Study On Analytical Methods For Some Kinds Of Partial Differential Equations

In this dissertation,we introduce some attractive ideas of variational iteration method(VIM),homotopy perturbation method(HPM)and Laplace transformation to solve the nu-merous type of partial differential equations.Furthermore,these analytical methods find that VIM provides successive approximations through implementing the iterations in the correction functional,and HPM overcomes the nonlinear terms that frequently arise in expressing the non-linear phenomena such that the approximate solutions converge rapidly to the exact solutions.Some partial differential equations like Klein-Gordon equation,Sine-Gordon equation,mod-ified Boussinesq equation,cubic Boussinesq equation and telegraph equations are introduced to demonstrate the efficiency of these proposed methods.This dissertation comprises of the following topics.In Chapter 1,we outline the background and research status.In Chapter 2,we reveal an innovative technique built on variational iteration method(VIM)and Laplace transformation which is called Modified Laplace variational iteration method(M-LVIM).Initially,we apply VIM and then Laplace transformation to calculate the Lagrange multiplier in recurrence relation.Furthermore,He’s polynomials are calculated by applying homotopy perturbation method(HPM),which overcomes the difficulty caused by the nonlinear terms.The proposed method points out that the derived solution exists without any lineariza-tion,discretization or any hypothesis.Some numerical models are demonstrated to show the compactness and ability of this method.In Chapter 3,we study an analytical approach for nonlinear vibration of shallow water wave equations.This approach is known as He-Laplace method which is coupled with variational iteration method and Laplace transformation.These results of this section have shown that He-Laplace method is very valid for nonlinear evolution equations,in particular the nonlinear vibration of these equations.In Chapter 4,a new approach for solving telegraph equations is suggested.He’s variational iteration method(VIM)and Laplace transformation are used to find the exact solution or an approximate solution of partial differential equations.The most distinct aspect of this method is that it doesn’t need to calculate integration for next iterations in recurrence relations,and doesn’t use convolution theorem to calculate the Lagrange multipliers in Laplace transformation.Moreover,because Laplace transform has some limitations to nonlinear terms,He’s polynomials via homotopy perturbation method(HPM)is introduced to bring down the computational work in nonlinear terms.The results obtained by proposed method indicates that this approach is easy to implement and converges rapidly to the exact solution.Several problems are illustrated to demonstrate the accuracy and stability of this method.