Gauss-Jacobi-type quadrature rules for fractional directional integrals

Fractional directional integrals are the extensions of the Riemann–Liouville fractional integrals from one- to multi-dimensional spaces and play an important role in extending the fractional differentiation to diverse applications. In numerical evaluation of these integrals, the weakly singular kernels often fail the conventional quadrature rules such as Newton–Cotes and Gauss–Legendre rules. It is noted that these kernels after simple transforms can be taken as the Jacobi weight functions which are related to the weight factors of Gauss–Jacobi and Gauss–Jacobi–Lobatto rules. These rules can evaluate the fractional integrals at high accuracy. Comparisons with the three typical adaptive quadrature rules are presented to illustrate the efficacy of the Gauss–Jacobi-type rules in handling weakly singular kernels of different strengths. Potential applications of the proposed rules in formulating and benchmarking new numerical schemes for generalized fractional diffusion problems are briefly discussed in the final remarking section.postprin

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