Journal of Applied Mathematics

Journal of Applied Mathematics / 2015 / Article

Research Article | Open Access

Volume 2015 |Article ID 357861 | https://doi.org/10.1155/2015/357861

Mohamed S. Mohamed, Khaled A. Gepreel, Faisal A. Al-Malki, Maha Al-Humyani, "Approximate Solutions of the Generalized Abel’s Integral Equations Using the Extension Khan’s Homotopy Analysis Transformation Method", Journal of Applied Mathematics, vol. 2015, Article ID 357861, 9 pages, 2015. https://doi.org/10.1155/2015/357861

Approximate Solutions of the Generalized Abel’s Integral Equations Using the Extension Khan’s Homotopy Analysis Transformation Method

Academic Editor: Md Sazzad Hossien Chowdhury
Received09 Sep 2014
Accepted26 Dec 2014
Published04 Mar 2015

Abstract

User friendly algorithm based on the optimal homotopy analysis transform method (OHATM) is proposed to find the approximate solutions to generalized Abel’s integral equations. The classical theory of elasticity of material is modeled by the system of Abel integral equations. It is observed that the approximate solutions converge rapidly to the exact solutions. Illustrative numerical examples are given to demonstrate the efficiency and simplicity of the proposed method. Finally, several numerical examples are given to illustrate the accuracy and stability of this method. Comparison of the approximate solution with the exact solutions shows that the proposed method is very efficient and computationally attractive. We can use this method for solving more complicated integral equations in mathematical physical.

1. Introduction

An integral equation is defined as an equation in which the unknown function to be determined appears under the integral sign. The subject of integral equations is one of the most useful mathematical tools in both pure and applied mathematics. It has enormous applications in many physical problems. Many initial and boundary value problems associated with ordinary differential equation (ODE) and partial differential equation (PDE) can be transformed into problems of solving some approximate integral equations. Abel’s equation is one of the integral equations derived directly from a concrete problem of physics, without passing through a differential equation. This integral equation occurs in the mathematical modeling of several models in physics, astrophysics, solid mechanics, and applied sciences. The great mathematician Niels Abel gave the initiative of integral equations in 1823 in his study of mathematical physics [14]. In 1924, generalized Abel’s integral equation on a finite segment was studied by Zeilon [5]. The different types of Abel integral equation in physics have been solved by Pandey et al. [6], Kumar and Singh [7], Kumar et al. [8], Dixit et al. [9], Yousefi [10], Khan and Gondal [11], and Li and Zhao [12] by applying various kinds of analytical and numerical methods.

The development of science has led to the formation of many physical laws, which, when restated in mathematical form, often appear as differential equations. Engineering problems can be mathematically described by differential equations, and thus differential equations play very important roles in the solution of practical problems. For example, Newton’s law, stating that the rate of change of the momentum of a particle is equal to the force acting on it, can be translated into mathematical language as a differential equation. Similarly, problems arising in electric circuits, chemical kinetics, and transfer of heat in a medium can all be represented mathematically as differential equations.

The main aim of this paper is to present analytical and approximate solution of integral equations by using new mathematical tool like optimal homotopy analysis transform method. The proposed method is coupling of the homotopy analysis method HAM and Laplace transform method. The HAM, first proposed in 1992 by Liao, has been successfully applied to solve many problems in physics and science [1318]. In recent years many authors have paid attention to study the solutions of linear and nonlinear partial differential equations by using various methods of approximate solutions such as HAM, HPM, and ADM combined with the Laplace transform [1927].

A typical form of an integral equation in is of the form where is called the kernel of integral equation (1) and and are the limits of integration. It can be easily observed that the unknown function appears under the integral sign. It is to be noted here that both the kernel and the function in (1) are given functions; and is a constant parameter. The prime objective of this text is to determine the unknown function that will satisfy (1) using a number of solution techniques. We will devote considerable efforts in exploring these methods to find solutions of the unknown function, so that we can deduce the exact solution of the integral equation (1).

