On spectral polar fractional Laplacian

Numerical solution of multidimensional time‐space fractional differential equations of distributed order with Riesz derivative

“…Proposition 1. [34] Let {𝜑 m } is the complete set of the orthonormal eigenfunctions respect to the eigenvalues 𝜆 2 m of the Laplacian (−Δ) over a bounded domain Ω subject to the homogeneous boundary conditions. Therefore…”

“…Definition [34] Let $$$ \left\{{\varphi}_m\right\} $$$ is the complete set of the orthonormal eigenfunctions respect to the eigenvalues $$$ {\lambda}_m^2 $$$ of the Laplacian $$$ \left(-\Delta \right) $$$ over a bounded domain $$$ \Omega $$$, that is, $$$$ {\displaystyle \begin{array}{ll}\left(-\Delta \right){\varphi}_m=& \kern0.2em {\lambda}_m^2{\varphi}_m,\kern0.5em \mathrm{on}\kern0.5em \Omega, \\ {}\mathrm{B}\left(\varphi \right)=& \kern0.2em 0,\kern0.5em \mathrm{on}\kern0.5em \mathrm{\partial \Omega },\end{array}} $$$$ in which symbol B $$$ \left(\varphi \right) $$$ is one of the standard three homogeneous boundary conditions. Suppose that …”

“…Proposition [34] Let $$$ \left\{{\varphi}_m\right\} $$$ is the complete set of the orthonormal eigenfunctions respect to the eigenvalues $$$ {\lambda}_m^2 $$$ of the Laplacian $$$ \left(-\Delta \right) $$$ over a bounded domain $$$ \Omega $$$ subject to the homogeneous boundary conditions. Therefore $$$$ {\left(-\Delta \right)}^{\frac{\alpha }{2}}f=\left\{\begin{array}{cc}{\left(-\Delta \right)}^nf,& \mathrm{if}\kern0.5em \alpha =2n,n=0,1,2,\dots, \\ {}{\left(-\Delta \right)}^{\frac{\alpha }{2}-n}{\left(-\Delta \right)}^nf,& \mathrm{if}\kern0.5em n-1<\frac{\alpha }{2}<n,n=1,2,\dots, \\ {}\sum \limits_{m=1}^{\infty }{\lambda}_m^{\alpha}\left\langle f,...$$…”

An efficient numerical method for the time-fractional distributed order nonlinear Klein–Gordon equation with shifted fractional Gegenbauer multi-wavelets method

“…By applying the Laplace transform to both sides of equation (26), we obtain: By means of equations ( 13) and (14) for equation (28), one has…”

“…Therefore, it is absolutely necessary to pay attention to the use of numerical methods. Many authors and researchers have been proposed different numerical methods for the numerical solution of partial differential equations with fractional distributed order operators, such as standard quadrature method [15], implicit finite difference method [16], compact difference method [17], implicit numerical method [18], weighted and shifted Grünwald difference method [19], finite element method [20], Chebyshev collocation method [21], Petrov-Galerkin and spectral collocation methods [22], mid-point quadrature method [23], Legendre wavelets method [24], improved meshless method [25], finite volume method [26] and combination of alternating direction implicit difference, Laplace transform and Hankel transform [27], matrix transfer technique [28], fourth-order compact difference scheme [29], Chebyshev cardinal polynomials [30], and extrapolation method [31]. Recently, the use of wavelets has led to better numerical methods in fractional calculus.…”