New development of homotopy analysis method for non-linear integro-differential equations with initial value problems

: pp. 842–859
Received: August 11, 2022
Accepted: September 04, 2022

Mathematical Modeling and Computing, Vol. 9, No. 4, pp. 842–859 (2022)

Faculty of Ocean Engineering Technology and Informatics, University Malaysia Terengganu; DSc Doctorate, Faculty of Applied Mathematics and Intellectual Technologies, National University of Uzbekistan

Homotopy analysis method (HAM) was proposed by Liao in 1992 in his PhD thesis for non-linear problems and was applied in many different problems of mathematical physics and engineering.  In this note, a new development of homotopy analysis method (ND-HAM) is demonstrated for non-linear integro-differential equation (NIDEs) with initial conditions.  Practical investigations revealed that ND-HAM leads an easy way how to find initial guess and it approaches the exact solution faster than the standard HAM, modified HAM (MHAM), new modified of HAM (mHAM) and more general method of HAM (q-HAM).  Uniqueness solution of the problem and convergence of ND-HAM are proved in the Banach space.  Finally, two examples are illustrated to show the accuracy and validity of the proposed method.  Five methods are compared in each example.

  1. Liao S. J.  The proposed homotopy analysis technique for the solution of nonlinear problems.  PhD thesis, Shanghai Jiao Tong University (1992).
  2. Liao S.-J.  An explicit, totally analytic approximation of Blasius' viscous flow problems.  International Journal of Non-Linear Mechanics.  34 (4), 759–778 (1999).
  3. Liao S. J.  Beyond Perturbation: Introduction to the Homotopy Analysis Method.  Boca Raton, Chapman and Hall/CRC Press (2003).
  4. Liao S. J.  Homotopy Analysis Method In Nonlinear Differential Equations.  Higher education press. Springer Berlin, Heidelberg (2011).
  5. Bataineh A. S., Noorani M. S. M., Hashim I.  Modified homotopy analysis method for solving systems of second-order BVPs.  Communications in Nonlinear Science and Numerical Simulation.  14 (2), 430–442 (2009).
  6. Ayati Z., Biazar J., Gharedaghi B.  The Application of Modified Homotopy Analysis Method For Solving Linear and Non-Linear Inhomogeneous Klein–Gordon Equations.  Acta Universitatis Apulensis.  39, 31–40 (2014).
  7. Yin X.-B., Kumar S., Kumar D.  A modified homotopy analysis method for solution of fractional wave equations.  Advances in Mechanical Engineering.  7 (12), 1–8 (2015).
  8. Ziane D., Cherif H.  Modified Homotopy Analysis Method For Non-linear Fractional Partial Differential Equations.  International Journal of Analysis and Applications.  14 (1), 77–87 (2017).
  9. Eshkuvatov Z. K., Laadjal Z., Ismail S.  Numerical treatment of nonlinear mixed Volterra–Fredholm integro-differential equations of fractional order.  AIP Conference Proceedings.  2365, 020006 (2021).
  10. Fan T., You X.  Optimal homotopy analysis method for nonlinear differential equations in the boundary layer.  Numerical Algorithms.  62, 337–354 (2013).
  11. Mabood F., Md Ismail A. I., Hashim I.  Application of Optimal Homotopy Asymptotic Method for the Approximate Solution of Riccati Equation.  Sains Malaysiana.  42 (6), 863–867 (2013).
  12. Saberi–Nadjafi J., Saberi–Jafari H.  Comparison of Liao's optimal HAM and Niu's one-step optimal HAM for solving integro-differential equation.  Journal of Applied Mathematics and Bioinformatics.  1 (2), 85–98 (2011).
  13. El-Tawil M. A., Huseen S. N.  The q-Homotopy Analysis Method (q-HAM).  Int. J. of Appl. Math. and Mech.  8 (15), 51–75 (2012).
  14. El-Tawil M. A., Huseen S. N.  On Convergence of q-Homotopy Analysis Method.  International Journal of Contemporary Mathematical Sciences.  8 (10), 481–497 (2013).
  15. Huseen Sh. N., Ayay N. M.  A New Technique of The q-Homotopy Analysis Method for Solving Non-Linear Initial Value Problems.  Journal of Progressive Research in Mathematics (JPRM).  14 (1), 2292–2307 (2018).
  16. Huseen Sh. N.  A Comparative Study of q-Homotopy Analysis Method and Liao's Optimal Homotopy Analysis Method.  Advances in Computer and Communication.  1 (1), 36–45 (2020).
  17. Huseen Sh. N., Grace S. R., El-Tawil M. A.  The Optimal q-Homotopy Analysis Method (Oq-HAM).  International Journal of Computers and Technology.  11 (8), 2859–2866 (2013).
  18. Batiha B., Noorani M. S. M., Hashi I.  Numerical solutions of the nonlinear integro-differential equations.  Int. J. Open Problems Compt. Math.  1 (1), 34–42 (2008).
  19. Hosseini M. M.  Taylor-successive approximation method for solving nonlinear integral equations.  Journal of Advanced Research in Scientific Computing.  1 (2), 1–13 (2009).
  20. Al-Khaled K., Allan F.  Decomposition method for solving nonlinear integro-differential Equations.  Journal of Applied Mathematics and Computing.  19 (1–2), 415–425 (2005).
  21. El-Sayed S. M., Abdel–Aziz M. R.  A comparison of Adomian's decomposition method and wavelet–Galerkin method for solving integro-differential equations.  Applied Mathematics and Computation. 136 (1), 151–159 (2003).
  22. Xie L.-J.  A New Modification of Adomian Decomposition Method for Volterra Integral Equations of the Second Kind.  Journal of Applied Mathematics.  2013, 795015 (2013).
  23. Behiry S. H., Mohamed S. I.  Solving high-order nonlinear Volterra–Fredholm integro-differential equations by differential transform method.  Natural Science.  4 (8), 581–587 (2012).
  24. Avudainayagam A., Vani C.  Wavelet–Galerkin method for integro-differential equations.  Applied Numerical Mathematics.  32 (3), 247–254 (2000).
  25. Ebadi G., Rahimi–Ardabili M. Y., Shahmorad S.  Numerical solution of the nonlinear Volterra integro-differential equations by the Tau method.  Applied Mathematics and Computation. 188 (2), 1580–1586 (2007).
  26. Eshkuvatov Z. K., Khadijah M. H., Taib B. M.  Modified HPM for high-order linear fractional integro-differential equations of Fredholm–Volterra type.  Journal of Physics: Conference Series.  1132, 012019 (2018).
  27. Al-Hayan W.  Solving $n$th-Order Integro-Differential Equations Using the Combined Laplace Transform–Adomian Decomposition Method.  Applied Mathematics.  4 (6), 882–886 (2013).
  28. Aloev R. D., Eshkuvatov Z. K., Davlatov S. O., Nik Long N. M. A.  Sufficient condition of stability of finite element method for symmetric T-hyperbolic systems with constant coefficients.  Computers & Mathematics with Applications.  68 (10), 1194–1204 (2014).
  29. Kanwal R. P.  Linear Integral Equations. Theory and Technique.  Academic Press, Inc (London), LTD. (1971).
  30. Cherruault Y.  Convergence of Adomian's method.  Kybernetes.  18 (2), 31–38 (1989).