A fractional B-spline collocation method for the numerical solution of fractional predator-prey models

We present a collocation method based on fractional B-splines for the solution of fractional differential problems. The key-idea is to use the space generated by the fractional B-splines, i.e., piecewise polynomials of noninteger degree, as approximating space. Then, in the collocation step the fractional derivative of the approximating function is approximated accurately and efficiently by an exact differentiation rule that involves the generalized finite difference operator. To show the effectiveness of the method for the solution of nonlinear dynamical systems of fractional order, we solved the fractional Lotka-Volterra model and a fractional predator-pray model with variable coefficients. The numerical tests show that the method we proposed is accurate while keeping a low computational cost

A variable-order fractal derivative model for anomalous diffusion

This paper pays attention to develop a variable-order fractal derivative model for anomalous diffusion. Previous investigations have indicated that the medium structure, fractal dimension or porosity may change with time or space during solute transport processes, results in time or spatial dependent anomalous diffusion phenomena. Hereby, this study makes an attempt to introduce a variable-order fractal derivative diffusion model, in which the index of fractal derivative depends on temporal moment or spatial position, to characterize the previous mentioned anomalous diffusion (or transport) processes. Compared with other models, the main advantages in description and the physical explanation of new model are explored by numerical simulation. Further discussions on the dissimilitude such as computational efficiency, diffusion behavior, and heavy tail phenomena of the new model, and variable-order fractional derivative model are also offered

Diffusive approximation of a time-fractional Burger's equation in nonlinear acoustics

A fractional time derivative is introduced into the Burger's equation to model losses of nonlinear waves. This term amounts to a time convolution product, which greatly penalizes the numerical modeling. A diffusive representation of the fractional derivative is adopted here, replacing this nonlocal operator by a continuum of memory variables that satisfy local-in-time ordinary differential equations. Then a quadrature formula yields a system of local partial differential equations, well-suited to numerical integration. The determination of the quadrature coefficients is crucial to ensure both the well-posedness of the system and the computational efficiency of the diffusive approximation. For this purpose, optimization with constraint is shown to be a very efficient strategy. Strang splitting is used to solve successively the hyperbolic part by a shock-capturing scheme, and the diffusive part exactly. Numerical experiments are proposed to assess the efficiency of the numerical modeling, and to illustrate the effect of the fractional attenuation on the wave propagation.Comment: submitted to Siam SIA

Numerical Solution of Fractional Diffusion-Wave Equation with two Space Variables by Matrix Method

Mathematics Subject Classi¯cation 2010: 26A33, 65D25, 65M06, 65Z05.In the present paper we solve space-time fractional diffusion-wave equation with two space variables, using the matrix method. Here, in particular, we give solutions to classical diffusion and wave equations and fractional diffusion and wave equations with different combinations of time and space fractional derivatives. We also plot some graphs for these problems with the help of MATLAB routines

A semi–analytical study of diffusion type multi–term time fractional partial differential equation

This work suggested algorithm for the solution of multi–term time fractional partial differential equation by the application of homotopy analysis fractional Sumudu transform method based on iterative process. The method is cumulation of Sumudu transform and homotopy analysis method. Also, the multi-term time fractional partial differential equation represented in the form of system of fractional partial differential equations as per certain conditions of fractional derivatives. The Caputo fractional order derivatives are taken for the multi–term time fractional partial differential equations. Numerical examples are discussed for the support of theory and the approximate solution compared with exact solutions at the integer value of derivatives.Emerging Sources Citation Index (ESCI)MathScinetScopu

Semianalytic Solution of Space-Time Fractional Diffusion Equation

We study the space-time fractional diffusion equation with spatial Riesz-Feller fractional derivative and Caputo fractional time derivative. The continuation of the solution of this fractional equation to the solution of the corresponding integer order equation is proved. The series solution of this problem is obtained via the optimal homotopy analysis method (OHAM). Numerical simulations are presented to validate the method and to show the effect of changing the fractional derivative parameters on the solution behavior