A Unified Petrov–Galerkin Spectral Method and Fast Solver for Distributed-Order Partial Differential Equations

Abstract

Fractional calculus and fractional-order modeling provide effective tools for modeling and simulation of anomalous diffusion with power-law scalings. In complex multi-fractal anomalous transport phenomena, distributed-order partial differential equations appear as tractable mathematical models, where the underlying derivative orders are distributed over a range of values, hence taking into account a wide range of multi-physics from ultraslow-to-standard-to-superdiffusion/wave dynamics. We develop a unified, fast, and stable Petrov–Galerkin spectral method for such models by employing Jacobi poly-fractonomials and Legendre polynomials as temporal and spatial basis/test functions, respectively. By defining the proper underlying distributed Sobolev spaces and their equivalent norms, we rigorously prove the well-posedness of the weak formulation, and thereby, we carry out the corresponding stability and error analysis. We finally provide several numerical simulations to study the performance and convergence of proposed scheme.

Appendix A Entries of Spatial Stiffness Matrix

Here, we provide the computation of entries of the spatial stiffness matrix by performing an affine mapping $\vartheta$ from the standard domain $\mu ^\mathrm {{stn}}_j \in [-1,1]$ to $\mu _j \in [\mu _j^{\max },\mu _j^{\min }]$.

Lemma A.1

The total spatial stiffness matrix $S^{{\rm Tot}}_{j}$ is symmetric and its entries can be exactly computed as

$$\begin{aligned} S^{\, \mathrm {{Tot}}}_{j}=c_{l_j} \times S_{l}^{\varrho _j} + c_{r_j} \times S_{r}^{\varrho _j}-\kappa _{l_j} \times S_{l}^{\rho _j}-\kappa _{r_j} \times S_{r}^{\rho _j}, \end{aligned}$$

(A1)

where $j=1,2,\cdots ,d$.

Proof

Regarding the definition of the stiffness matrix, we have

$$\begin{aligned} \{ S_{l}^{\varrho _j} \}_{r,n}&= \int _{-1}^{1} \int _{\mu _j^{\min }}^{\mu _j^{\max }} \varrho _j(\mu _j^{}) {}^{}_{-1}{\mathcal {D}}_{\xi _j}^{\mu _j^{}} \Big (\,\phi ^{}_n ( x_j ) \Big ) {}^{}_{\xi _j}{\mathcal {D}}_{1}^{\mu _j^{}} \Big (\varPhi ^{}_r ( x_j ) \Big ) \, {\text{d}}{x_j},\nonumber \\&= \beta _1 \int _{-1}^{1} \int _{-1}^{1} \varrho _j\big (\vartheta (\mu _j^{{\rm stn}})\big ) {}^{}_{-1}{\mathcal {D}}_{\xi _j}^{\mu _j^{{\rm stn}}} \Big (\, P_{n+1}({\xi _j}) - P_{n-1}({\xi _j}) \Big )\nonumber \\&\quad \times {}^{}_{{\xi }_j}{\mathcal {D}}_{1}^{\mu _j^{{\rm stn}}} \Big (\, P_{k+1}({\xi _j}) - P_{k-1}({\xi _j}) \Big ) \,{\text{d}}{\xi _j} ,\nonumber \\&= \beta _1 \Big (\,\widetilde{S}^{\, \varrho _j}_{r+1,n+1}-\widetilde{S}_{r+1,n-1}^{\, \varrho _j}-\widetilde{S}_{r-1,n+1}^{\, \varrho _j}+\widetilde{S}_{r-1,n-1}^{\, \varrho _j} \Big ), \quad \quad \end{aligned}$$

