Elsevier

Applied Mathematics and Computation

Volume 307, 15 August 2017, Pages 311-320
Applied Mathematics and Computation

A simple algorithm for exact solutions of systems of linear and nonlinear integro-differential equations

https://doi.org/10.1016/j.amc.2017.03.013Get rights and content

Abstract

A Simple algorithm is used to achieve exact solutions of systems of linear and nonlinear integro- differential equations arising in many scientific and engineering applications. The algorithm does not need to find the Adomain Polynomials to overcome the nonlinear terms in Adomain Decomposition Method (ADM). It does not need to create a homotopy with an embedding parameter as in Homotopy Perturbation Method (HPM) and Optimal Homotopy Asymptotic Method (OHAM). Unlike VIM, it does not need to find Lagrange Multiplier. In this manuscript no restrictive assumptions are taken for nonlinear terms. The applied algorithm consists of a single series in which the unknown constants are determined by the simple means described in the manuscript. The outcomes gained by this algorithm are in excellent concurrence with the exact solution and hence proved that this algorithm is effective and easy. Four systems of linear and nonlinear integro-differential equations are solved to prove the above claims and the outcomes are compared with the exact solutions as well as with the outcomes gained by already existing methods.

Introduction

Systems of integro-differential equations arise in mathematical modeling of many phenomena, such as activity of interacting inhibitory and excitatory neurons [1], nano-hydrodynamics [2], wind ripple in the desert [3], drop wise condensation [4]. The problems in science and engineering physical phenomena such as currents in electric fields and magnetic fields, pulses in biological chains, industrial mathematics, oscillation theory and control theory of financial mathematics, fluid dynamics mass and heat are modeled by integro-differential equations [5], [6]. Presently much attention is paid by the researchers to solve such problems. Different methods are applied to solve such problems numerically as well as analytically [7], [8], [9], [10], [11], [12], [13]. Various polynomials like; Taylor, Chebyshev, Daftardar-Jaffari and Adomain’s are used to solve systems of linear and nonlinear integro-differential equations [14], [15], [16], [17], [18]. Different matrices like; Bernstein operational matrix of derivatives, operational matrix with Block-Pulse functions are applied to solve the same equations [19], [20]. The application of the above techniques can give good results but at the same time it creates various difficulties; like, constructing a homotopy in HPM, OHAM and the solution of the corresponding algebraic equations. Calculation of Adomian polynomials to overcome the nonlinear terms in ADM and calculate Lagrange multiplier in VIM, respectively. In 2008, Tahmasbi and Fard [14] have used a new and simple method called the Power Series Method (PSM) to solve Volterra integral equations. Wazwaz et al. used VIM and ADM to solve the Volterra integral form of the Lane–Emden equations with initial values and boundary conditions [12], [17]. Ebrahimi and Rashidinia used Collocation Method for Volterra–Fredholm Integral and Integro-Differential Equations [21]. Armand el al. give Numerical solution of the system of Volterra integral equations of the first kind [22]. Hesameddint and Asadolahifard [23] solve Systems of Linear Volterra Integro-Differential Equations by Using Sinc-Collocation Method. Biazar et al. used a Strong Method for solving systems of Integro-Differential Equations. Aguirre-Hernández et al. have used polynomial and topological methods to study systems of differential equations as mentioned in [24], [25], [26]. In the present paper we apply a series algorithm to solve systems of linear and nonlinear integro-differential equations. The algorithm consist of few steps explained in the coming section which converges easily to the exact solution. The applied algorithm is simple in learning and easy to apply. The paper is divided into five parts. First and second parts consist of introduction related with the manuscript and applied algorithm. Third part is devoted to the application of the algorithm to four systems of linear and nonlinear integro-differential equations. In the fourth part the results and errors are explained in tabular form and finally, conclusions are given in the fifth part. We have used math type and mathematica 7.0 for calculations and numerical simulations.

Section snippets

Introduction of the applied algorithm

Consider a system of Integral equations of the form μi(r)=fi(r)+0rκi(r,s)ϕi(μ1(s),μ2(s),μ3(s),,μm(s))ds,where κi(r,s),i=1,2,3,,m are kernels of integral equations and μi(r),i=1,2,3,,m=[μ1,μ2,,μm]T are unknown solutions to be calculated fi(r) are real valued functions and ϕi(μ(s)) are linear or nonlinear function of μ1(s),μ2(s),μ3(s),,μm(s).

Let the solution of (1) be given as: μi(r)=j=0mαijrj,with initial conditions αi0=μi(0),i=1,2,,m and also αj=αij,j=0,1,2,3,,m.

The coefficients in the

Application of the proposed algorithm to systems of linear and nonlinear integro-differential equations forming in linear and nonlinear phenomena

Problem 1

Consider the system of Volterra Integral Equations of the First kind as follows [22] 0r(μ1+(rs)μ1μ2)ds+3/41/2r1/2r21/12r4+er+1/4e2r=00r(μ2+(rs)μ1μ2)ds5/41/2r1/2r21/12r4+er+1/4e2r=0,with exact solutions μ1(r)=r+er and μ2(r)=rer. Now we proceed the steps for the solution mentioned in Section 2. Taking derivatives twice we gained: rμ11r2er+e2rμ1μ2=0rμ21r2+er+e2rμ1μ2=0,with initial conditions μ1(0)=1,μ2(0)=1. Taking integration on both sides we obtained: μ110r(1+r2+ere2r)

In this section the results and the errors are compared with the exact solution and also with the other methods

Conclusions

A series Algorithm is applied to systems of linear and nonlinear Integro-differential equations which gives very good and reliable results.The algorithm is quite simple and short. It gives almost exact solution.This algorithm has great potential to solve systems of linear and nonlinear problems in short as well as in broad intervals.The beauty of the technique is; less calculation, less use of computer memory, economical in terms of computer power, and involve no tedious calculations. The

References (26)

  • S. Krystyna et al.

    A higher-order finite difference method for solving a system of integro-differential equations

    J. Comput. Appl. Math.

    (2000)
  • E.Y. (Agadjanov)

    An efficient algorithm for solving integro-differential equations system

    Appl. Math. Comput.

    (2007)
  • J. Nadjafi et al.

    The variational iteration method: A highly promising method for solving the system of integro-differential equations

    Comput. Math. Appl.

    (2007)
  • Cited by (5)

    View full text