Exact solutions to linear and nonlinear wave and diffusion equations - AL-Jawary, Majeed (International Journal of Applied Mathematical Research, 2015)
In the present paper, the new iterative method proposed by Daftardar-Gejji and Jafari (NIM or DJM) [V. Daftardar-Gejji, H. Jafari, An iterative method for solving non linear functional equations, J. Math. Anal. Appl. 316 (2006) 753-763] is used for solving linear and nonlinear wave and diffusion equations. In this iterative method the solution is obtained in the series form that converge to the exact solution with easily computed components. The results demonstrate that the method has many merits such as being derivative-free, overcome thedifficulty arising in calculating Adomian polynomials to handle the nonlinear terms in Adomian Decomposition Method (ADM). It does not require to calculate Lagrange multiplier as in Variational Iteration Method (VIM) and no needs to construct a homotopy and solve the corresponding algebraic equations as in Homotopy Perturbation Method (HPM). A comparison with some existing techniques such as ADM, HPM and VIM also presented, which shows that the DJM is effective and convenient to use and overcomes the difficulties arising in existing techniques. The results show that the present method is very effective and simple and provide the analytic solutions. The software used for the calculations in this study was MATHEMATICAÂ®8.0.
Keywords: Diffusion equations; Exact solution; New iterative method; Wave equations.