2. Basic Idea of Optimal Homotopy Analysis Transform Method

In order to elucidate the solution procedure of the optimal homotopy analysis transform method, we consider the following integral equations of second kind:

Now operating the Laplace transform on both sides in (2), we get

We define the nonlinear operator where is an embedding parameter and is the real function of and . By means of generalizing the traditional homotopy methods, the great mathematician Liao [13, 14] constructed the zero order deformation equation where is a nonzero auxiliary parameter, is an auxiliary function, is an initial guess of , and is an unknown function. It is important that one has great freedom to choose auxiliary thing in OHATM. Obviously, when and , it holds respectively. Thus, as increases from to , the solution varies from the initial guess to the solution. Expanding in Taylor’s series with respect to , we have where

If the auxiliary linear operator, the initial guess, the auxiliary parameter , and the auxiliary function are properly chosen, series (7) converges at and we have which must be one of the solutions of the original integral equations. Define the vectors

Differentiating (6)  -times with respect to the embedding parameter , then setting , and finally dividing them by , we obtain the th order deformation equation where In this way, it is easy to obtain for , at th order; we have When we get an accurate approximation of original equation (2).

Many recently references such as [2830] have applied the homotopy analysis method to nonlinear ODEs and discussed the optimization method to find out the optimal convergence control parameters by minimum of the square residual error integrated in the whole region. Their approach is based on the square residual error. Let denote the square residual error of governing equation (2) and it is expressed as where and the optimal value of is given by a nonlinear algebraic equation as

3. Numerical Results

In this section, we discuss the implementation of our proposed algorithm and investigate its accuracy by applying the homotopy analysis transform method. The simplicity and accuracy of the proposed method are illustrated through the following numerical examples by computing the absolute error: where is the exact solution of the integral equations and is the approximate solution of the integral equations.

To demonstrate the effectiveness of the HATM algorithm, we have discussed above, several examples for some integral equations in this section. Here all the results are calculated by using the symbolic calculus software Mathematica 7.

Example 1. Consider the following system of Abel’s integral equations [31]: with the initial condition and the exact solution where with the property that which implies that
Taking Laplace transform of (18) of both sides subject to the initial condition, we get
We now define the nonlinear operator as and then the th order deformation equation is given by Taking inverse Laplace transform of (26), we get where with the assumption .
Let us take the initial approximation as The other components are given by
Proceeding in this manner, the rest of the components for can be completely obtained and the series solutions are thus entirely determined. The solution of the problem is given as however, mostly, the results given by the Laplace decomposition method and homotopy analysis transform method converge to the corresponding numerical solutions in a rather small region. But, different from those two methods, the homotopy analysis transform method provides us with a simple way to adjust and control the convergence region of solution series by choosing a proper value for the auxiliary ; if we select , then
The above result is in complete agreement with [31].
The graphical comparison between the exact solution and the approximate solution which obtained by the HATM at . It can be seen that the solution obtained by the present method nearly identical to the exact solution. The above result is in complete agreement with [31].

Example 2. In this example, we considered the following system of Abel integral equations of second kind as with the initial condition with the exact solution where with the property that which implies that
Taking Laplace transform of (33) of both sides subject to the initial condition, we get
We now define the nonlinear operator as and then the th order deformation equation is given by
Taking inverse Laplace transform of (41), we get where with the assumption .
Let us take the initial approximation as The other components are given by
Hence the solution of (41) is given as The homotopy analysis transform method provides us with a simple way to adjust and control the convergence region of solution series by choosing a proper value for the auxiliary parameter ; if we select , then the above result is in complete agreement with [31].
In general, by means of the so-called -curve, it is straightforward to choose an appropriate range for which ensures the convergence of the solution series. To study the influence of on the convergence of solution, the -curves of and are sketched, as shown in Figures 5 and 6 (see Table 1). For better presentation, these valid regions have been listed in Table 2. The absolute error is exhibited in Figure 4 and also from Figures 1 to 6 we show the graphical comparison between the exact solution and the approximate solution obtained by the OHATM. It can be seen that the solution obtained by the present method nearly identical to the exact solution. The above result is in complete agreement with [31].


The values of -curve derived from Figures 2 and 3

≤−0.4
≤−0.4


The values of -curve derived from Figures 5 and 6

≤−0.8
≤−0.94

4. Conclusions

The main aim of this work is to provide the system of Abel integral equations of the second kind which has been studied by the optimal homotopy analysis transform method OHATM. The OHATM is more suitable than other analytic methods. OHATM is coupling of homotopy analysis and Laplace transform method. The new modification is a powerful tool to search for solutions of Abel’s integral equation. An excellent agreement is achieved. The proposed method is employed without using linearization, discretization, or transformation. It may be concluded that the OHATM is very powerful and efficient in finding the analytical solutions for a wide class of differential and integral equations. The approximate solution of this system is calculated in this form of series whose components are computed by applying a recursive relation. Results indicate that the solution obtained by this method converges rapidly to an exact solution. The graphs plotted confirm the results.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

