Elsevier

Applied Mathematics and Computation

Volume 339, 15 December 2018, Pages 272-285
Applied Mathematics and Computation

Numerical solution of three-dimensional Volterra–Fredholm integral equations of the first and second kinds based on Bernstein’s approximation

https://doi.org/10.1016/j.amc.2018.07.021Get rights and content

Abstract

A new and efficient method is presented for solving three-dimensional Volterra–Fredholm integral equations of the second kind (3D-VFIEK2), first kind (3D-VFIEK1) and even singular type of these equations. Here, we discuss three-variable Bernstein polynomials and their properties. This method has several advantages in reducing computational burden with good degree of accuracy. Furthermore, we obtain an error bound for this method. Finally, this method is applied to five examples to illustrate the accuracy and implementation of the method and this method is compared to already present methods. Numerical results show that the new method provides more efficient results in comparison with other methods.

Introduction

The multi-dimensional Volterra–Fredholm integral equations can be arisen in many branches of sciences and provide an important tool for modeling many problems in mathematics, physics, and engineering. These equations appear in fracture mechanics, aerodynamics, the theory of porous filtering, antenna problems in electromagnetic theory, in the quantum effects of electromagnetic fields in the blackbody whose interior is filled by a Kerr nonlinear crystal, in the description of the three-dimensional structure of water around globular solutes, and in the study of a traveling wave solution for a mathematical model describing the population change influenced by a uniformly changing environment (see [4], [5], [13], [25]). Numerical Solution of the three-dimensional integral equations is very significant since they appear in the mathematical modeling. Because these equations are usually difficult to solve practically, the aim of the present research is to develop an accurate method to solve the problem numerically. In this method, we approximate our unknown function with Bernstein’s approximation, which will be introduced in the following. One of the advantages of this method is that not only we can get good numerical solutions for second and first kinds of three-dimensional Volterra–Fredholm integral equations, but also we can implement Bernstein’s approximation method on these equations with singularity simply and get acceptable solutions for these kinds of equations too.

Several numerical methods for approximating the solution of linear and nonlinear three-dimensional integral equations and specially three-dimensional Volterra–Fredholm integral equations exist in the literature [2], [7], [8], [16], [17], [20], [24]. Also, Bernstein polynomials are studied by many authors and applied to solve different problems; for example, see [1], [3], [6], [10], [11], [12], [14], [15], [18], [19], [21], [22], [23]. In the presented paper, we apply Bernstein polynomials to solve 3D-VFIEK1, 3D-VFIEK2 and even singular type of these equations.

The paper is organized as follows: in Section 2, we will introduce the Bernstein’s approximation. In Sections 3 and 4, we will perform it on integral equations 2D-VFIEK2 and 3D-VFIEK1, respectively, and demonstrate the solving process by discretization. Also, we find an error bound for proposed method in Section 5. Section 6 offers five examples to show efficiently of approximating the solution of these kinds of integral equations with Bernstein’s approximation method.

Section snippets

Definition and properties

The Bernstein polynomials of degree (m1, m2, m3) are defined on Ω=[a1,b1]×[a2,b2]×[a3,b3] as follows: Pm1,m2,m3,i1,i2,i3(x,y,z)=(m1i1)(m2i2)(m3i3)(b1a1)m1(b2a2)m2(b3a3)m3(xa1)i1(b1x)m1i1(ya2)i2(b2y)m2i2(za3)i3(b3z)m3i3,where i1=0,1,,m1, i2=0,1,,m2, i3=0,1,,m3 and m1, m2, m3 are arbitrary positive integers. We can write Pm1,m2,m3,i1,i2,i3(x,y,z)=Pm1,i1(x)Pm2,i2(y)Pm3,i3(z),where Pm1,i1(x)=(m1i1)(b1a1)m1(xa1)i1(b1x)m1i1,i1=0,1,,m1,Pm2,i2(y)=(m1i1)(b2a2)m2(ya2)i2(b2y)m2i2,i2

Solving integral equations 3D-VFIEK2 by 3D-BPs

Let us consider the three-dimensional Volterra–Fredholm integral equations of the second kind (3D-VFIEK2) of the form f(x,y,z)=g(x,y,z)+λ1a1xa2ya3zk1(x,y,z,s,t,r)f(s,t,r)drdtds+λ2a1b1a2b2a3b3k2(x,y,z,s,t,r)f(s,t,r)drdtds,where (x, y, z) ∈ Ω and λ1,λ2R. In Eq. (4), the functions g(x, y, z), k1(x, y, z, s, t, r) and k2(x, y, z, s, t, r) are given continuous functions and f(x, y, z) is an unknown function. For numerically solving of this kind of integral equation, we approximate the unknown

Solving integral equations 3D-VFIEK1 by 3D-BPs

Let us consider the three-dimensional Volterra–Fredholm integral equations of the first kind (3D-VFIEK1) of the form g(x,y,z)=λ1a1xa2ya3zk1(x,y,z,s,t,r)f(s,t,r)drdtds+λ2a1b1a2b2a3b3k2(x,y,z,s,t,r)f(s,t,r)drdtds,where (x, y, z) ∈ Ω and λ1,λ2R. In Eq. (9), the functions g(x, y, z), k1(x, y, z, s, t, r) and k2(x, y, z, s, t, r) are given continuous functions and f(x, y, z) is an unknown function. For numerically solving of this kind of integral equation, we approximate the unknown function f

Error bound estimation of the method

We give the following theorems about uniform convergence and error bound of the Bernstein approximation (3) for f(x,y,z).

Lemma 1

see [6]

Let x ∈ [a, b] and Pm,i(x) be defined by (2), then i=0mikPm,i(x)=j=1ktjcj(x),where t1=1 and for j=2,3,,k: tj=n=2j(1)jn(nk11)(n1)!(jn)!,and for j=1,2,,k: cj(x)=m(m1)(mj+1)(ba)j(xa)j.

Lemma 2

see [6]

The Bernstein polynomials have the following properties:

  • (i)

    i=0m(xix)Pm,i(x)=0,

  • (ii)

    i=0m(xix)2Pm,i(x)=1m(xa)(bx),

  • (iii)

    i=0m(xix)3Pm,i(x)=2m2(xa)(bx)(a+b2x),

for any x ∈ [a, b] and xi=a

Numerical examples

To demonstrate the efficiency and the practicability of the proposed method, based on three-dimensional Bernstein polynomials, we consider the following five examples. The numerical experiments are carried out for ɛ=0.01 and m1=m2=m3=m,mN. To compare the presented method with the other methods, the examples are selected from other papers. All examples in this section are tested using Maple 2016. The results are displayed in Table 1, Table 2, Table 3, Table 4, Table 5. Comparison between

Conclusion

In this paper, we used Bernstein’s approximation to approximate the solution of three-dimensional Volterra–Fredholm integral equations of the second kind, first kind and even singular type of these equations. In this method, we approximate our unknown function with Bernstein’s approximation. Our achieve results in this paper show that Bernstein’s approximation method for solving three-dimensional Volterra–Fredholm integral equations of the second kind, first kind and even singular type of these

Acknowledgments

The authors are grateful to the reviewers for their valuable comments and suggestions.

References (25)

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