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Legendre spectral collocation method for distributed and Riesz fractional convection–diffusion and Schrödinger-type equation


The numerical analysis of the temporal distributed and spatial Riesz fractional problem (TDSRFP) is presented in this work. To address the two independent variables, the suggested technique employs a completely spectral Legendre collocation approach. For the current model, our technique is proven to be more accurate, efficient, and practical. The results confirmed that the spectral scheme is exponentially convergent.


As it is more appropriate for modeling many real-world situations than the classical derivative [13], the notion of fractional derivatives [49] has become one of the most important topics in applied mathematics. The major reason is that it is widely utilized in fields like chemistry [10], biology [11], physics [12], engineering, and finance. In the mathematical literature, fractional derivatives are defined in a variety of ways, including the Riemann–Liouville and Caputo fractional senses.

The anomalous diffusion [13, 14] and relaxation processes have been thoroughly described using distributed fractional problems, which are an extension of single-term and multi-term problems. Chechkin et al. [15] were the first to list them, and they were thereafter widely used to characterize the phenomena of diffraction and relaxation, i.e., the heating and cooling of objects in a thermal or magnetic field. Mashayekhi and Razzaghi [16] solved temporal distributed fractional differential equations using the Riesz fractional derivative, which is one of the key definitions of the spatial fractional derivative in quantum mechanics [17]. To solve quantum mechanical problems, block-pulse functions and Bernoulli polynomials, as well as Fourier transformations, have been employed. Chen et al. [18] solved the one-dimensional temporal distributed fractional reaction-diffusion equation with unbounded domain, while the replicating kernel technique [19] was used to solve the temporal distributed fractional diffusion equation with variable coefficients. To solve one- and multi-dimensional distributed-order diffusion equations, spectral Galerkin [20] and collocation [2125] have been used. Liu et al. [26] used a finite volume technique, Fan and Liu [27] used a finite element method, and Jia and Wang [28] utilized a fast finite difference method for distributed fractional differential problems in space variables. Another case is the use of anomalous diffusion, the issue in which the Riesz derivative indicates nonlocality and is used to represent the dependency of diffusion concentration on route. Fractional differential equations with the Riesz derivative are required to describe this type of phenomena. Riesz derivatives are two-sided fractional operators that have both left and right derivatives. This capability is especially useful for fractional modeling on finite domains. There is limited literature on fractional differential equations with the Riesz derivative. Chen et al. [29] investigated the existence of Riesz-fractional differential equations in the Caputo sense. In [30], the time-space Riesz fractional advection-diffusion equations were solved using the finite difference method. Riesz–Caputo variational optimal problems have been discussed by Agrawal [31]. Noether’s theorem for Riesz–Caputo fractional variational problems has been established by Frederico and Torres [32]. Almeida [33] investigated optimality criteria for Riesz–Caputo variational problems. Various numerical techniques [3438] have been developed to handle Riesz Riemann–Liouville derivative fractional issues. Wang et al. [39] also used a second-order finite difference approach to solve the linear Riesz-distributed advection-dispersion issue. Fan et al. [27] mentioned the finite element technique for two-dimension linear Riesz-distributed diffusion problem on an irregular convex domain. To solve Riesz-distributed problems, finite volume techniques were utilized in [26, 40].

Spectral techniques [4144] are effective tools for solving many sorts of differential [45, 46] and integral equations encountered in science and engineering [4750]. Because explicit analytical solutions to space and/or time-fractional differential equations are in most circumstances impossible to acquire, creating efficient numerical methods is a high priority. There has been significant growth in fractional differential and integral equations due to their high-order accuracy. Compared to the effort put into analyzing finite difference schemes in the literature for solving the fractional-order differential equations, only a little work has been put into developing and analyzing global spectral schemes [5153]. The primary goal of this work is to develop the Gauss–Lobatto Legendre collection technique (GLLCT) and the Gauss–Radu shifted Legendre collection technique (GRSLCT) for handling spatial and temporal variables.

The following is a description of the paper’s structure. The numerical approach for solving the TDSRFP is presented in Sect. 2. Section 3 solves and analyzes three cases to demonstrate the method’s efficiency and correctness. The major conclusions are outlined in Sect. 4.

