1. Introduction
Many important problems in various sciences, such as engineering, physics, and biology, have been reduced to integral and integro-differential equations in general. The importance of these integro-differential equations increases, in particular if they are associated with delay. In these types of equations, it is not possible to obtain a single solution, but rather several solutions, the best of which can be chosen. Since these equations are difficult to solve analytically, there are many different semi-analytical approaches and numerical approaches for solving integro-differential Equations (IDEs). Aghazadeh and Khajehnasiri in [
1], used block-pulse functions for solving two-dimensional IDEs. Tari and Shahmorad, in [
2], introduced differential transform method for solving system of IDEs. Hussain et al., in [
3], applied the variational iteration method to solve two-dimensional partial integro-differential equations (PIDEs). Hamoud et al., in [
4], studied the numerical solution of IDEs by Adomian decomposition method. Khajehnasiri, in [
5], presented triangular function to obtain the solution of two-dimensional IDEs. Mirzaee et al., in [
6], applied Bernstein polynomials for solving PIDEs. Rivaz et al., in [
7], used Chebyshev polynomials to obtain the solution of IDEs. Behzadi, in [
8], used some iterative techniques for solving Volterra-Fredholm integro-differential equations (VFIDEs). Ahmed and Elzaki, in [
9], studied the solution of IDEs by difference numerical methods. Pandey, in [
10], introduced the finite difference strategy to obtain the solution of Fredholm integro-differential Equations (FIDEs). Abdel-Aty et al., in [
11], used the optimal axillary function method for solving FIDEs. Al-Bugami in [
12], applied the Toeplitz matrix and product Nystrom methods for solving two-dimensional FIDEs with singular kernels. Abdou et al., in [
13], applied the Adomian decomposition method for solving fractional IDEs. Abdou and Elsayed, in [
14], studied the existence of a unique solution of the fractional IDEs in Hilbert space. ALmostafa et al., in [
15], introduced the Aboodh and Double Aboodh transform methods to obtain the solution of PIDEs. Al-Bugami, in [
16], used the Homotopy analysis strategy and Adomian decomposition strategy to obtain the numerical solution of FIDEs.
Recently, difference integro-differential equations (DeIDEs) drew the interest of numerous scholars, such as Saadatmandi and Dehghan, in [
17], who introduced the numerical solution of DeIDEs using the Tau method. Sezer et al. [
18] presented the Chebyshev collocation method for solving DeIDEs, while in [
19], Sezer et al. used the Bernoulli polynomial method for solving DeIDEs. Moreover, Sezer et al. [
20] used the Taylor expansion approach to obtain the solution of mixed DeIDEs. Furthermore, Sezer and Mollaoglu [
21] applied Gegenbauer polynomials for solving high-order linear DeIDEs. Finally, Sezer et al. [
22] applied Lucas polynomials to obtain the solution of functional DeIDEs. Özkol and Yavuz, in [
23], used the differential transform method for solving DeIDEs.
In the present paper, the solution of 2D-MDeIDE, is investigated by using the separation of variables and BPM. Abdelkawy et al., in [
24], used the Bernoulli collocation method for solving hyperbolic telegraph equations. Sezer and Dascioglu, in [
25], studied the solution of high-order generalized pantograph equations by BPM. Mirzaee, in [
26], used BPM for solving integral-algebraic equations. Bhrawy et al., in [
27], presented BPM for solving two-dimensional mixed Volterra–Fredholm integral equations (MVFIEs). Toutounian et al., in [
28], applied BPM to obtain the solution of complex differential equations.
In the late twenty-first century, some authors began to study the effect of the time function in solving all kinds of integral equations (IEs). For more information, see [
29,
30,
31,
32]. Three-dimensional MVFIEs are solved computationally by Mahdy et al., in [
33]. Mahdy and Mohamed, in [
34], used Lucas polynomials to obtain the solution of Cauchy IEs. Mahdy et al. [
35] applied a Chelyshkov polynomial approach to solve the first class of two-dimensional nonlinear Volterra integral equations.
