محمد ازادمرزابادي

The collocation method, a spectral technique widely applied in solving ordinary, partial, and fractional differential equations, was investigated within this study. The research introduced a novel fractional isolocal (computational spectral) method tailored to tackle a specific set of fractional penetration-transmission problems. This approach involves expressing the approximate solution as a truncated series utilizing fractional Lagrange functions. Notably, the derivative matrices of fractional orders δ and δ+1 proved highly effective and efficient. Numerical results showcased the method's effectiveness at interpolation points derived from the roots of fractional Jacobi polynomials and extremum roots of fractional Jacobi polynomials. However, increasing the degree of approximation N significantly augmented the coefficient matrix's dimensionality, particularly for the δ + 1 order derivative and equally spaced points, sometimes resulting in divergence for large N values. Hence, as a novel research endeavor, potential modifications might include incorporating suitable preconditioners to mitigate the matrix growth issue with increasing N. Additionally, the study examined the fractional Sturm-Liouville problem method for approximating fractional diffusion equations in two-dimensional space. Various knot types, including Jacobi polynomials, first-type Chebyshev, second-type Chebyshev, and Legendre polynomials, were explored. These knots were employed to construct fundamental Lagrange polynomials for spectral approximations, with expansion coefficients determined at collocation points. The efficacy of these four knot types in the Lagrange spectral collocation method was assessed through two examples of the fractional diffusion equation in two-dimensional space. Among the considered knot types, first-type Chebyshev knots exhibited the least 106 accuracy, while Jacobi knots demonstrated the highest accuracy. eva‎luation against exact solutions involved comparing absolute positional errors with maximum absolute error.

معادلات ديفرانسيل كسري , روش هم محلي كسري , روش شبه طيفي كسري , ماتريس مشتق , ماتريس انتگرال , روش طيفي لاگرانژ , معادله انتشار كسري دو بعدي

Fractional differential equations , Fractional collocation method , Fractional quasi-spectral method , derivative matrix , Integral matrix , Lagrange spectral method , Two-dimensional fractional diffusion equation

Mohammad Azad Marzabadi

Dr. Jalil Rashidinia