Solving nonlinear two-dimensional Volterra integro-differential equations by block-pulse functions

Introduction

An area of increasing scientific interest over the past decades is the study of Volterra integro-differential equation. This equation is encountered in various applications such as physics, mechanics, and applied science [ ¹ – ⁴ ].

A general form of the Volterra integral equation can be written as

with given supplementary conditions, where u ( t , x ) is an unknown function which should be determined; g ( t , x ) and k ( t , s , x , y ) are analytical functions, respectively[ ⁵ ]. In this paper, we consider the nonlinear function F ( u ( s , y )) in the following form

where p is a positive integer. With regard to the fact that every finite interval can be transformed to [0,1] by linear map, without loss of generality, we can consider A ₁ = A ₂ =1.

As we know, the block-pulse functions (BPFs) presented by Harmuth [ ⁶ ] are a powerful mathematical tool for solving various kinds of integral equations. These functions are a set of orthogonal functions with piecewise constant values which are defined in the time interval [0, T ₁ ]as

where i =0,…, m −1 with m as a positive integer.

The solution of Fredholm and Volterra integral equations of the second kind have been approximated using BPFs in [ ⁷ ]. Maleknejad and Mahmoudi in [ ⁸ ] have applied a combination of Taylor and block-pulse functions to solve linear Fredholm integral equation. The BPFs and Lagrange interpolating polynomials have been used to approximate the solution of Volterra’s population model by Marzban et al. [ ⁹ ]. Recently, Maleknejad and Mahdiani have applied two dimensional (2D-BPFs) for solving nonlinear mixed Volterra-Fredholm integral equations [ ¹⁰ ]. In this paper, we use 2D-BPFs to approximate the solution of Equation ¹ .

This paper is organized as follows. In section ‘Properties of the 2D-BPFs’, the definition and some properties of the 2D-BPFs are presented. The 2D-BPFs are applied to solve Equation ¹ in ‘Applying the method’ section. The error analysis of the proposed method has been investigated in section ‘The error analysis’. Some numerical results have been presented in section ‘Numerical results’ to show accuracy and efficiency of the proposed method. Finally, some concluding remarks are given in ‘Conclusion’ section.

Properties of the 2D-BPFs

We usually call the block-pulse functions containing two variables as two-dimensional block-pulse functions. An ( m ₁ m ₂ ) set of 2D-BPFs are defined in region t ∈[0, T ₁ )and x ∈[0, T ₂ ) as

where i ₁ =1,2,…, m ₁ and i ₂ =1,2,…, m ₂ with positive integer values for m ₁ , m ₂ , and h1=T1m1,h2=T2m2 . There are some properties for 2D-BPFs, e.g. disjointness, orthogonality, and completeness.

1. Disjointness

The two-dimensional block-pulse functions are disjoined with each other, i.e.

2. Orthogonality

The two-dimensional block-pulse functions are orthogonal with each other, i.e.

in the region of t ∈ [0, T ₁ ) and x ∈[0, T ₂ ) where i ₁ , j ₁ =1,2,…, m ₁ and i ₂ , j ₂ =1,2,…, m ₂ .

3. Completeness

For every f ∈ L ² ([0, T ₁ )×[0, T ₂ )) when m ₁ and m ₂ go to infinity, Parseval identity holds:

where

The set of 2D-BPFs may be written as a ( m ₁ m ₂ ) vector Φ ( t , x ):

where ( t , x ) ∈[0, T ₁ )×[0, T ₂ ).From the above representation and disjointness property, it follows that

where V is an m ₁ m ₂ vector and V~=diag(V). Moreover, it can be clearly concluded that for every ( m ₁ m ₂ ) × ( m ₁ m ₂ ) matrix A

where  is an m ₁ m ₂ vector with elements equal to the diagonal entries of matrix A .

Applying the method

In this section, we solve the nonlinear two-dimensional Volterra integro-differential equations using 2D-BPFs. As we have shown before, we can write

where the m ₁ m ₂ vectors U , G , Λ , U _x , U _t , U _xx , U _tt , U _tx and matrix K are the BPF coefficients of u ( t , x ), g ( t , x ),[ u ( s , y )] ^p , u _x ( t , x ), u _t ( t , x ), u _xx ( t , x ), u _tt ( t , x ), u _tx ( t , x ) and k ( x , y , s , t ), respectively; Λ is a column vector, the elements of which are p th power of the elements of the vector U . Now, consider the following equation:

Using the proposed equations in section ‘Properties of the 2D-BPFs’ to approximate the partial derivatives, we have

If we put B=KΛ~Υ , then it can be written from Equation ¹² ,

Hence, we have

Now, using Equations ²⁶ , ²⁷ , ²⁹ , ³⁰ , ³² and ³³ , we can obtain a nonlinear system, where the solution can be obtained from Newton-Raphson method.

The error analysis

Here, we investigate the representation error of a differentiable function f ( t , x ) when it is represented in a series of 2D-BPFs over the region D =[0,1)×[0,1). For this, we briefly review and use some results from [ ¹⁰ , ¹¹ ]. For details, see the mentioned references. We put m ₁ = m ₂ = m , so h1=h2=1m .

We define the representation error between f ( t , x )and its 2D-BPF expansion over every subregion Di1,i2 as follows:

where

Using mean value theorem, it can be shown that

where ∥ f ^′ ( t , x )∥≤ M [ ¹⁰ , ¹¹ ]. Error between f ( t , x ) and its 2D-BPF expansion, f _m ( t , x ), over the region D can be obtained as follows:

Using Equations ³⁶ and ³⁷ , it can be shown that (see [ ¹⁰ , ¹¹ ])

Hence, ∥e(t,x)∥=O(1m) . Similar to the proposed method in [ ¹⁰ , ¹¹ ], suppose that f ( t , x ) is approximated by

we get f̄i1,i2 , the approximation of fi1,i2 , and

Then, from Equation ³⁸ for (t,x)∈Di1,i2 , we have

Therefore, from Equation ³⁹ , it can be shown that

For an error estimation, reconsider the following nonlinear two-dimensional Volterra integro-differential equation

Let emp(t,x)=up(t,x)−ump(t,x) be the error function of the approximate solution u _m ( t , x ) to u ( t , x ), where u ( t , x ) is the exact solution of Equation ⁴⁰ . Then, we consider

where R _m ( t , x ) is the perturbation function that depends on u _m ( t , x ), ( u _xx ( t , x )) _m , ( u _tx ( t , x )) _m and ( u _tt ( t , x )) _m . It can be obtained by substituting u _m ( t , x ), ( u _xx ( t , x )) _m , ( u _tx ( t , x )) _m and ( u _tt ( t , x )) _m into Equation ⁴⁰ as

Subtracting (41) from (40) gives

Finally, the proposed method in this paper can be applied to approximate e _m ( t , x ) in Equation ⁴² .

Numerical results

In this section, three examples are given to show the the accuracy of the proposed method. For the all examples, we consider the supplementary conditions from the exact solution. The absolute error is computed for m = m ₁ = m ₂ terms of 2D-BPF series in all examples. All computations are implemented in MATLAB software on a personal computer.

Conclusion

In this paper, we have successfully approximated the solution of the form (1) of nonlinear Volterra integro-differential equations. To this end, we have used some orthogonal functions called block-pulse functions. Moreover, the error of the proposed method is analyzed. For more investigation, some examples have been presented. As the numerical results showed, the proposed method is an effective method to solve the Volterra integro-differential equations.