African Journal of
Mathematics and Computer Science Research

  • Abbreviation: Afr. J. Math. Comput. Sci. Res.
  • Language: English
  • ISSN: 2006-9731
  • DOI: 10.5897/AJMCSR
  • Start Year: 2008
  • Published Articles: 254

Full Length Research Paper

A new approach to homotopy perturbation method for solving systems of Volterra integral equations of first kind

M. S. Ahamed
  • M. S. Ahamed
  • Department of Mathematics, Rajshahi University of Engineering and Technology, Rajshahi-6204, Bangladesh.
  • Google Scholar
M. Kamrul Hasan
  • M. Kamrul Hasan
  • Department of Mathematics, Rajshahi University of Engineering and Technology, Rajshahi-6204, Bangladesh.
  • Google Scholar
M. S. Alam
  • M. S. Alam
  • Department of Mathematics, Rajshahi University of Engineering and Technology, Rajshahi-6204, Bangladesh.
  • Google Scholar


  •  Received: 18 May 2017
  •  Accepted: 07 July 2017
  •  Published: 31 August 2017

 ABSTRACT

In this article, He’s homotopy perturbation method was applied in a variant way to solve the system of Volterra integral equations of first kind. The results reveal that the proposed approach is very efficient for handling such system of integral equations. Some examples are given to show the ability of the proposed modification.

 

Key words: Integral equations, Volterra integral equations of first kind, homotopy perturbation method.


 INTRODUCTION

The application of homotopy perturbation method (HPM) (He, 1999) was studied by many scientist and engineers because this method continuously reduces a nonlinear problem into a set of linear one which is easy to handle. To handle wide variety of linear and nonlinear problems, the method was further modified and improved by He (2000, 2003, 2004). HPM has been used to solve various types of integral equations with diverse variations. In this paper, a variant approach based on HPM was proposed to solve the system of Volterra integral equations of first kind of the form:
 
 
where  are known functions,   are the kernels and  are  linear  or  nonlinear  functions  of .  Exact  or approximate solution of integral equations has great importance because it has wide applications in scientific research. Many researchers (Golbabai and Keramati, 2008; Biazar et al., 2009; Eslami, 2014a; Biazar et al., 2012; Biazar and Mostafa, 2011a, b; Rabbani et al., 2007; Maleknejad et al., 2007; Tahmasbi and Fard, 2008; Odibat, 2008; Maleknejad and Najafi, 2011; Biazar and Eslami, 2011; Eslami and Mirzazadeh, 2014; Eslami, 2014b; Biazar and Eslami, 2010a, b, c; 2011) have solved various types of integral equations by several methods. Biazar et al. (2008, 2009) solved system of Volterra integral equations of type (1) by HPM. Using operational matrix with block-pulse functions, Babolian and Masouri (2008) solved this type of equations. Applying Adomian method, Biazar et al. (2003) presented the solution of system of Volterra integral equations of the first kind.Making a simple   modification   on    HPM,  Ghorbani and Saberi-Nadjafi (2008) solved nonlinear integral equations. Biazar and Eslami (2010a, b, c) also solved the equations of type (1) by differential transform method. Volterra integral equations were also studied by several researchers with various numerical techniques. Masouri et al. (2010) presented the numerical solution of Volterra integral equation of the first kind by an expansion-iterative method. Using Runge-Kutta method, Maleknejad and Shahrezaee (2004) solved the equations of type (1). Armand and Gouyandeh (2014) presented numerical solutions of the system of Volterra integral equations. Applying decomposition method, Ngarasta et al. (2009) solved Volterra integral equations system. Saeedi et al. (2013) solved some nonlinear Volterra integral equations of the first kind numerically.


 BASIC IDEAS OF HE’S HPM

To illustrate the basic ideas of He’s HPM, let us consider an integral or differential operator  such that:
 

 


 BASIC IDEA OF THE NEW APPROACH

To explain the new approach, let us consider the system of Volterra integral equations of type (1). In the new approach, we split  into infinite sums as follows:
 
 
Examples
 
Here, the proposed approach was applied to obtain exact solutions of some linear and nonlinear system of Volterra integral equations of first kind.
 
Consider the system of linear Volterra integral equations of first kind (Maleknejad et al., 2007):
 
 
Example
 
 
 

 


 CONCLUSION

Based on HPM, an analytic approach for solving the system of Volterra type integral equations of first kind was developed. Evaluating the examples, it was observed that the proposed approach is straightforward in calculations and very effective in both linear and nonlinear cases. Moreover, in most of the cases, it gives exact solutions at the first approximation.   


 CONFLICT OF INTERESTS

The authors have not declared any conflict of interests.



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