Numerical Solution for IVP in Volterra Type Linear Integrodifferential Equations System

Abstract

A method is proposed to determine the numerical solution of system of linear Volterra integrodifferential equations (IDEs) by using Bezier curves. The Bezier curves are chosen as piecewise polynomials of degree n, and Bezier curves are determined on [𝑡₀,𝑡_𝑓] by 𝑛+1 control points. The efficiency and applicability of the presented method are illustrated by some numerical examples.

1. Introduction

Integrodifferential equations (IDEs) have been found to describe various kinds of phenomena, such as glass forming process, dropwise condensation, nanohydrodynamics, and wind ripple in the desert (see [1, 2]).

There are several numerical and analytical methods for solving IDEs. Some different methods are presented to solve integral and IDEs in [3, 4]. Maleknejad et al. [5] used rationalized Haar functions method to solve the linear IDEs system. Linear IDEs system has been solved by using Galerkin methods with the hybrid Legendre and block-Pulse functions on interval [0,1) (see [6]). Yusufoğlu [7] presented an application of He's homotopy perturbation (HPM) method to solve the IDEs system. He's variational iteration method has been used for solving two systems of Volterra integrodifferential equations (see [8]). Arikoglu and Ozkol [9] presented differential transform method (DTM) for integrodifferential and integral equation systems. He's homotopy perturbation (HPM) method was proposed for system of integrodifferential equations (see [10]). A numerical method based on interpolation of unknown functions at distinct interpolation points has been introduced for solving linear IDEs system with initial values (see [11]). Recently, Biazar introduced a new modification of homotopy perturbation method (NHPM) to obtain the solution of linear IDEs system (see [12]). Taylor expansion method has been used for solving IDEs (see [13, 14]). Rashidinia and Tahmasebi developed and modified Taylor series method (TSM) introduced in [15] to solve the system of linear Volterra IDEs.

In the present work, we suggest a technique similar to the one which was used in [16] for solving the system of linear Volterra IDEs in the following form: ^𝑛𝛼_𝑖=1^𝑚𝑖_𝑗=0𝑝_𝑚𝑖𝑗(𝑡)𝑦_𝑖^(𝑗)(𝑡)+^𝑛_𝑖=1^𝑡_𝑡0𝑘_𝑚𝑖(𝑡,𝑥)^𝛼𝑚𝑖_𝑗=0𝑦_𝑖^(𝑗)(𝑥)𝑑𝑥=𝑓_𝑚(𝑡),𝑚=1,2,…,𝑛,𝑡₀≤𝑡≤𝑡_𝑓, (1) with the initial conditions 𝑦_𝑖⁽⁰⁾𝑡₀=𝑐_𝑖0,𝑦_𝑖⁽¹⁾𝑡₀=𝑐_𝑖1,…,𝑦^{(𝛼𝑚𝑖}_𝑖⁻¹⁾𝑡₀=𝑐_{𝑖(𝛼𝑚𝑖−1)}, (2) where 𝑦_𝑖^(𝑗)(𝑡) stands for 𝑗th-order derivative of 𝑦_𝑖(𝑡). 𝑓_𝑚(𝑡), 𝑘_𝑚𝑖(𝑡,𝑥), and 𝑝_𝑚𝑖𝑗(𝑡) are known functions (𝑚,𝑖=1,2,…,𝑛;𝑗=0,1,…,𝛼_𝑚𝑖), and 𝑡₀, 𝑡_𝑓, and 𝑐_𝑖𝑗 (𝑖=1,2,…,𝑛;𝑗=0,1,…,𝛼_𝑚𝑖−1) are appropriate constants.

The current paper is organized as follows. In Section 2, function approximation will be introduced. Numerical examples will be stated in Section 3. Finally, Section 4 will give a conclusion briefly.

2. Function Approximation

Our strategy is to use Bezier curves to approximate the solutions 𝑦_𝑖(𝑡), for 1≤𝑖≤𝑛, which are given below. Define the Bezier polynomials of degree 𝑁 that approximate, respectively, the actions of 𝑦_𝑖(𝑡) over the interval [𝑡₀,𝑡_𝑓] as follows: 𝑦_𝑖(𝑡)=^𝑁_𝑟=0𝑎^𝑖_𝑟𝐵_𝑟,𝑁𝑡−𝑡₀ℎ, (3) where ℎ=𝑡_𝑓−𝑡₀ and 𝑎_𝑟 is the control point of Bezier curve, and 𝐵_𝑟,𝑁𝑡−𝑡₀ℎ=𝑁𝑟1ℎ^𝑁𝑡_𝑓−𝑡^𝑁−𝑟𝑡−𝑡₀^𝑟 (4) is the Bernstein polynomial of degree 𝑁 over the interval [𝑡₀,𝑡_𝑓] (see [17]). By substituting (3) in (2), 𝑅_𝑚(𝑡) can be defined for 𝑡∈[𝑡₀,𝑡_𝑓] as 𝑅_𝑚(𝑡)=^𝑛𝛼_𝑖=1^𝑚𝑖_𝑗=0𝑝_𝑚𝑖𝑗(𝑡)𝑦_𝑖^(𝑗)(𝑡)+^𝑛_𝑖=1^𝑡₀𝑘_𝑚𝑖(𝑡,𝑥)^𝛼𝑚𝑖_𝑗=0𝑦_𝑖^(𝑗)(𝑥)𝑑𝑥−𝑓_𝑚(𝑡),𝑚=1,2,…,𝑛, (5) where (2) is satisfied. The convergence was proved in the approximation with Bezier curves when the degree of the approximate solution, 𝑁, tends to infinity (see [18]).

