Multistage Homotopy Analysis Method for Solving Nonlinear Integral Equations

In this paper, we present an efficient modification of the homotopy analysis method (HAM) that will facilitate the calculations. We then conduct a comparative study between the new modification and the homotopy analysis method. This modification of the homotopy analysis method is applied to nonlinear integral equations and mixed Volterra-Fredholm integral equations, which yields a series solution with accelerated convergence. Numerical illustrations are investigated to show the features of the technique. The modified method accelerates the rapid convergence of the series solution and reduces the size of work

Modified Variational Iteration Method for Second Order Initial Value Problems

In this paper, we introduce a modified variational iteration method for second order initial value problems by transforming the integral of iteration process. The main advantages of this modification are that it can overcome the restriction of the form of nonlinearity term in differential equations and improve the iterative speed of conventional variational iteration method. The method is applied to some nonlinear second order initial value problems and the numerical results reveal that the modified method is accurate and efficient for second order initial value problems

Analysis of Nonlinear Dynamic Behaviour of Nanobeam resting on Winkler and Pasternak Foundations

Dynamic modeling of nanobeam under stretching and two-parameter foundation effects result in nonlinear equations that are very difficult to find exact analytical solutions. In this study, variation iteration method is used to develop approximate analytical solutions to nonlinear vibration analysis of nanobeam under the effects of stretching and Winkler and Pasternak foundations. The governing equation of motion for the nanotube was derived based on Euler-Bernoulli beam theory. The developed approximate analytical solutions for the governing equation are validated the results of other methods of analysis, are also used to carry out effects of some model parameters on the dynamic behaviour of the nanobeam.  These analytical solutions can serve as a starting point for a better understanding of the relationship between the physical quantities in the problems as it provides clearer insights to understanding the problems in comparison with numerical methods

He’s polynomials method for analytical solutions of telegraph equation

In this paper, He’s polynomials solution method (HPSM) is fully utilized for solving telegraph equation. The proposed HPSM is technically presented and applied to homogeneous linear form the telegraph equation. The results are expressed in closed form with good agreement compared to those in literature thereby attesting to the efficiency and reliability of the method as proposed. The HPSM remarked to be less time consuming with high level of accuracy. As such, it can serve as alternative to other methods

Perturbation Iteration Transform Method for the Solution of Newell-Whitehead-Segel Model Equations

In this study, a computational method referred to as Perturbation Iteration Transform Method (PITM), which is a combination of the conventional Laplace Transform Method (LTM) and the Perturbation Iteration Algorithm (PIA) is applied for the solution of Newell-Whitehead- Segel Equations (NWSEs). Three unique examples are considered and the results obtained are compared with their exact solutions graphically. Also, the results agree with those obtained via other semi-analytical methods viz: New Iterative Method and Adomian Decomposition Method. This present method proves to be very efficient and reliable. Mathematica 10 is used for all the computations in this stud

A Handy Approximation Technique for Closedform and Approximate Solutions of Time- Fractional Heat and Heat-Like Equations with Variable Coefficients

In this paper, we propose a handy approximation technique (HAT) for obtaining both closed-form and approximate solutions of time-fractional heat and heat-like equations with variable coefficients. The method is relatively recent, proposed via the modification of the classical Differential Transformation Method (DTM). It devises a simple scheme for solving the illustrative examples, and some similar PDEs. Besides being handy, the results obtained converge faster to their exact forms. This shows that this modified DTM (MDTM) is very efficient and reliable. It involves less computational work, even without given up accuracy. Therefore, we strongly recommend it for solving both linear and nonlinear time-fractional partial differential equations (PDEs) with applications in other aspects of pure and applied sciences, management, and finance

Generalized Solutions of Wick-type Stochastic KdV-Burgers Equations Using Exp-function Method

Variable coefficients and Wick-type stochastic KdV-Burgers equations are researched. Expfunction method is proposed to present soliton and periodic wave solutions for variable coefficients KdVBurgers equation. Generalized white noise functional solutions for Wick-type stochastic KdV-Burgersequations are showed via Hermite transform and white noise analysis.Keywords: KdV-Burgers equation; Exp-function method; Wick product; Hermite transform; White noise

Rational and multi-wave solutions to nonlinear evolution equations by means of the Exp-function method

In this paper, we present a new application of the Exp-function method to carry out the integration of nonlinear evolution equations in terms of multi-wave and rational solutions. To elucidate the solution procedure, we analytically investigate the Sharma-Tasso-Olver equation and the fifth-order Korteweg de Vries equation. Unlike Hirota's method, our procedure does not require the bilinear formalism of the equations studied

Exp-function Method for Wick-type Stochastic Combined KdV-mKdV Equations

Exp-function method is proposed to present soliton and periodic wave solutions for variable coefficients combined KdV- mKdV equation. By means of Hermite transform and white noise analysis, we consider the variable coefficients and Wick-type stochastic combined KdV-mKdV equations. As a result, we can construct new and more general formal solutions. These solutions include exact stochastic soliton and periodic wave solutions.Keywords: combined KdV-mKdV equation, Exp-function method, Wick product, Hermite transform, White noise