Solving fractional diffusion and fractional diffusion-wave equations by Petrov-Galerkin finite element method

In the last few years, it has become highly evident that fractional calculus has been widely used in several areas of science. Because of this fact, their numerical solutions also have become urgently important. In this manuscript, numerical solutions of both the fractional diffusion and fractional diffusion-wave equations have been obtained by a Petrov-Galerkin finite element method using quadratic B-spline base functions as trial functions and linear B-spline base functions as the test functions. In those equations, fractional derivatives are used in terms of the Caputo sense. While the L1 discretizaton formula has been applied to fractional diffusion equation, the L2 discretizaton formula has been applied to the fractional diffusion-wave equation. Finally, the error norms L2 and L∞ have been calculated for testing the accuracy of the proposed scheme.Publisher's Versio

A Galerkin finite element method to solve fractional diffusion and fractional diffusion-wave equations

In the present study, numerical solutions of the fractional diffusion and fractional diffusion-wave equations where fractional derivatives are considered in the Caputo sense have been obtained by a Galerkin finite element method using quadratic B-spline base functions. For the fractional diffusion equation, the L1 discretizaton formula is applied, whereas the L2 discretizaton formula is applied for the fractional diffusion-wave equation. The error norms L 2 and L ∞ are computed to test the accuracy of the proposed method. It is shown that the present scheme is unconditionally stable by applying a stability analysis to the approximation obtained by the proposed scheme

Collocation solutions for the time fractional telegraph equation using cubic B-spline finite elements

In this study, we investigate numerical solutions of the fractional telegraph equation with the aid of cubic B-spline collocation method. The fractional derivatives have been considered in the Caputo forms. The L1and L2 formulae are used to discretize the Caputo fractional derivative with respect to time. Some examples have been given for determining the accuracy of the regarded method. Obtained numerical results are compared with exact solutions arising in the literature and the error norms L2 and L∞ have been computed. In addition, graphical representations of numerical results are given. The obtained results show that the considered method is effective and applicable for obtaining the numerical results of nonlinear fractional partial differential equations (FPDEs)

An application of finite element method for a moving boundary problem

The Stefan problems called as moving boundary problems are defined by the heat equation on the domain 0 < x < s(t). In these problems, the position of moving boundary is determined as part of the solution. As a result, they are non-linear problems and thus have limited analytical solutions. In this study, we are going to consider a Stefan problem described as solidification problem. After using variable space grid method and boundary immobilization method, collocation finite element method is applied to the model problem. The numerical solutions obtained for the position of moving boundary are compared with the exact ones and the other numerical solutions existing in the literature. The newly obtained numerical results are more accurate than the others for the time step Δt = 0.0005, it is also seen from the tables, the numerical solutions converge to exact solutions for the larger element numbers

Novel Exact Solutions of the Extended Shallow Water Wave and the Fokas Equations

In this study, a Sine-Gordon expansion method for obtaining novel exact solutions of extended shallow water wave equation and Fokas equation is presented. All of the equations which are under consideration consist of three or four variable. In this method, first of all, partial differential equations are reduced to ordinary differential equations by the help of variable change called as travelling wave transformation, then Sine Gordon expansion method allows us to obtain new exact solutions defined as in terms of hyperbolic trig functions of considered equations. The newly obtained results showed that the method is successful and applicable and can be extended to a wide class of nonlinear partial differential equations

Novel Exact Solutions of the Extended Shallow Water Wave and the Fokas Equations

In this study, a Sine-Gordon expansion method for obtaining novel exact solutions of extended shallow water wave equation and Fokas equation is presented. All of the equations which are under consideration consist of three or four variable. In this method, first of all, partial differential equations are reduced to ordinary differential equations by the help of variable change called as travelling wave transformation, then Sine Gordon expansion method allows us to obtain new exact solutions defined as in terms of hyperbolic trig functions of considered equations. The newly obtained results showed that the method is successful and applicable and can be extended to a wide class of nonlinear partial differential equations

A numerical treatment based on Haar wavelets for coupled KdV equation

In this paper, numerical solutions of one dimensional coupled KdV equation has been investigated by Haar Wavelet method. Time derivatives given in this equation are discretized by finite differences and nonlinear terms appearing in the equations are linearized by some linearization techniques and space derivatives are discretized by Haar wavelets. For examining the performance of the proposed method, single soliton solution and conserved quantities of some test problems are used. Also, error analysis of numerical scheme is investigated and numerical results are compared with some results already existing in the literature

A study on the improved tan(

In this study, the improved tan(ϕ (ξ) /2)-expansion method (ITEM), one of the improved expansion methods, has been applied to (3+1)- dimensional Jimbo Miwa and Sharma-Tasso-Olver equations using symbolic computation. With the aid of the method, many new and abundant analytical solutions have been obtained. The newly obtained results show that ITEM is a new and significant technique for solving nonlinear differential equations which plays an important role on fluids mechanics, engineering and many physics fields