In this paper, we will use the successive approximation method for solving Fredholm integral equation of the second kind using Maple18. By means of this method, an algorithm is successfully established for solving the non-linear Fredholm integral equation of the second kind. Finally, several examples are presented to illustrate the application of the algorithm and results appear that this method is very effective and convenient to solve these equations.
KeywordsNonlinear Fredholm Integral Equation of the Second KindSuccessive Approximation MethodMaple18
He, J.H. (1999) Homotopy Perturbation Technique. Computer Methods in Applied Mechanics and Engineering, 178, 257-262. https://doi.org/10.1016/S0045-7825(99)00018-3
He, J.H. (2000) A Coupling Method of a Homotopy Technique and a Perturbation Technique. International Journal of Non-Linear Mechanics, 35, 37-43. https://doi.org/10.1016/S0020-7462(98)00085-7
He, J.H. (2003) Homotopy Perturbation Method: A New Non-Linear Analytical Technique. Applied Mathematics and Computation, 135, 73-79. https://doi.org/10.1016/S0096-3003(01)00312-5
He, J.H. (2004) Asymptotology by Homotopy Perturbation Method. Applied Mathematics and Computation, 156, 591-596. https://doi.org/10.1016/j.amc.2003.08.011
He, J.H. (2005) Application of Homotopy Perturbation Method to Nonlinear Wave Equations. Chaos, Solitons and Fractals, 26, 695-700. https://doi.org/10.1016/j.chaos.2005.03.006
He, J.H. (2006) Homotopy Perturbation Method for Solving Boundary Value Problems. Physics Letters A, 350, 87-88. https://doi.org/10.1016/j.physleta.2005.10.005
He, J.H. (2006) New Interpretation of Homotopy Perturbation Method. International Journal of Modern Physics B, 18, 2561-2568. https://doi.org/10.1142/S0217979206034819
Balakumar, V. and Murugesan, K. (2013) Numerical Solution of Systems of Linear Volterra Integral Equations Using Block-Pulse Functions, Malaya. Journal of Mathematic, 1, 77-84.
Burden, R.L. and Douglas Faires, J. (2005) Numerical Analysis. 8th Edition, Thomson Brooks/Cole, Belmont.
Burton, T.A. (2005) Volterra Integral and Differential Equations. 2nd Edition, Mathematics in Science & Engineering, Elsevier, Amsterdam, 202.
Berenguer, M.I., Gamez, D., Garralda-Guillem, A.I., Ruiz Galan, M. and Serrano Perez, M.C. (2011) Biorthogonal Systems for Solving Volterra Integral Equation Systems of the Second Kind. Journal of Computational and Applied Mathematics, 235, 1875-1883. https://doi.org/10.1016/j.cam.2010.07.011
Biazar, J. and Eslami, M. (2011) Modified HPM for Solving Systems of Volterra Integral Equation of the Second Kind. Journal of King Saud University-Science, 23, 35-39. https://doi.org/10.1016/j.jksus.2010.06.004
Linz, P. (1985) Analytical and Numerical Methods for Volterra Equations. Studies in Applied Mathematics 7, SIAM, Philadelphia. https://doi.org/10.1137/1.9781611970852
Mirzaee, F. (2010) Numerical Computational Solution of the Linear Volterra Integral Equations System via Rationalized Hear Functions. Journal of King Saud University-Science, 4, 265-268. https://doi.org/10.1016/j.jksus.2010.05.010
Wazwaz, A.M. (2011) Linear and Nonlinear Integral Equation: Methods and Applications. Springer, Berlin. https://doi.org/10.1007/978-3-642-21449-3
Effati, S. and Noori Skandari, M.H. (2012) Optimal Control Approach for Solving Linear Volterra Integral Equations. International Journal of Intelligent Systems and Applications, 4, 40-46. https://doi.org/10.5815/ijisa.2012.04.06
Maturi, D.A. (2014) Numerical Solution of System of Two Nonlinear Volterra Integral Equations. International Journal of Computers & Technology, 12, 3967-3975. https://doi.org/10.24297/ijct.v12i10.2989
Maturi, D.A., Bajamal, A.Z. and Al-Gethami, B.M. (2014) Numerical Solution of Volterra Integral Equation of Second Kind Using Implicit Trapezoidal. Journal of Advances in Mathematics, 8, 1540-1553.
Maturi, D.A. (2014) Adomian Decomposition Method of Fredholm Integral Equation of the Second Kind Using Maple. Journal of Advances in Mathematics, 9, 1868-1875.
