# Homotopy analysis method for solving nonlinear diffusion equation with convection term

## Authors

• Bewar Mahmood Mathematics Department, Faculty of Science, University of Zakho
• Saad A. Manaa
• Fadhil H. Easif

2014-07-06

## Abstract

In this article the homotopy analysis method (HAM) is used to find a numerical solution for the nonlinear diffusion equation with convection term. The numerical results obtained by using this method compared with the exact solution, by solving numerical example shows that (HAM) is accurate and close to the exact solution.

Keywords: Homotopy Analysis Method, Nonlinear Diffusion Equation with Convection Term.

## References

S.J. Liao,” The proposed homotopy analysis method techniques for the solution of nonlinear problems”, Thesis Shanghai, Jiao Tong University Shanghai, (1992).

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

J.D. Cole,” Perturbation Method in Applied Mathematics”, Blaisdell, Waltham, MA, (1968).

J.H. He,” Homotopy perturbation technique”, Comput. Methods Appl, Mech, Engrg, 178, (1999), 257-262.

S.J. Liao,” On the homotopy analysis method for nonlinear problems”, App. Math. Comput. 147, (2004), 499-513.

M. Ayub,A. Rasheed,T. Hayat,” Exact flow of a third grade fluid past a porous plate using homotopy analysis”, Int. J. Eng. Sci, 41, (2003), 2091-2103.

T. Hayat,M. Khan,” Exact flow of a third grade fluid past a porous plate using homotopy analysis”, Int. J. Eng. Sci, 56, (2005), 1012-1029.

M. Hayat,M. Khan," Homotopy solutions for a generalized second grade fluid past a porous plate”, Nonlinear Dynam, 42, (2005), 395-405.

M. Sajad,T. Hayat,S. Asghar,” On the analytic solution of the steady flow of a fourth grade fluid. Phys.”, Lett. A, 355, (2006), 18-26.

B. Raftari,” Application of He's homotopy perturbation method and variational iteration method for nonlinear partial integro-differential equations”, World Applied Sciences Journal, 7, 4, (2009), 399-404.

B. Raftari,” Numerical solutions of the linear volterra integro-differential equations: homotopy perturbation method and finite Difference method”, World Applied Sciences Journal, 9, (2010), 7-12.

M. Matinfar,M. Saeidy,” The homotopy perturbation method for solving higher dimensional initial boundary value problems of variable coefficients”, World Applied Sciences Journal, 5, 1, (2009),72-80.

Y. Mahmoudi,E.M. Kazemian,” The Homotopy Analysis Method for Solving the Kuramoto-Tsuzuki Equation ”,World Applied Sciences Journal, 21, 12, (2013), 1776-1781.

Ch. Xiurong,Yu. Jiaju,” Homotopy Analysis Method for a Class of Holling Model with the Functional Reaction”, the Open Automation and Control Systems Journal, 5, (2013), 150-153.

R. Cherniha,M. Servo,” Lie and non-lie symmetries of nonlinear di_usion equations with convection term ”, Symmetry in Nonlinear Mathematical Physics, 2, (1997), 444-449.

W.F. Ames,” Nonlinear Partial Differential Equations in Engineering “, Academic Press, New York, (1972).

J.Crank,” The Mathematics of Diffusion “, Oxford University Press, Oxford, (1975).

R. Aris,” The Mathematical Theory of Diffusion and Reaction in Permeable Catalysts “, Oxford University Press, Oxford, (1975).

J.D. Murray,” Mathematical Biology “, Springer, Berlin, (1989).

M.A. El-Tawi,H.N.Hassan,” A new application of using homotopy analysis method for solving stochastic quadratic nonlinear diffusion equation “Int. J. of Appl Math and Mech, 9, 16, (2013), 35-55.

S.J. Liao,” Notes on the homotopy analysis method: some definitions and theories “, Communications in Nonlinear Science Numerical Simulation, 14, (2009), 983-997.

M. Sajid,T. Hayat,S. Asghar,” Non-similar solution for the ax symmetric flow of a third-grade fluid over radially stretching sheet “, Acta Mechanica,189, (2007), 193-205.

V.F. Zaitsev,A. Polyanim,” Handbook of Nonlinear Partial Di_erential Equations “, Chapman and Hall/CRC Press, Boca Raton, (2004).