References

  1. A.-M. Wazwaz, Linear and Nonlinear Integral Equations Methods and Applications, Springer, Heidelberg, Germany, 2011. View at: Publisher Site | MathSciNet
  2. A.-M. Wazwaz, A First Course in Integral Equations, World Scientific Publishing, River Edge, NJ, USA, 1997. View at: Publisher Site | MathSciNet
  3. R. Gorenflo and S. Vessella, Abel Integral Equations, Springer, Berlin, Germany, 1991. View at: MathSciNet
  4. A. M. Wazwaz and M. S. Mehanna, “The combined Laplace-Adomian method for handling singular integral equation of heat transfer,” International Journal of Nonlinear Science, vol. 10, no. 2, pp. 248–252, 2010. View at: Google Scholar | MathSciNet
  5. N. Zeilon, “Sur quelques points de la theorie de l'equation integrale d'Abel,” Arkiv för Matematik, Astronomi Och Fysik, vol. 18, pp. 1–19, 1924. View at: Google Scholar
  6. R. K. Pandey, O. P. Singh, and V. K. Singh, “Efficient algorithms to solve singular integral equations of Abel type,” Computers & Mathematics with Applications, vol. 57, no. 4, pp. 664–676, 2009. View at: Publisher Site | Google Scholar | MathSciNet
  7. S. Kumar and O. P. Singh, “Numerical inversion of the abel integral equation using homotopy perturbation method,” Zeitschrift fur Naturforschung—Section A, vol. 65, no. 8, pp. 677–682, 2010. View at: Google Scholar
  8. S. Kumar, O. P. Singh, and S. Dixit, “Homotopy perturbation method for solving system of generalized Abel's integral equations,” Applications and Applied Mathematics, vol. 6, no. 11, pp. 2009–2024, 2011. View at: Google Scholar | MathSciNet
  9. S. Dixit, O. P. Singh, and S. Kumar, “A stable numerical inversion of generalized Abel's integral equation,” Applied Numerical Mathematics, vol. 62, no. 5, pp. 567–579, 2012. View at: Publisher Site | Google Scholar | MathSciNet
  10. S. A. Yousefi, “Numerical solution of Abel's integral equation by using Legendre wavelets,” Applied Mathematics and Computation, vol. 175, no. 1, pp. 574–580, 2006. View at: Publisher Site | Google Scholar | MathSciNet
  11. M. Khan and M. A. Gondal, “A reliable treatment of Abel's second kind singular integral equations,” Applied Mathematics Letters, vol. 25, no. 11, pp. 1666–1670, 2012. View at: Publisher Site | Google Scholar | MathSciNet
  12. M. Li and W. Zhao, “Solving Abel's type integral equation with Mikusinski's operator of fractional order,” Advances in Mathematical Physics, vol. 2013, Article ID 806984, 4 pages, 2013. View at: Publisher Site | Google Scholar | MathSciNet
  13. S. J. Liao, “On the homotopy analysis method for nonlinear problems,” Applied Mathematics and Computation, vol. 147, no. 2, pp. 499–513, 2004. View at: Publisher Site | Google Scholar | MathSciNet
  14. S. Liao, “Comparison between the homotopy analysis method and homotopy perturbation method,” Applied Mathematics and Computation, vol. 169, no. 2, pp. 1186–1194, 2005. View at: Publisher Site | Google Scholar | MathSciNet
  15. K. M. Hemida, K. A. Gepreel, and M. S. Mohamed, “Analytical approximate solution to the time-space nonlinear partial fractional differential equations,” International Journal of Pure and Applied Mathematics, vol. 78, no. 2, pp. 233–243, 2012. View at: Google Scholar
  16. K. M. Hemida and M. S. Mohamed, “Numerical simulation of the generalized Huxley equation by homotopy analysis method,” Journal of Applied Functional Analysis, vol. 5, no. 4, pp. 344–350, 2010. View at: Google Scholar | MathSciNet
  17. H. A. Ghany and M. S. Mohammed, “White noise functional solutions for Wick-type stochastic fractional KdV-Burgers-KURamoto equations,” Chinese Journal of Physics, vol. 50, no. 4, pp. 619–626, 2012. View at: Google Scholar | MathSciNet
  18. K. A. Gepreel and M. S. Mohamed, “Analytical approximate solution for nonlinear space—time fractional Klein—gordon equation,” Chinese Physics B, vol. 22, no. 1, Article ID 010201, 2013. View at: Publisher Site | Google Scholar