Maturi, D.A. (2014) Application of Adomian Decomposition Method for Solving of Fredholm Integral Equation of the Second Kind. European Journal of Science and Engineering, 9, 1-9.
Sorkun, H.H. and Yalcinbas, S. (2010) Approximate Solutions of Linear Volterra Integral Equation Systems with Variable Coefficients. Applied Mathematical Modeling, 34, 3451-3464. https://doi.org/10.1016/j.apm.2010.02.034
Maturi, D.A. and Malaikah, H. (2018) Numerical Solution of System of Three Nonlinear Volterra Integral Equation Using Implicit Trapezoidal. Journal of Mathematics Research, 10, 44. https://doi.org/10.5539/jmr.v10n1p44
Ostvoar, A.A. and Hasina, M. (1999) Solving nth-Order Integro-Differential Equations Using the Combined Laplace Transfor-Successive Approximations Method. Communications on Advanced Computational Science with Applications, 178, 257-262.
Khan, F., Mustafa, G., Omar, M. and Komal, H. (2017) Numerical Approach Based on Bernstein Polynomials for Solving Mixed Volterra-Fredholm Integral Equations. AIP Advances, 7, 125-123. https://doi.org/10.1063/1.5008818
Huanga, G.X. and Yin, F. (2011) An Inverse Eigenproblem and an Associated Approximation Problem for Generalized Reflexive and Anti-Reflexive Matrices. Journal of Computational and Applied Mathematics, 235, 2888-2895. https://doi.org/10.1016/j.cam.2010.12.016
Behiry, S.H., Abd-Elmonem, R.A. and Gomaa, A.M. (2010) Discrete Adomian Decomposition Solution of Nonlinear Fredholm Integral Equation. Ain Shams Engineering Journal, 1, 97-101. https://doi.org/10.1016/j.asej.2010.09.009
Biazar, J., Ghanbari, B. and Porshokouhi, M.G. (2011) He’s Homotopy Perturbation Method: A Strongly Promising Method for Solving Non-Linear Systems of the Mixed Volterra-Fredholm Integral Equations. Computers and Mathematics with Applications, 61, 1016-1023. https://doi.org/10.1016/j.camwa.2010.12.051
Ayati, Z. and Biazar, J. (2015) On the Convergence of Homotopy Perturbation Method. Journal of the Egyptian Mathematical Society, 23, 424-428. https://doi.org/10.1016/j.joems.2014.06.015
Biazar, J. and Ghanbari, B. (2012) The Homotopy Perturbation Method for Solving Neutral Functional-Differential Equations with Proportional Delays. Journal of King Saud University-Science, 24, 33-37. https://doi.org/10.1016/j.jksus.2010.07.026
Wu, G.C., Baleanu, D. and Deng, Z.G. (2015) Variational Iteration Method as a Kernel Constructive Technique. Applied Mathematical Modelling, 39, 4378-4384. https://doi.org/10.1016/j.apm.2014.12.032
Dong, C., Chen, Z. and Jiang, W. (2013) A Modified Homotopy Perturbation Method for Solving the Nonlinear Mixed Volterra-Fredholm Integral Equation. Journal of Computational and Applied Mathematics, 239, 359-366. https://doi.org/10.1016/j.cam.2012.09.003
Turkyilmazoglu, M. (2010) A Note on the Homotopy Analysis Method. Applied Mathematics Letters, 23, 1226-1230. https://doi.org/10.1016/j.aml.2010.06.003
Akram, G. and Sadaf, M. (2017) Application of Homotopy Analysis Method to the Solution of Ninth Order Boundary Value Problems in AFTI-F16 Fighters. Journal of the Association of Arab Universities for Basic and Applied Sciences, 24, 149-155. https://doi.org/10.1016/j.jaubas.2016.08.002
Eldeberky, Y. (2011) Modeling Spectra of Breaking Waves Propagating Overa Beach. Ain Shams Engineering Journal, 2, 71-77. https://doi.org/10.1016/j.asej.2011.07.002
Daoud, Y. and Khidir, A.A. (2018) Modified Adomian Decomposition Method for Solving the Problem of Boundary Layer Convective Heat Transfer. Propulsion and Power Research, 7, 231-237. https://doi.org/10.1016/j.jppr.2018.05.005
Ali, L., Islam, S., Gul, T., Khan, I. and Dennis, L.C.C. (2016) New Version of Optimal Homotopy Asymptotic Method for the Solution of Nonlinear Boundary Value Problems in Finite and Infinite Intervals. Alexandria Engineering Journal, 55, 2811-2819. https://doi.org/10.1016/j.aej.2016.07.013