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The presented work aims to develop a novel technique for obtaining the solution of linear and nonlinear Partial Differential Equations (PDEs). This technique is based on applying a collocation method with the aid of Bernoulli polynomials and the use of such an algorithm to solve different types of PDEs. The method applies the regular finite difference scheme to the main problem and transforms it into an algebraic system. The obtained system is then solved, the unknown coefficient is acquired, and an approximate solution for the problems is achieved. Some test results of famous equations, including the telegraph, viscous Burger, and modified Burger equations, are tested to ensure that the provided algorithm is effective and robust. In addition, a comparison is provided with other recent techniques from the literature. The current technique proves to have high accuracy concerning the error measure and through graphical representation of the solution. [ABSTRACT FROM AUTHOR]

The current study proposes an eficient numerical method to evaluate the effects of the variable heat ux, viscous dissipation, and slip velocity on the viscous Casson Heat Transfer (CHT), considering the unsteady stretching sheet that took into account the effect of heat generation or absorption. In this respect, Finite Element Method (FEM) was employed to solve the resulting ODEs that described the problem. The effects of the factors governing the HT such as unsteadiness parameter, slip velocity parameter, Casson parameter, local Eckert number, heat generation parameter, and Prandtl number were studied. In addition, the local skin friction coeficient and local Nusselt number on the stretching sheet were also computed. Finally, the obtained solutions confirmed that the proposed procedure could be an easy and eficient tool for finding a solution to such uid models. [ABSTRACT FROM AUTHOR]

In this research work, we propose a new fractional numerical algorithm to obtain the numerical or exact solutions of generalizedCaputo type fractional-order differential equations of order 0 < θ ≤ 1. For finding the numerical or exact solutions by the proposed technique, we use the solutions of integer-order differential equations. Generalization of the proposed scheme to finite systems is also introduced. At the last, we derive the numerical simulations of some specific equations along with the solution of a computer virus model to illustrate the applications of our results. The proposed scheme is very effective and easy to apply on different kind of systems. [ABSTRACT FROM AUTHOR]

Byline: M. Inc, M. Ergut, D.J. Evans In this paper, we study a reliable numerical approximation of the linear and non-linear Klein-Gordon equations by using the Adomian's decomposition method. The solution is calculated in the form of a series with easily computable components. We prove the convergence of this method applied to the non-linear Klein-Gordon equation. The numerical results demonstrate the effectiveness of the method for these types of equations.

In this paper, a geometric approach is proposed to the integration of the system of Volterra integro-differential equations (IDEs). To do this, the equivalent system of ordinary differential equations of IDEs are obtained and converted into a Lie type augmented dynamical system. Then we construct the group preserving scheme (GPS) on the system of Volterra IDEs which is formulated by an exponential mapping to keep the group properties of . Some linear and nonlinear examples are solved as a description for the power and efficiency of GPS. Finally, in order to demonstrate the validity and efficiency of the present method, the convergency is numerically discussed and accuracy of results are compared with the reported results in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we study general solutions of the new fifth order nonlinear evolution and the Burgers KP equations with the aid of the two variables (G'/G, 1/G)-expansion method. The kink, bell-shaped solitary wave, periodic and singular periodic solutions are obtained. Finally, the numerical simulations add to these obtained solutions. [ABSTRACT FROM AUTHOR]