The rest of this essay is structured as follows: in
Section 2, separation of variables is applied to transform the 2D-MDeIDE into a one-dimensional FDeIDE. The separation of variables depends on the physical meanings of the problem, and each of them has its own uses, whether these uses are in basic sciences or in economics. Therefore, the authors covered this problem scientifically, as the technique of separating the variables has two important bases. The first is that the time of the free (given) function is the same as the time of the unknown function that needs to be known. The second is that the time of the unknown function differs from the time of the known function.
In
Section 3, we describe how we used the Bernoulli polynomial method, as a numerical method, to obtain a system of (N + 1) linear algebraic equations with (N + 1) unknown in position. It is known that the product formula of the algebraic system depends on the separation of variables technique. The authors used Bernoulli’s method to obtain (N + 1) from linear algebraic equations, so the error in this case is the absolute difference between the analytical solution and the numerical solution. Since the analytical solution is not known in this type of problem, the error is defined as the algebraic equation following the last equation calculated. In
Section 4, the authors solve some numerical examples that correspond to the chosen appropriate time. Through computer programs, the authors were able to obtain numerical results for each example as well as the resulting error for each case.
In the last section, a summary of the important conclusions is presented, which includes in each example the highest and lowest error values, and the relationship of the authors’ choice of time to the increase or decrease of the error.
Consider the following nth difference integro-differential equations with variable coefficients:
under (
n − 1)determines the initial boundary conditions. For the difference equations and their initial boundary conditions, please see references [
17,
18,
19,
20]).
where
, and
are known continuous functions, the coefficients
are constants, and
is the unknown function.
This type of problem associated with the mixed condition of the study of hysteresis cannot find a single solution. However, the only solution that can be found when certain conditions are imposed on the free function is the given function. It is known that the unknown function behaves the same way as the known function. Thus, when imposing certain conditions on the known function, they will apply to the unknown function, and it will have the only solution that applies and agrees with it.
2. Separation of Variables
In mathematical physics problems, we find that researchers have been concerned with finding the indefinite potential function, which is related to time and location. Thus, researchers have been able to use a variety of techniques to utilize the unknown function. One such method is the time division, which converts the mixed equation in position and time into an algebraic system of mixed equations in position only (Alalyani et al. [
29]). Here, the authors apply a new separation method to discuss the solution of the mixed boundary value problem (1), (2) utilizing the coefficients of the space functions. In this case, these time coefficients will take the form of an integral operator of the Volterra type. This scheme enables the authors to select the known function of time more easily, according to the nature of the problem at hand and the space used. Thus, the time required to obtain the required results can be chosen. There are two cases for the function of the time when we apply separation of variables:
Case (I): when the function of time for the known function (free term) and unknown function is the same, this means that the function of time is known and we use the following technique (Alhazmi and Abdou [
36]). Assume that the unknown and known functions in Equation (
1), respectively take the following forms:
where
is a known function. Hence, the formula (1) yields:
Under the conditions:
where
It is noted that by using this separation technique, the authors were able to obtain the mixed problem (4) and (5) in position. Furthermore, the coefficients of this problem were transformed into an integral term in time (6). The integral term after knowing the time can be determined.
Case (II): when the functions of time for the known and the unknown functions are different, we present the following technique (Abdou [
37]).
where
are unknown functions and
are known functions.
Thus, the formula (1) yields:
Separating the variables, we have:
which represent Volterra integral equation in time, and
which represent FDeIDE only in position.
Various methods, whether numerical methods or analytical methods, can be used to solve Volterra integral equation with a continuous kernel. After using one of these methods, multiplying the result by the function of position, the solution of the problem is completely determined.
3. Bernoulli Polynomial Method
In this section, the BPM is applied for solving the difference integro-differential Equations (4) and (10).
The Bernoulli polynomials (BP)
of degree
ℓ satisfy the following relations:
where
are Bernoulli numbers and the first few Bernoulli numbers are defined as [
38]:
The BP
of degree
ℓ are constructed from the following relation:
The first few BP are defined as follows:
The BP satisfy the following relations:
- (1)
- (2)
- (3)
Case (I): The approximate solution of Equation (
4) is truncated by BP as follows:
where
are the unknown Bernoulli coefficients and
are BP, which is defined by (12).