  19. K. A. Gepreel and S. Omran, “Exact solutions for nonlinear partial fractional differential equations,” Chinese Physics B, vol. 21, no. 11, pp. 110204–110211, 2012. View at: Publisher Site | Google Scholar
  20. K. A. Gepreel, “The homotopy perturbation method applied to the nonlinear fractional Kolmogorov-Petrovskii-PISkunov equations,” Applied Mathematics Letters, vol. 24, no. 8, pp. 1428–1434, 2011. View at: Publisher Site | Google Scholar | MathSciNet
  21. M. A. Herzallah and K. A. Gepreel, “Approximate solution to time—space fractional cubic nonlinear Schrodinger equation,” Applied Mathematical Modeling, vol. 36, no. 11, pp. 5678–5685, 2012. View at: Publisher Site | Google Scholar
  22. Y. Khan, N. Faraz, S. Kumar, and A. A. Yildirim, “A coupling method of homotopy method and Laplace transform for fractional modells,” UPB Science Bulletin, Series A: Applied Mathematics & Physics, vol. 74, no. 1, pp. 57–68, 2012. View at: Google Scholar
  23. J. Singh, D. Kumar, S. Kumar, and S. Kapoor, “New homotopy analysis transform algorithm to solve Volterra integral equation,” Ain Shams Engineering Journal, vol. 5, no. 1, pp. 243–246, 2014. View at: Publisher Site | Google Scholar
  24. M. M. Khader, S. Kumar, and S. Abbasbandy, “New homotopy analysis transform method for solving the discontinued problems arising in nanotechnology,” Chinese Physics B, vol. 22, no. 11, Article ID 110201, 2013. View at: Publisher Site | Google Scholar
  25. A. S. Arife, S. K. Vanani, and F. Soleymani, “The laplace homotopy analysis method for solving a general fractional diffusion equation arising in nano-hydrodynamics,” Journal of Computational and Theoretical Nanoscience, vol. 10, no. 1, pp. 33–36, 2013. View at: Google Scholar
  26. M. Khan, M. A. Gondal, I. Hussain, and S. K. Vanani, “A new comparative study between homotopy analysis transform method and homotopy perturbation transform method on a semi infinite domain,” Mathematical and Computer Modelling, vol. 55, no. 3-4, pp. 1143–1150, 2012. View at: Publisher Site | Google Scholar | MathSciNet
  27. Z. M. Odibat, “Differential transform method for solving Volterra integral equation with separable kernels,” Mathematical and Computer Modelling, vol. 48, no. 7-8, pp. 1144–1149, 2008. View at: Publisher Site | Google Scholar | MathSciNet
  28. S. M. Abo-Dahab, M. S. Mohamed, and T. A. Nofal, “A one step optimal homotopy analysis method for propagation of harmonic waves in nonlinear generalized magnetothermoelasticity with two relaxation times under influence of rotation,” Abstract and Applied Analysis, vol. 2013, Article ID 614874, 14 pages, 2013. View at: Publisher Site | Google Scholar
  29. K. A. Gepreel and M. S. Mohamed, “An optimal homotopy analysis method nonlinear fractional differential equation,” Journal of Advanced Research in Dynamical and Control Systems, vol. 6, no. 1, pp. 1–10, 2014. View at: Google Scholar | MathSciNet
  30. M. S. Mohamed, “Application of optimal HAM for solving the fractional order logistic equation,” Applied and Computational Mathematics, vol. 3, no. 1, pp. 27–31, 2014. View at: Publisher Site | Google Scholar
  31. A. Jafarian, P. Ghaderi, K. Alireza, and D. Aleanu, “Analytical treatment of system of Abel integral equations by homotopy analysis method,” Romanian Reports in Physics, vol. 66, no. 3, pp. 603–611, 2014. View at: Google Scholar

Copyright © 2015 Mohamed S. Mohamed et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


More related articles

 PDF Download Citation Citation
 Download other formatsMore
 Order printed copiesOrder
Views2219
Downloads858
Citations

Related articles

Article of the Year Award: Outstanding research contributions of 2020, as selected by our Chief Editors. Read the winning articles.