## Approximate solutions of the generalized Abel’s integral equations using the extension Khan’s homotopy analysis transformation method.(English)Zbl 1343.65149

Summary: 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.

### MSC:

 65R20 Numerical methods for integral equations 45E10 Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type)
Full Text:

### References:

 [1] Wazwaz, A.-M., Linear and Nonlinear Integral Equations Methods and Applications (2011), Heidelberg, Germany: Springer, Heidelberg, Germany · Zbl 1227.45002 · doi:10.1007/978-3-642-21449-3 [2] Wazwaz, A.-M., A First Course in Integral Equations (1997), River Edge, NJ, USA: World Scientific Publishing, River Edge, NJ, USA · Zbl 0924.45001 · doi:10.1142/3444 [3] Gorenflo, R.; Vessella, S., Abel Integral Equations (1991), Berlin, Germany: Springer, Berlin, Germany · Zbl 0717.45002 [4] Wazwaz, A. M.; Mehanna, M. S., The combined Laplace-Adomian method for handling singular integral equation of heat transfer, International Journal of Nonlinear Science, 10, 2, 248-252 (2010) · Zbl 1215.65206 [5] Zeilon, N., Sur quelques points de la theorie de l’equation integrale d’Abel, Arkiv för Matematik, Astronomi Och Fysik, 18, 1-19 (1924) [6] Pandey, R. K.; Singh, O. P.; Singh, V. K., Efficient algorithms to solve singular integral equations of Abel type, Computers & Mathematics with Applications, 57, 4, 664-676 (2009) · Zbl 1165.45303 · doi:10.1016/j.camwa.2008.10.085 [7] Kumar, S.; Singh, O. P., Numerical inversion of the abel integral equation using homotopy perturbation method, Zeitschrift fur Naturforschung—Section A, 65, 8, 677-682 (2010) [8] Kumar, S.; Singh, O. P.; Dixit, S., Homotopy perturbation method for solving system of generalized Abel’s integral equations, Applications and Applied Mathematics, 6, 11, 2009-2024 (2011) [9] Dixit, S.; Singh, O. P.; Kumar, S., A stable numerical inversion of generalized Abel’s integral equation, Applied Numerical Mathematics, 62, 5, 567-579 (2012) · Zbl 1241.65122 · doi:10.1016/j.apnum.2011.12.008 [10] Yousefi, S. A., Numerical solution of Abel’s integral equation by using Legendre wavelets, Applied Mathematics and Computation, 175, 1, 574-580 (2006) · Zbl 1088.65124 · doi:10.1016/j.amc.2005.07.032 [11] Khan, M.; Gondal, M. A., A reliable treatment of Abel’s second kind singular integral equations, Applied Mathematics Letters, 25, 11, 1666-1670 (2012) · Zbl 1253.65202 · doi:10.1016/j.aml.2012.01.034 [12] Li, M.; Zhao, W., Solving Abel’s type integral equation with Mikusinski’s operator of fractional order, Advances in Mathematical Physics, 2013 (2013) · Zbl 1269.45003 · doi:10.1155/2013/806984 [13] Liao, S. J., On the homotopy analysis method for nonlinear problems, Applied Mathematics and Computation, 147, 2, 499-513 (2004) · Zbl 1086.35005 · doi:10.1016/s0096-3003(02)00790-7 [14] Liao, S., Comparison between the homotopy analysis method and homotopy perturbation method, Applied Mathematics and Computation, 169, 2, 1186-1194 (2005) · Zbl 1082.65534 · doi:10.1016/j.amc.2004.10.058 [15] Hemida, K. M.; Gepreel, K. A.; Mohamed, M. S., Analytical approximate solution to the time-space nonlinear partial fractional differential equations, International Journal of Pure and Applied Mathematics, 78, 2, 233-243 (2012) · Zbl 1247.65110 [16] Hemida, K. M.; Mohamed, M. S., Numerical simulation of the generalized Huxley equation by