Jalil Manafian

Bio: Jalil Manafian is an academic researcher from University of Tabriz. The author has contributed to research in topics: Soliton & Nonlinear system. The author has an hindex of 31, co-authored 126 publications receiving 3255 citations. Previous affiliations of Jalil Manafian include Islamic Azad University & Amirkabir University of Technology.

Solving nonlinear fractional partial differential equations using the homotopy analysis method

Abstract: In this article, the homotopy analysis method is applied to solve nonlinear fractional partial differential equations. On the basis of the homotopy analysis method, a scheme is developed to obtain the approximate solution of the fractional KdV, K(2,2), Burgers, BBM-Burgers, cubic Boussinesq, coupled KdV, and Boussinesq-like B(m,n) equations with initial conditions, which are introduced by replacing some integer-order time derivatives by fractional derivatives. The homotopy analysis method for partial differential equations of integer-order is directly extended to derive explicit and numerical solutions of the fractional partial differential equations. The solutions of the studied models are calculated in the form of convergent series with easily computable components. The results of applying this procedure to the studied cases show the high accuracy and efficiency of the new technique. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010

The Solution of the Variable Coefficients Fourth-Order Parabolic Partial Differential Equations by the Homotopy Perturbation Method

Abstract: In this work, the homotopy perturbation method proposed by Ji-Huan He [1] is applied to solve both linear and nonlinear boundary value problems for fourth-order partial differential equations The numerical results obtained with minimum amount of computation are compared with the exact solution to show the efficiency of the method The results show that the homotopy perturbation method is of high accuracy and efficient for solving the fourth-order parabolic partial differential equation with variable coefficients The results show also that the introduced method is a powerful tool for solving the fourth-order parabolic partial differential equations

New soliton solution to the longitudinal wave equation in a magneto-electro-elastic circular rod

Abstract: This paper examines the effectiveness of an integration scheme which called the extended trial equation method (ETEM) in exactly solving a well-known nonlinear equation of partial differential equations (PDEs). In this respect, the longitudinal wave equation (LWE) that arises in mathematical physics with dispersion caused by the transverse Poisson’s effect in a magneto-electro-elastic (MEE) circular rod, which a series of exact traveling wave solutions for the aforementioned equation is formally extracted. Explicit new exact solutions are derived in different form such as dark solitons, bright solitons, solitary wave, periodic solitary wave, rational function, and elliptic function solutions of the longitudinal wave equation. The movements of obtained solutions are shown graphically, which helps to understand the physical phenomena of this longitudinal wave equation. Many other such types of nonlinear equations arising in non-destructive evaluation of structures made of the advanced MEE material can also be solved by this method.

Novel solitary wave solutions for the (3+1)-dimensional extended Jimbo–Miwa equations

Abstract: This paper retrieves new periodic solitary wave solutions for the ( 3 + 1 ) -dimensional extended Jimbo–Miwa equations, based on the Hirota bilinear method, by utilizing Maple software. As a result, the Hirota bilinear method is successfully employed and acquired several classes of solitary wave solutions in terms of a new combination of exponential function, trigonometric function and hyperbolic functions. All solutions have been verified back into its corresponding equation by Maple. We depicted the physical explanation of the extracted solutions with the free choice of the different parameters by plotting some 3D and 2D illustrations. Finally, it has been illustrated that the executed method is robust and more efficient than other methods and the obtained solutions are trustworthy in the applied sciences.

Direct Methods in Soliton Theory (非線形現象の取扱いとその物理的課題に関する研究会報告)

Abstract: The main purpos e of this chapter is to present a direct and systematic way of finding exact solutions and Backlund transformations of a certain class of nonlinear evolution equations. The nonlinear evolution equations are transformed, by changing the dependent variable(s), into bilinear differential equations of the following special form $$ F\left( {\frac{\partial }{{\partial t}} - \frac{\partial }{{\partial {t^1}}},\frac{\partial }{{\partial x}} - \frac{\partial }{{\partial {x^1}}}} \right)f(t,x)f({t^1},{x^1}){|_{t = {t^1},x = {x^1}}} = 0 $$ , which we solve exactly using a kind of perturbational approach.