Rational spectral methods for PDEs involving fractional Laplacian in unbounded domains

Many PDEs involving fractional Laplacian are naturally set in unbounded domains with underlying solutions decay very slowly, subject to certain power laws. Their numerical solutions are under-explored. This paper aims at developing accurate spectral methods using rational basis (or modified mapped Gegenbauer functions) for such models in unbounded domains. The main building block of the spectral algorithms is the explicit representations for the Fourier transform and fractional Laplacian of the rational basis, derived from some useful integral identites related to modified Bessel functions. With these at our disposal, we can construct rational spectral-Galerkin and direct collocation schemes by pre-computing the associated fractional differentiation matrices. We obtain optimal error estimates of rational spectral approximation in the fractional Sobolev spaces, and analyze the optimal convergence of the proposed Galerkin scheme. We also provide ample numerical results to show that the rational method outperforms the Hermite function approach

The Numerical Technique Based on Shifted Jacobi-Gauss-Lobatto Polynomials for Solving Two Dimensional Multi-Space Fractional Bioheat Equations

    يتناول هذا البحث، الخوارزمية التقريبية لحل معادلة الحرارة الحيوية ثنائية البعد متعددة الرتبة الكسورية المكانية (M-SFBHE). سوف نوسع تطبيق طريقة التجميع لتقديم التقنية العددية لحل M-SFBHE مؤسسة على متعددات حدود جاكوبي- كاوس- لوباتو (SJ-GL-Ps) بالصيغة المصفوفية.  استخدمنا صيغة Caputo لتقريب المشتقة الكسرية و لإثبات فائدتها ودقتها, طبقنا الخوارزمية المقترحة على مثالين. النتائج العددية أظهرت أن النهج المستخدم فعال للغاية ويعطي دقة عالية وتقارب جيد.This article deals with the approximate algorithm for two dimensional multi-space fractional bioheat equations (M-SFBHE). The application of the collection method will be expanding for presenting a numerical technique for solving M-SFBHE based on “shifted Jacobi-Gauss-Labatto polynomials” (SJ-GL-Ps) in the matrix form. The Caputo formula has been utilized to approximate the fractional derivative and to demonstrate its usefulness and accuracy, the proposed methodology was applied in two examples. The numerical results revealed that the used approach is very effective and gives high accuracy and good convergence