homotopy analysis method, Journal of Applied Functional Analysis, 5, 4, 344-350 (2010) · Zbl 1196.65165 [17] Ghany, H. A.; Mohammed, M. S., White noise functional solutions for Wick-type stochastic fractional KdV-Burgers-KURamoto equations, Chinese Journal of Physics, 50, 4, 619-626 (2012) [18] Gepreel, K. A.; Mohamed, M. S., Analytical approximate solution for nonlinear space—time fractional Klein—gordon equation, Chinese Physics B, 22, 1 (2013) · doi:10.1088/1674-1056/22/1/010201 [19] Gepreel, K. A.; Omran, S., Exact solutions for nonlinear partial fractional differential equations, Chinese Physics B, 21, 11, 110204-110211 (2012) · doi:10.1088/1674-1056/21/11/110204 [20] Gepreel, K. A., The homotopy perturbation method applied to the nonlinear fractional Kolmogorov-Petrovskii-PISkunov equations, Applied Mathematics Letters, 24, 8, 1428-1434 (2011) · Zbl 1219.35347 · doi:10.1016/j.aml.2011.03.025 [21] Herzallah, M. A.; Gepreel, K. A., Approximate solution to time—space fractional cubic nonlinear Schrodinger equation, Applied Mathematical Modeling, 36, 11, 5678-5685 (2012) · Zbl 1254.65115 · doi:10.1016/j.apm.2012.01.012 [22] Khan, Y.; Faraz, N.; Kumar, S.; Yildirim, A. A., A coupling method of homotopy method and Laplace transform for fractional modells, UPB Science Bulletin, Series A: Applied Mathematics & Physics, 74, 1, 57-68 (2012) · Zbl 1245.65143 [23] Singh, J.; Kumar, D.; Kumar, S.; Kapoor, S., New homotopy analysis transform algorithm to solve Volterra integral equation, Ain Shams Engineering Journal, 5, 1, 243-246 (2014) · doi:10.1016/j.asej.2013.07.004 [24] Khader, M. M.; Kumar, S.; Abbasbandy, S., New homotopy analysis transform method for solving the discontinued problems arising in nanotechnology, Chinese Physics B, 22, 11 (2013) · doi:10.1088/1674-1056/22/11/110201 [25] Arife, A. S.; Vanani, S. K.; Soleymani, F., The laplace homotopy analysis method for solving a general fractional diffusion equation arising in nano-hydrodynamics, Journal of Computational and Theoretical Nanoscience, 10, 1, 33-36 (2013) [26] Khan, M.; Gondal, M. A.; Hussain, I.; Vanani, S. K., A new comparative study between homotopy analysis transform method and homotopy perturbation transform method on a semi infinite domain, Mathematical and Computer Modelling, 55, 3-4, 1143-1150 (2012) · Zbl 1255.65186 · doi:10.1016/j.mcm.2011.09.038 [27] Odibat, Z. M., Differential transform method for solving Volterra integral equation with separable kernels, Mathematical and Computer Modelling, 48, 7-8, 1144-1149 (2008) · Zbl 1187.45003 · doi:10.1016/j.mcm.2007.12.022 [28] Abo-Dahab, S. M.; Mohamed, M. S.; Nofal, T. A., 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, 2013 (2013) · Zbl 1470.35349 · doi:10.1155/2013/614874 [29] Gepreel, K. A.; Mohamed, M. S., An optimal homotopy analysis method nonlinear fractional differential equation, Journal of Advanced Research in Dynamical and Control Systems, 6, 1, 1-10 (2014) [30] Mohamed, M. S., Application of optimal HAM for solving the fractional order logistic equation, Applied and Computational Mathematics, 3, 1, 27-31 (2014) · doi:10.11648/j.acm.20140301.14 [31] Jafarian, A.; Ghaderi, P.; Alireza, K.; Aleanu, D., Analytical treatment of system of Abel integral equations by homotopy analysis method, Romanian Reports in Physics, 66, 3, 603-611 (2014)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.