Abstract
In this article, we implemented the Elzaki decomposition technique (EDM) to solve Volterra–Fredholm integro-differential equations of higher-order. Illustrations are used to test the technique’s accuracy and validity. Comparison among the acquired consequences by EDM and actual solutions have proven the power and accuracy of this technique. This technique is dependable and able to supply analytic remedies for solving such equations.
1. Introduction
This study is concerned with important problems in the sciences of engineering, which contain both integral and differential factors. In the early 20th century, this kind of equation was first presented by Volterra, where he studied genetic influences on population growth by developing the topic of integral and differential equations [1].
Many researchers, engineers, and scientists face problems of an electrical circuit and biological species when solving differential equations resulting from the spread of heat and mass that appear with increasing and decreasing generation rates.
In physics, engineering, and biology applications, one can find additional details about the origins of these equations as well as the behavior of advanced integral equations [2]. The Taylor collocation, Adomian decomposition, Chebyshev, Haar wavelet, successive approximation, Bernstein, homotopy perturbation, and other technologies are widely used to work with the IDEs of higher-order [3–13]. In recent years, Elzaki transform (ET) was introduced by Tarig Elzaki where it was used to solve different types of equations with integral, differential, partial integral, integro-differential, and fractional-order equations [14–20].
The motivation of the article is to enhance the analysis of the modified Elzaki transformation, known as the Elzaki decomposition method (EDM), to solve the higher-order IDE of Volterra–Fredholm of the formsubject towhere are constant values, are given forms with appropriate interval derivatives , and are nonlinear functions in .
2. Elzaki Transformation Method (ETM) with Their Properties
The Elzaki transform (ET) of , as in [21].
Since is the operator of Elzaki transform and the set is defined as can be finite or infinite, and is a real finite number.
If , then is called the inverse transform of Elzaki to and denoted by , where is the operator of inverse Elzaki transform. Table 1 shows the transformation of Elzaki for some basic functions [22].
Special properties of the Elzaki transform are as follows.
Theorem 1 (linearity, see [23]).. If and are Elzaki transform (ET) of and , respectively, then are constants.
Theorem 2 (differentiation, see [21]).. If the transform of Elzaki for given by , then
Theorem 3 (shift, see [23]).. If with Elzaki transform , then
Theorem 4 (convolution theorem, see [23])..
If and are functions with Elzaki transform and , then the convolution of isAlso, Elzaki transform is
Theorem 5 (integration, see [23]). Let F denote to the transform of Elzaki for , then the definite integral of ,has transform of Elzaki as
3. The Proposed Technique (EDM)
The proposed technique of the method (EDM) is as follows: we suppose that the (1) and (2) have sufficiently differentiable and unique solution . We use the Elzaki decomposition method to get an approximation for the integro-differential equations of high order.
Applying the ETM of both sides of (1), we get
Using Theorem 2, we get
Solving for , we get
Now, the EDM represents the solution of our problem as infinite series
Using the recursive relationship, the components E [y (z)] will be obtained, and the polynomials of the infinite series will be substituted for the nonlinear functions and .where and are the Adomian polynomials that are created using Adomian’s specialized algorithms [24].
Substituting equations (15) and (16) in (14), we have
Now,
The rest components of can be found using the previous solutions as
As a result, the components are identified by applying the inverse Elzaki transform of (18) and (19) to obtain
4. Numerical Examples
To apply the proposed technique (EDM), various examples of higher orders were taken, and to study the efficiency of the suggested technique, a comparison with the exact solution was made. The calculations of solutions for examples are performed using the Maple program.
Example 1. The third order linear IDE of Volterra–Fredholm issubject toApplying to both sides of (21) Elzaki transform, we getWe use the initial conditions together with Theorems 2 and 5 to obtainSolving the (15) for , we getSubstituting the series assumptions for as given in (15) and using the recursive relation, we getTaking the inverse transform of (26), we haveSo, the series solution will be the actual solution .
Example 2. The fourth-order nonlinear IDE of Volterra–Fredholm issubject toTaking the transformation of Elzaki to (28), we haveApplying the initial conditions together with Theorems 2–5, we obtainedNow, solving the (31) for , we getSubstituting the series assumptions for as given in (15) and using the recursive relation, we getwhere are given byWe apply for both sides of (33) the inverse Elzaki transform; we obtain general solution given by recursive relation aswhich leads toand so on.
Now, the solution for the (28) will be the following:which is the actual solution.
Table 2 indicates the evaluation of the numerical results and the exact solution, where the numerical solution exactly corresponds to the exact values as shown in Figure 1.
Example 3. The fifth order nonlinear IDE of Volterra–Fredholm iswhereTo both sides of (38), we take the Elzaki transform and we getApplying the initial conditions together with Theorems 2–5, we obtainedSolving the (41) for , we getSubstituting the series assumptions for as given in (15) and using the recursive relation, we getwhere are given byWe apply for both sides of (43) the inverse Elzaki transform; we obtain the general solution given by recursive relation aswhich leads toand so on.
Now, the solution for the (38) will be the following:which converges to the exact solution , as shown in Figure 2.
Table 3 indicates an assessment of the numerical outcomes and the exact solution, and Figure 2 shows that the approximations are very near to the actual values.
5. Conclusion
The purpose of this study is to offer an efficient approach for solving higher-order IDEs. This goal was achieved using the Elzaki decomposition method (EDM). The validity and the accuracy of our approach are checked by solving three examples. To explain the technique, plots of the behavior of the solutions are considered to make the shower that the EDM is a very powerful and efficient method to produce the approximate solution.
Data Availability
The data used to support the findings of this study are available from the corresponding authors upon request.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This work was supported by the Mosul University, College of Computer Sciences and Mathematics, Republic of Iraq.