Homotopy Analysis Method for Equations of the Type Δ2=b(x,y) and Δ2u=b(x,y,u)

In this paper, the homoto pyanalysis method (HAM) is presented to solve some of engineering problems. The homotopy analysis method is applied in obtaining exact solutions for equations of the type Δ2=b(x,y) and  Δ2u=b(x,y,u) on an elliptical domain. Exact solutions are presented for several examples involving to demon strate the applic ability and efficiency of HAM.

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

Yildirim, S. (2015) Homotopy Analysis Method for Equations of the Type Δ2=b(x,y) and Δ2u=b(x,y,u). Journal of Applied Mathematics and Physics, 3, 391-398. doi: 10.4236/jamp.2015.34049.

 [1] Liao, S.J. (2005) Comparison between the Homotopy Analysis Method and Homotopy Perturbation Method. Applied Mathematics and Computation, 169, 1186-1194. [2] Liao, S.J. (2004) On the Homotopy Analysis Method for Nonlinear Problems. Applied Mathematics and Computation, 147, 499-513. [3] Liao, S.J. (2010) An Optimal Homotopy-Analysis Approach for Strongly Nonlinear Differential Equations. Communications in Nonlinear Science and Numerical Simulation, 15, 2003-2016. [4] Liao, S.J. (2009) Notes on the Homotopy Analysis Method: Some Definitions and Theorems. Communications in Non-linear Science and Numerical Simulation, 14, 983-997. http://dx.doi.org/10.1016/j.cnsns.2008.04.013 [5] Liao, S.J. (2005) An Analytic Approach to Solve Multiple Solutions of a Strongly Nonlinear Problem. Applied Mathe- matics and Computation, 169, 854-865. [6] Liao, S.J. (2003) Beyond Perturbation: Introduction to the Homotopy Analysis Method. Chapman and Hall/CRC Press, Boca Raton. [7] Liao, S.J. (2012) Homotopy Analysis Method in Nonlinear Differential Equations. Springer, New York. [8] Liao, S.J. (1992) Proposed Homotopy Analysis Techniques for the Solution of Nonlinear Problems. Ph.D. Thesis, Shanghai Jiao Tong University, Shanghai. [9] Das, S., Vishal, K., Gupt, P.K. and Ray, S.S. (2011) Homotopy Analysis Method for Solving Fractional Diffusion Equation. International Journal of Applied Mathematics and Mechanics, 7, 28-37. [10] Song, L. and Zhang, H. (2007) Application of Homotopy Analysis Method to Fractional Kdv-Burgers-Kuramoto Equation. Physics Letters A, 367, 88-94. [11] Abdulaziz, O., Hashim, I. and Saif, A. Solutions of Time-Fractional PDEs by Homotopy Analysis Method. Differential Equations and Nonlinear Mechanics, 2008, Article ID: 686512. [12] Ganjiani, M. (2010) Solution of Nonlinear Fractional Differential Equations Using Homotopy Analysis Method. Applied Mathematical Modelling, 34, 1634-1641. http://dx.doi.org/10.1016/j.apm.2009.09.011 [13] Abidi, F. and Omrani, K. (2010) The Homotopy Analysis Method for Solving the Fornberg-Whitham Equation and Comparison with Adomian’s Decomposition Method. Computers & Mathematics with Applications, 59, 2743-2750. [14] Molabahrami, A. and Khani, F. (2009) The Homotopy Analysis Method to Solve the Burgers-Huxley Equation. Nonlinear Analysis: Real World Applications, 10, 589-600. http://dx.doi.org/10.1016/j.nonrwa.2007.10.014 [15] Ghanbari, B. (2104) An Analytical Study for (2+1)-Dimensional Schr?dinger Equation. The Scientific World Journal, 2014, Article ID: 438345. [16] Inc, M. (2007) On Exact Solution of Laplace Equation with Dirichlet and Neumann Boundary Conditions by Homotopy Analysis Method. Physics Letters A, 365, 412-415. [17] Patridge, P.W., Brebbia, C.A. and Wrobel, L.C. (1992) The Dual Reciprocity Boundary Element Method. CMP, Elsevier Applied Scince, New York.