Solving Nonlinear Integro-Differential Equations Using the Combined Homotopy Analysis Transform Method With Adomian Polynomials

Nahid Khanlari, Mahmoud Paripour


In this paper, we propose a reliable combination of the homotopy analysis method (HAM) and laplace transform-Adomian method to find the analytic approximate solution for nonlinear integro-differential equations. In this technique, the nonlinear term is replaced by its Adomian polynomials for the index \(k\), and hence the dependent variable components are replaced in the recurrence relation by their corresponding homotopy analysis transforms components of the same index. Thus, the nonlinear integro-differential equation can be easily solved with less computational work for any analytic nonlinearity due to the properties and available algorithms of the Adomian polynomials. The results show that the method is very simple and effective.


Nonlinear integro-differential equations; Homotopy analysis method; Laplace transform method; Adomian polynomials

Full Text:



S.H. Behiry, Nonlinear Integro-differential equations by differential transform method with Adomian polynomials, Mathematical Science Letters an International Journal 3 (2013), 209 – 221.

S.J. Liao and Y. Tan, A general approach to obtain series solutions of nonlinear differential equations, Studies in Applied Mathematics 119 (2007), 297 – 355.

S.J. Liao, On the homotopy analysis method for nonlinear problems, Applied Mathematics and Computation 147 (2004), 499 – 513.

S.J. Liao, Notes on the homotopy analysis method: Some Definitions and theorems, Communications in Nonlinear Science and Numerical Simulation 14(4) (2008), 983 – 997.

S.J. Liao, Beyond Perturbation: Introduction to the Homotopy Analysis Method, Chapman & Hall/CRC Press, Boca Raton (2003).

S.J. Liao, The Proposed Homotopy Analysis Technique for the Solution of Nonlinear Problems, PhD thesis, Shanghai Jiao Tong University (1992).

S.J. Liao, Homotopy analysis method: A new analytical technique for nonlinear problems, Commu. Nonlinear Sci. Numer. Simulation 2 (1997), 95 – 100.

A.M. Wazwaz, The combined Laplace transform-Adomian decomposition method for handling nonlinear Volterra integro-differential equations, Applied Mathematics Computation 216 (2010), 1304 – 1309.

Sh.S. Behzadi, Solving nonlinear Volterra-Fredholm integro-differential equations using the modified Adomian decomposition method, International Journal of Computer Mathematics 9 (2009), 321 – 331.

Y. Cherruault and G. Adomian, Decomposition methods: A new proof of convergence, Mathl. Comput. Modeling 18(12) (1993), 103 – 106.

N. Bildik and A. Konuralp, The use of variational iteration method, differential transform method and Adomian decomposition method for solving different types of nonlinear partial differential equations, International Journal of Nonlinear Sciences and Numerical Simulation 7(1) (2006), 65 – 70.

F. Awawdeh and A. Adawi, A numerical method for solving nonlinear integral equations, International Mathematical Forum 4(17) (2009), 805 – 817.

A.A. Kilbas and M. Saigo, On solution of nonlinear Abel Volterra integral equation, Journal of Mathematical Analysis and Application 229 (1999), 41 – 60.

S. Abbasbandy, Numerical solution of integral equation: Homotopy perturbation method and Adomian’s decomposition method, Applied Mathematics and Computation 173 (2006), 493 – 500.

D. Bugajewski, On BV-Solutions of some nonlinear integral equations, Integral Equations and Operator Theory 46 (2003), 387 – 398.

M. El-Shahed, Application of He’s homotopy perturbation method to Volterra’s integro-differential equation, International Journal of Nonlinear Sciences and Numerical Simulation 6 (2005), 163 – 168.

H. Brunner, M.R. Crisci, E. Russo and A. Recchio, A family of methods for Abel integrals for equations of the second kind, Journal of Computational and Applied Mathematics 34 (1991), 211 – 219.