Now, the residual function is defined over the interval [𝑡₀,𝑡_𝑓] as follows: 𝑅(𝑡)=^𝑡𝑓_𝑡0^𝑛_𝑚=1𝑅_𝑚(𝑡)²𝑑𝑡, (6) where ‖⋅‖ is the Euclidean norm. Our aim is to solve the following problem over the interval [𝑡₀,𝑡_𝑓]: min𝑅(𝑡)s.t.𝑦_𝑖⁽⁰⁾𝑡₀=𝑐_𝑖0,𝑦_𝑖⁽¹⁾𝑡₀=𝑐_𝑖1,…,𝑦^{(𝛼𝑚𝑖}_𝑖⁻¹⁾𝑡₀=𝑐_{𝑖(𝛼𝑚𝑖−1)}. (7)

The mathematical programming problem (7) can be solved by many subroutine algorithms, and we used Maple 12 to solve this optimization problem.

Remark 1. In Chapter 1 of [19], it was proved that 𝑁 satisfies 𝑆𝑁>𝛿²𝜖, (8) where 𝑆=‖𝑦_𝑖(𝑡)‖, and because of this reason that 𝑦_𝑖(𝑡) is uniformly continuous on [𝑡₀,𝑡_𝑓], we have 𝑠,𝑡∈[𝑡₀,𝑡_𝑓] that |𝑡−𝑠|<𝛿 and −(𝜖/2)<𝑦_𝑖(𝑡)−𝑦_𝑖(𝑠)<𝜖/2, for more details see [19].

3. Applications and Numerical Results

Consider the following examples which can be solved by using the presented method.

Example 1. Consider a system of third-order linear Volterra IDEs on the interval [0,1] (see [4]): 𝑦₁^′′(𝑡)+𝑡²𝑦₁(𝑡)−𝑦₂^′′(𝑡)+^𝑡₀(𝑡−𝑥)𝑦₁(𝑥)+𝑦₂(𝑥)𝑑𝑥=𝑔₁(𝑡),4𝑡³𝑦^′₁(𝑡)+6𝑡²𝑦₁(𝑡)+𝑦₂^′′′(𝑡)+^𝑡₀𝑦₁(𝑥)+(𝑡+𝑥)𝑦₂(𝑥)𝑑𝑥=𝑔₂(𝑡), (9) with the initial conditions 𝑦₁(0)=𝑦^′₁(0)=1, 𝑦₂(0)=𝑦₂^′′(0)=0, and 𝑦^′₂(0)=1, where 𝑔₁(𝑡)=2+𝑡²𝑒^𝑡(𝑡)(𝑡),𝑔−𝑡−cos+sin₂(𝑡)=7sin(𝑡)−(1+2𝑡)cos(𝑡)+𝑒^𝑡1+4𝑡²+4𝑡³+𝑡−1. (10)

The exact solution of this system is 𝑦₁(𝑡)=𝑒^𝑡, 𝑦₂(𝑡)=sin(𝑡).

With 𝑁=5, the approximated solutions for 𝑦₁(𝑡) and 𝑦₂(𝑡) are shown, respectively, in Figures 1 and 2, and the computed errors are shown in Table 1 which show the high accuracy of the proposed method.

Table 1 

Computed errors for Example 1.

Figure 1 

The graph of approximated 𝑦1(𝑡) for Example 1.

Figure 2 

The graph of approximated 𝑦2(𝑡) for Example 1.

Example 2. Consider the following system of linear Volterra IDEs equations as follows (see [4]): −𝑦^′₁−12𝑡𝑦₁+32𝑦₂=52−𝑡−272𝑡²+𝑡⁴+32−1+2𝑡²−12−3𝑡+4𝑡³+^t₋₁𝑦₁−3𝑡𝑦₂𝑡𝑑𝑥,₂𝑦₁+𝑦^′₂−𝑡𝑦₂=25+3𝑡+3𝑡³−85𝑡⁵+𝑡²−3𝑡+4𝑡³+^𝑡₋₁(2𝑡+𝑥)𝑦₁+3𝑥²𝑦₂𝑑𝑥, (11) under the conditions 𝑦₁(0)=0 and 𝑦₂(0)=−1, with the exact solution 𝑦₁(𝑡)=4𝑡³−3𝑡,  𝑦₂(𝑡)=2𝑡²−1.

With 𝑁=5, the computed errors are shown in Table 2 which show the high accuracy of the proposed method.

Table 2 

Computed errors for Example 2.

4. Conclusions

In this paper, Bernstein's approximation is used to approximate the solution of linear Volterra IDEs. In this method, we approximate our unknown function with Bernstein's approximation. The present results show that Bernstein's approximation method for solving linear Volterra IDEs is very effective and simple, and the answers are trusty, and their accuracy is high, and we can execute this method in a computer simply. The numerical examples support this claim.

Acknowledgment

The authors are very grateful to the referees for their valuable suggestions and comments that improved the paper.