained numerical results are compared with the existing exact solutions and are depicted in terms of plots to reveal its precision and reliability.
Collapse
Affiliation(s)
- S Narayanamoorthy
- Department of Mathematics, Bharathiar University, Coimbatore, Tamil Nadu 641 046 India
| | - S P Sathiyapriya
- Department of Mathematics, Bharathiar University, Coimbatore, Tamil Nadu 641 046 India
| |
Collapse
|
3
|
Filobello-Nino U, Vazquez-Leal H, Rashidi MM, Sedighi HM, Perez-Sesma A, Sandoval-Hernandez M, Sarmiento-Reyes A, Contreras-Hernandez AD, Pereyra-Diaz D, Hoyos-Reyes C, Jimenez-Fernandez VM, Huerta-Chua J, Castro-Gonzalez F, Laguna-Camacho JR. Laplace transform homotopy perturbation method for the approximation of variational problems. SPRINGERPLUS 2016; 5:276. [PMID: 27006884 PMCID: PMC4779117 DOI: 10.1186/s40064-016-1755-y] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 11/04/2015] [Accepted: 02/12/2016] [Indexed: 01/23/2023]
Abstract
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.
Collapse
Affiliation(s)
- U Filobello-Nino
- Facultad de Instrumentación Electrónica, Universidad Veracruzana, Circuito Gonzalo Aguirre Beltrán S/N, 91000 Xalapa, Veracruz Mexico
| | - H Vazquez-Leal
- Facultad de Instrumentación Electrónica, Universidad Veracruzana, Circuito Gonzalo Aguirre Beltrán S/N, 91000 Xalapa, Veracruz Mexico
| | - M M Rashidi
- Shanghai Key Lab of Vehicle Aerodynamics and Vehicle Thermal Management Systems, Tongji University, 4800 Cao An Rd., Jiading, Shanghai, 201804 China ; ENN-Tongji Clean Energy Institute of Advanced Studies, Shanghai, China
| | - H M Sedighi
- Department of Mechanical Engineering, Shahid Chamran University, Ahvaz, Iran
| | - A Perez-Sesma
- Facultad de Instrumentación Electrónica, Universidad Veracruzana, Circuito Gonzalo Aguirre Beltrán S/N, 91000 Xalapa, Veracruz Mexico
| | - M Sandoval-Hernandez
- Doctorado en Ciencia, Cultura y Tecnología, Universidad de Xalapa, Km 2 Carretera Xalapa-Veracruz, 91190 Xalapa, Veracruz Mexico
| | - A Sarmiento-Reyes
- National Institute for Astrophysics, Optics and Electronics, Luis Enrique Erro #1, Sta. María Tonantzintla, 72840 Puebla, Mexico
| | - A D Contreras-Hernandez
- Facultad de Instrumentación Electrónica, Universidad Veracruzana, Circuito Gonzalo Aguirre Beltrán S/N, 91000 Xalapa, Veracruz Mexico
| | - D Pereyra-Diaz
- Facultad de Instrumentación Electrónica, Universidad Veracruzana, Circuito Gonzalo Aguirre Beltrán S/N, 91000 Xalapa, Veracruz Mexico
| | - C Hoyos-Reyes
- Facultad de Instrumentación Electrónica, Universidad Veracruzana, Circuito Gonzalo Aguirre Beltrán S/N, 91000 Xalapa, Veracruz Mexico
| | - V M Jimenez-Fernandez
- Facultad de Instrumentación Electrónica, Universidad Veracruzana, Circuito Gonzalo Aguirre Beltrán S/N, 91000 Xalapa, Veracruz Mexico
| | - J Huerta-Chua
- Facultad de Ingeniería Electrónica y Comunicaciones, Universidad Veracruzana, Venustiano Carranza S/N, Col. Revolución, 93390 Poza Rica, Veracruz Mexico
| | - F Castro-Gonzalez
- Facultad de Instrumentación Electrónica, Universidad Veracruzana, Circuito Gonzalo Aguirre Beltrán S/N, 91000 Xalapa, Veracruz Mexico
| | - J R Laguna-Camacho
- Facultad de Ingeniería Mecánica Eléctrica, Universidad Veracruzana, Venustiano Carranza S/N, Col. Revolución, 93390 Poza Rica, Veracruz Mexico
| |
Collapse
|