A. Golbabai and B. Keramati, Modified homotopy perturbation method for solving Fredholm integral equations, Chaos Solitons & Fractals 37(5) (2008), 1528 – 1537.

S. Yalcina, Taylor polynomial solution of nonlinear Volterra-Fredholm integral equations, Applied Mathematics and Computation 127 (2002), 195 – 206.

A. Borhanifar and R. Abazari, Differential transform method for a class of nonlinear integrodifferential equations with derivative type kernel, Canadian Journal on Computing in Mathematics, Natural Sciences, Engineering and Medicine 3 (2012), 1 – 6.

Y. Khan, An efficient modification of the Laplace decomposition method for nonlinear equations, International Journal of Nonlinear Sciences and Numerical Simulation 10 (2009), 1373 – 1376.

A. Golbabai and B. Keramati, Modified homotopy perturbation method for solving Fredholm integral equations, Chaos Solitons & Fractals 37(5) (2008), 1528 – 1537.

Y. Khani and N. Faraz, A new approach to differential-difference equations, Journal of Advanced Research in Differential Equations 2 (2010), 1 – 12.

S. Islam, Y. Khani and N. Faraz, Numerical solution of logistic differential equations by using the Laplace decomposition method, World Applied Sciences Journal 8(39) (2010), 1100 – 1105.

K. Maleknejad, B. Basirat and E. Hashemizadeh, Hybrid Legendre polynomials and Block-Pulse functions approach for nonlinear Volterra-Fredholm integro-differential equations, Computers and Mathematics with Applications 61(9) (2011), 2821 – 2828.

M. Hashmi, N. Khan and S. Iqbal, Optimal homotopy asymptotic method for solving nonlinear Fredholm integral equations of second kind, Applied Mathematics and Computation 218 (2012), 10982 – 10989.

A.M. Wazwaz, A comparison study between the modified decomposition method and traditional method, Applied Mathematics and Computation 181 (2006), 1703 – 1712.

B. Ghanbari, The convergence study of the homotopy analysis method for solving nonlinear Volterra-Fredholm Integro-differential equations, The Scientific World Journal 2014 (2014), 1 – 7.

M. Mohamed, K. Gepreel, M. Alharthi and R. Alotabi, Homotopy analysis transform method for Integro-differential equations, General Mathematics Notes 32(1) (2016), 32 – 48.

S. Kumar, J. Singh, D. Kumar and S. Kapoor, New homotopy analysis transform algorithm to solve Volterra integral equation, Ain Shams Engineering 5 (2014), 243 – 246.

S. Noeiaghdam, Ei. Zarei and H. Barzegar, Homotopy analysis transform method for solving Abel’s integral equations of the first kind, Ain Shams Engineering 7 (2016), 483 – 495.

S. Abbasbandy, T. Hayat, A. Alsaedi and M.M. Rashidi, Numerical and analytical solutions for Falkner-Skan flow of MHD Oldroyd-B fluid, International Journal of Numerical Methods Heat Fluid Flow 24(2) (2014), 390 – 401.

K. Hemida and M.S. Mohamed, Numerical simulation of the generalized Huxley equation by homotopy analysis method, Journal of Applied Functional Analysis 5(4) (2010), 344 – 350.

S. Abbasbandy, R. Naz, T. Hayat and A. Alsaedi, Numerical and analytical solutions for Falkner-Skan flow of MHD Maxwell fluid, Applied Mathematics and Computation 242 (2014), 569 – 575.

K.A. Gepreel and M.S. Mohamed, Analytical approximate solution for nonlinear space-time fractional Klein Gordon equation, Chinese Physics B 22(1) (2013), 010201-6.

A.M. Wazwaz, A new algorithm for calculating Adomian polynomials for nonlinear operators, Applied Mathematics and Computation 111 (2000), 53 – 69.



  • There are currently no refbacks.

eISSN 0975-8607; pISSN 0976-5905