Similarly
and
are defined as follows:
Substituting from (15)–(17) into (4), we obtain:
By using the collocation points:
Thus, we obtain the following system of
linear algebraic equations with
unknowns:
with the conditions
It can be seen in Equations (20) and (21) that the time function on the right side of the mixed condition (21) fully affects its determination. Therefore, the form of the general solution to Equation (
20) will depend on the form of the time function.
Case (II): Similarly as in case (I), the approximate solution of Equation (
10) after using the collocation points (19) is obtained by solving the following SLAE:
with the conditions
To get the general solution form, firstly, the Volterra Integral Equation (
9) is solved to find the time function
. Secondly, the position equation
is solved using the given initial conditions. Finally, we obtain the general solution by multiplying the time function by the position function
4. Numerical Results
In this section, numerical examples are presented to illustrate the above strategy. Here, the authors are concerned with the four examples of the effect of the time function on the numerical solutions of the problem to be solved. The authors also show the effect of this time on the resulting error function. all numerical results are implemented by Maple software.
Example 1. Consider the third-order MDeIDE:under the conditions:where In this example, we use the two techniques of separation.
Case (I): If the time function for the known and the unknown functions are the same, i.e.,
, we can apply separation of variables and BPM to obtain the approximate solution of Equation (
24) when
for different values of
t as follows:
Case (II): Substituting (7) into (24) and separating variables, we obtain the following Volterra integral equation:
and the FDeIDE in position:
The solution of Equation (
25) is obtained by using Laplace transformation in the following form:
Moreover the approximate solution of Equation (
26) is obtained by using BPM when
in the form:
Thus, the approximate solution of Equation (
24) is given by:
Absolute errors in cases (I) and (II) are presented in
Table 1, while a comparison between the errors for different values of
t is presented in
Figure 1 and
Figure 2.
From the above results, it is clear that the first case when the function of time for the known and unknown functions equal the results is better than the second case when the functions of the time for the known and unknown functions are different.
Note that choosing the time of the known function is the same as measuring the unknown function, resulting in less error and faster calculations. Therefore, in the following examples, we will apply this rule.
Example 2. Consider the second-order MDeIDE:under the conditions: Comparing (27) with (1), then we have
First, separation of variables is applied for the MDeIDE (27), then we obtain the following FDeIDE: Now, BPM can be applied to the FDeIDE (28) for different values of time and different values on The approximate solutions for are given in the forms: The approximate solutions for are given in the forms: The approximate solution for are given in the forms: In Example 2, we deduce that the integral operator of time is
The corresponding errors with respect to the approximate solution for
at different times
, and
are considered in
Table 2). The approximate solutions for the same times
, and
and position divisions
are computed in
Figure 3.
Figure 4 describes the arithmetic error at time
,
Figure 5 at time
, and
Figure 6 at time
In addition, for the same integral operator
,
Table 3 describes the arithmetic error for
at different times
(
Figure 7),
(
Figure 8) and
(
Figure 9).
Example 3. Consider the third-order MDeIDE:under the conditions:where Approximate solutions of Equation (
29),
and 6:
Approximate solutions of Equation (
29),
and 6:
Approximate solutions of Equation (
29),
and 6:
In Example 3, the time integral operator is in the form
. We compute the corresponding errors at
and
and position divisions
are computed in
Table 4 and
Table 5, respectively. The approximate solutions corresponding to the same times
and
and position divisions
are considered in
Figure 10.
Figure 11,
Figure 12 and
Figure 13 descript the error at
, and
and position divisions
, respectively. In addition,
Figure 14,
Figure 15 and
Figure 16 describe the error at
, and
and position divisions
, respectively.
Example 4. Consider the third-order MDeIDE:under the conditions:where Similarly to Examples 1, 2, and 3, the approximate solutions of Equation (30), , and 5 are given by: Approximate solutions of Equation (30), , and 5 Approximate solutions of Equation (30), , and 5 In Example 4, for
and time
The corresponding error for
, and 5 are considered in
Table 6, respectively, while for time
and for
, and 5, the arithmetic errors are computed in
Table 7, respectively. The approximation solutions at
, and
and position divisions
are computed at
Figure 17, while
Figure 18,
Figure 19 and
Figure 20 describe the arithmetic errors for
; and
, respectively. Moreover, at
see
Figure 21, and at
see
Figure 22. At time
, and
see
Figure 23,
Figure 24 and
Figure 25, respectively.