The Modified Variational Iteration Method on the Newell-Whitehead-Segel Equation Using He’s polynomials

In this paper, we apply a modified version of the variational iteration method (MVIM) for solving Newell-Whitehead-Segel equation. The Newell-Whitehead-Segel equation models the interaction of the effect of the diffusion term with the nonlinear effect of the reaction term which appeared in the investigation of fluid dynamics. The proposed modification is made by introducing He’s polynomials in the correction functional of the variational iteration method. The use of Lagrange multiplier coupled with He’s polynomials are the clear advantages of this technique over the decomposition method Keywords: Variational Iteration Method, He’s Polynomials, Newell-Whitehead-Segel equation, nonlinear differential equation

Analytical Solution of Multi-Pantograph Delay Differential Equations Via Sumudu Decomposition Method

In this paper, we apply Sumudu Decomposition Method (SDM) to solve the multi-pantograph delay differential equations with constant coefficients. Three problems are resolved to show the effectiveness and consistency of the SDM. The obtained results by this method provide solutions in a series form and in few terms. This technique successfully determines the convergence of the solution. Keywords: Sumudu Decomposition Method (SDM), Multi-Pantograph Delay Differential Equation

Variational Iteration Method for Mixed Type Integro-Differential Equations

In this paper, we approximate Lagrange multipliers to solve integro differential equations of mixed type those are linear first and second order. It is observed that use of approximate Lagrange multipliers reduces the iteration and give faster results as compare to other techniques. Numerical examples support this idea. Keywords Approximate Lagrange Multiplier, Variational Iteration Method, Mixed Type Integro-Differential Equations

Application of Homotopy Perturbation Method to Nonlinear System of PDEs

In this study, Homotopy Perturbation Method (HPM) is used to obtain the analytically exact solution of linear and nonlinear systems of partial differential equations (PDEs). The efficiency and accuracy of HPM are demonstrated through several test examples. HPM yields solutions in convergent series forms with easily computable terms. Generally, the closed form of the exact solution is obtained without any noise terms. Keywords: Homotopy Perturbation Method, Linear and Nonlinear system of PDEs

On Higher Order Boundary Value Problems Via Power Series Approximation Method

In this work, a relatively new technique called Power Series Approximation Method (PSAM) is applied for the numerical approximate solution of non-linear higher order boundary value problems. Several examples are given to illustrate the efficiency and implementation of the method. The proposed method is efficient and effective on the experimentation as compared with the exact solutions. Numerical results are included to demonstrate the reliability and efficiency of the methods. Graphical representation of the obtained results reconfirms the potential of the suggested method. Keywords: Power series, nonlinear problems, boundary value problem, numerical simulatio

An Efficient Sumudu Decomposition Method to Solve System of Pantograph Equations

This paper is the witness of the coupling of decomposition method with the efficient Sumudu transform known as Sumudu decomposition method to build up the exact solutions of the linear and nonlinear system of Pantograph model equations. Three mathematical models are tested to elucidate effectiveness of the method. The obtained numerical results re-confirm the potential of the proposed method. In nonlinear cases this method uses He’s Polynomials for solving the non-linear terms. It is observed that suggested scheme is highly reliable and may be extended to other highly nonlinear delay differential models. Keywords: Decomposition method, Sumudu transform, System of multi-Pantograph delay differential equations, He’s polynomial

Numerical Treatment of Non-Linear Fuzzy Integral Equations by Homotopy Perturbation Method

The main purpose of this paper is to present an approximation method for solving fuzzy integral equation. The solution of various types of non-linear fuzzy integral equations like non-linear fuzzy Volterra integral equation, non-linear fuzzy Fredholm integral equation and non-linear able fuzzy integral equation is determined by an advanced iterative approach the homotopy perturbation method. The method is discussed in details and it is illustrated by solving some numerical examples. Keywords: Homotopy perturbation method, non-linear fuzzy Volterra integral equations, non-linear fuzzy Fredholm integral equation, non-linear Abel fuzzy integral equations

Decomposition Method for Kdv Boussinesq and Coupled Kdv Boussinesq Equations

This paper obtains the solitary wave solutions of two different forms of Boussinesq equations that model the study of shallow water waves in lakes and ocean beaches. The decomposition method using He’s polynomials is applied to solve the governing equations. The travelling wave hypothesis is also utilized to solve the generalized case of coupled Boussinesq equations, and, thus, an exact soliton solution is obtained. The results are also supported by numerical simulations. Keywords: Decomposition Method, He’s polynomials, cubic Boussinesq equation, Coupled Boussinesq equation

On Inhomogeneous Fractional Partial Differential Equations

In this paper, a coupling method of Laplace transform and Homotopy analysis method is applied for solving various inhomogeneous fractional partial differential equations.The proposed algorithm presents a procedure of construct the base function and gives a high order deformation equation in simple form. The purpose of using the Laplace transform is to overcome the deficiency that is mainly caused by unsatisfied conditions in the other analytical techniques. The scheme is tested for some examples to demonstrate the capability of LHAM for fractional partial differential equations. Keywords: Laplace homotopy analysis method; homotopy analysis method; fractional differential equations; modified Riemann-Liouville derivative; Wave equation; Burger’s equation; Klein-Gorden equation

On FIDEs System by Modified Sumudu Decomposition Method

In this paper, the technique of modified Sumudu decomposition method has been employed to solve a system of Fredholm integro-differential equations with initial conditions. Two examples are discussed to show applicability, reliability and the performance of the modified sumudu decomposition method. This study showed the capability, simplicity and effectiveness of the modified approach. Keywords: Modified Sumudu decomposition method; System of Fredholm integro-differential equations