Numerical Methods for the Fractional Laplacian: a Finite Difference-quadrature Approach

The fractional Laplacian $(-\Delta)^{\alpha/2}$ is a non-local operator which depends on the parameter $\alpha$ and recovers the usual Laplacian as $\alpha \to 2$. A numerical method for the fractional Laplacian is proposed, based on the singular integral representation for the operator. The method combines finite difference with numerical quadrature, to obtain a discrete convolution operator with positive weights. The accuracy of the method is shown to be $O(h^{3-\alpha})$. Convergence of the method is proven. The treatment of far field boundary conditions using an asymptotic approximation to the integral is used to obtain an accurate method. Numerical experiments on known exact solutions validate the predicted convergence rates. Computational examples include exponentially and algebraically decaying solution with varying regularity. The generalization to nonlinear equations involving the operator is discussed: the obstacle problem for the fractional Laplacian is computed.Comment: 29 pages, 9 figure

Discrete monotone method for space-fractional nonlinear reaction–diffusion equations

A discrete monotone iterative method is reported here to solve a space-fractional nonlinear diffusion–reaction equation. More precisely, we propose a Crank–Nicolson discretization of a reaction–diffusion system with fractional spatial derivative of the Riesz type. The finite-difference scheme is based on the use of fractional-order centered differences, and it is solved using a monotone iterative technique. The existence and uniqueness of solutions of the numerical model are analyzed using this approach, along with the technique of upper and lower solutions. This methodology is employed also to prove the main numerical properties of the technique, namely, the consistency, stability, and convergence. As an application, the particular case of the space-fractional Fisher’s equation is theoretically analyzed in full detail. In that case, the monotone iterative method guarantees the preservation of the positivity and the boundedness of the numerical approximations. Various numerical examples are provided to illustrate the validity of the numerical approximations. More precisely, we provide an extensive series of comparisons against other numerical methods available in the literature, we show detailed numerical analyses of convergence in time and in space against fractional and integer-order models, and we provide studies on the robustness and the numerical performance of the discrete monotone method. © 2019, The Author(s).Russian Foundation for Basic Research, RFBR: 19-01-00019Consejo Nacional de Ciencia y Tecnología, CONACYT: A1-S-45928The first author would like to acknowledge the financial support of the National Council for Science and Technology of Mexico (CONACYT). The second (and corresponding) author acknowledges financial support from CONACYT through grant A1-S-45928. ASH is financed by RFBR Grant 19-01-00019

New Advancements in Pure and Applied Mathematics via Fractals and Fractional Calculus

This reprint focuses on exploring new developments in both pure and applied mathematics as a result of fractional behaviour. It covers the range of ongoing activities in the context of fractional calculus by offering alternate viewpoints, workable solutions, new derivatives, and methods to solve real-world problems. It is impossible to deny that fractional behaviour exists in nature. Any phenomenon that has a pulse, rhythm, or pattern appears to be a fractal. The 17 papers that were published and are part of this volume provide credence to that claim. A variety of topics illustrate the use of fractional calculus in a range of disciplines and offer sufficient coverage to pique every reader's attention