(A2)

where $\beta _1= {\widetilde{\sigma }}_r\,\sigma _n \, \Big (\frac{\mu _j^{\max }-\mu _j^{\min }}{2}\Big )$ and

$$\begin{aligned} \widetilde{S}_{r,n}^{\varrho _j}&= \,\int _{-1}^{1}\int _{-1}^{1} \varrho _j\big (\vartheta (\mu _j^{{\rm stn}})\big ) {}^{}_{-1}{\mathcal {D}}_{\xi _j}^{\mu _j^{{\rm stn}}} \Big (\, P_{n} (\xi _j) \Big ) {}^{}_{\xi _j}{\mathcal {D}}_{1}^{\mu _j^{{\rm stn}}} \Big (\, P_{r} (\xi _j) \Big ) \, {\text {d}}{\xi _j} \, {\text {d}}\mu _j^{{\rm stn}} \\&= \int _{-1}^{1} \varrho _j\big ( \vartheta (\mu _j^{{\rm stn}})\big )\, \frac{\Gamma (r+1)}{\Gamma (r-\mu _j^{{\rm stn}}+1)}\, \frac{\Gamma (n+1)}{\Gamma (n-\mu _j^{{\rm stn}}+1)} \,\\&\quad \times \int _{-1}^{1} {(1-\xi _j^2)}^{-\mu _j^{{\rm stn}}} \, P^{ -\mu _j^{{\rm stn}},\mu _j^{{\rm stn}}}_{r} \, P^{ \mu _j^{{\rm stn}},-\mu _j^{{\rm stn}}}_{n} {\text {d}}{\xi _j}\, {\text {d}}\mu _j^{{\rm stn}}. \end{aligned}$$

$\widetilde{S}_{r,n}^{\varrho _j}$ can be computed accurately using Gauss–Legendre (GL) quadrature rules as

$$\begin{aligned} \widetilde{S}_{r,n}^{\varrho _j^{stn}}&= \sum _{q=1}^{Q} \frac{\Gamma (r+1)}{\Gamma (r-\mu _j^{stn}|_{q}+1)} \frac{\Gamma (n+1)}{\Gamma (n-\mu _j^{stn}|_{q}+1)} \varrho _j|_{q}\, w_q \, \nonumber \\& \quad \times\int _{-1}^{1} {(1-\xi _j^2)}^{-\mu _j^{stn}|_{q}} \, P^{ -\mu _j^{stn}|_{q},\mu _j^{stn}|_{q}}_{r}(\xi _j) \, P^{ \mu _j^{stn}|_{q},-\mu _j^{stn}|_{q}}_{n}(\xi _j) {\text{d}}{\xi _j}, \quad \end{aligned}$$

(A3)

in which $\mathcal {Q} \geqslant {\mathcal {M}}_j +2$ represents the minimum number of GL quadrature points $\{\mu _j^{stn}|_{q}\}_{q=1}^{\mathcal {Q}}$ for exact quadrature, and $\{w_q\}_{q=1}^{Q}$ are the corresponding quadrature weights. Exploiting the property of the Jacobi polynomials where $P^{\alpha , \beta }_n(-\xi _j) = (-1)^n P^{ \beta ,\alpha }_n(\xi _j)$, we have $\widetilde{S}_{r,n}^{\, \varrho _j^{{\rm stn}}}=(-1)^{(r+n)}\,\widetilde{S}_{n,r}^{\, \varrho _j^{{\rm stn}}}$. Following [50], ${\widetilde{\sigma }}_r$ and $\sigma _n$ are chosen, such that $(-1)^{(n+r)}$ is canceled. Accordingly, $\{ S_{l}^{\varrho _j}\}_{n,r}=\{S_{l}^{\varrho _j}\}_{r,n}=\{ S_{r}^{\varrho _j}\}_{r,n}=\{S_{r}^{\varrho _j}\}_{r,n}$ due to the symmetry of $S_{l}^{\varrho _j}$ and $S_{r}^{\varrho _j}$. Similarly, we get $\{S_{l}^{\rho _j}\}_{n,r}=\{ S_{l}^{\rho _j}\}_{r,n}=\{ S_{r}^{\rho _j}\}_{n,r}=\{ S_{r}^{\rho _j}\}_{r,n}$. Eventually, we conclude that the stiffness matrix $S^{\, \varrho _j}_{l}$, $S^{\, \varrho _j}_{r}$, $S^{\, \rho _j}_{l}$, $S^{\, \rho _j}_{r}$, and thereby $\{S^\mathrm {{Tot}}_{j}\}_{n,r}$ as the sum of symmetric matrices is symmetric. $\square$