Spectral collocation treatment

Distributed and Riesz fractional convection–diffusion equation

To tackle the TDSRFP of the form, GLLCT and GRLCT are suggested.

$$ \int _{0}^{1}\kappa (\mu ){}_{0}^{c} \mathcal{D}^{ \mu }_{t}\mathcal{U}(x,t)\,d\mu +\varepsilon (- \Delta )^{ \frac{\delta }{2}}\mathcal{U}(x,t)=\Lambda \bigl(x,t,\mathcal{U}(x,t) \bigr),\quad (x,t)\in \Lambda ^{\bullet } \times \Lambda ^{\diamond }, $$

where \(\Lambda ^{\bullet }\equiv [-1,1]\) and \(\Lambda ^{\diamond } \equiv [0,t_{\mathrm{end}}]\). Related to

$$ \begin{aligned} &\mathcal{U}(x,0)=\Theta _{1}(x),\quad x\in \Lambda ^{ \bullet }, \\ &\mathcal{U}(0,t)=\Theta _{2}(t), \qquad \mathcal{U}( x_{\mathrm{end}},t)= \Theta _{3}(t),\quad t\in \Lambda ^{\diamond } . \end{aligned} $$

To convert the TDSRFP into a nonlinear algebraic system, the GLLCT and GRSLCT are used. The truncated solution is written as

$$ \mathcal{U}_{\mathcal{N},\mathcal{M}}(x,t)=\sum _{ \substack{r_{1}=0,\dots ,\mathcal{N}\\ r_{2}=0,\dots ,\mathcal{M} }} \varsigma _{r_{1},r_{2}} \mathcal{G}_{r_{1}}(x)\mathcal{G}^{t_{\mathrm{end}}}_{r_{2}}(t), $$

where \(\mathcal{G}_{r_{1}}(x)\) and \(\mathcal{G}^{t_{\mathrm{end}}}_{r_{2}}(t)\) are the Legendre and shifted Legendre polynomials, see [54, 55] for more details. \((-\Delta )^{\frac{\delta }{2}}\mathcal{U}(x,t)\) is calculated as follows:

$$ \begin{aligned} (-\Delta )^{\frac{\delta }{2}} \mathcal{U}(x,t)&=- \frac{1}{2 \cos (\frac{\pi \delta }{2} )} \bigl({}_{-1}^{ c} \mathcal{D}^{\delta }_{x}\mathcal{U}(x,t)+{} _{x}^{c} \mathcal{D}^{\delta }_{1} \mathcal{U}(x,t) \bigr) \\ &=-\frac{1}{2 \cos (\frac{\pi \delta }{2} )} \biggl(\sum_{ \substack{r_{1}=0,\dots ,\mathcal{N}\\ r_{2}=0,\dots ,\mathcal{M} }} \varsigma _{r_{1},r_{2}} \bigl({}_{-1}^{ c} \mathcal{D}^{\delta }_{x} \mathcal{G}_{r_{1}}(x)+{} _{x}^{c}\mathcal{D}^{\delta }_{1} \mathcal{G}_{r_{1}}(x) \bigr)\mathcal{G}^{t_{\mathrm{end}}}_{r_{2}}(t) \biggr) \\ &=\sum_{ \substack{r_{1}=0,\dots ,\mathcal{N}\\ r_{2}=0,\dots ,\mathcal{M} }} \varsigma _{r_{1},r_{2}} \mathcal{G}^{\Delta }_{r_{1}}(x)\mathcal{G}^{t_{\mathrm{end}}}_{r_{2}}(t), \end{aligned} $$

where \(\mathcal{G}^{\Delta }_{r_{1}}(x)=- \frac{1}{2 \cos (\frac{\pi \delta }{2} )} ({}_{-1}^{ c}\mathcal{D}^{\delta }_{x}\mathcal{G}_{r_{1}}(x)+{} _{x}^{c} \mathcal{D}^{\delta }_{1}\mathcal{G}_{r_{1}}(x) )\), see [56, 57] for Riesz fractional derivative definition.

The Caputo fractional derivative \({}_{0}^{c}\mathcal{D}^{\mu }_{t}\mathcal{U}(x,t)\) is computed as

$$ {}_{0}^{c} \mathcal{D}^{\mu }_{t}\mathcal{U}(x,t)=\sum _{ \substack{r_{1}=0,\dots ,\mathcal{N}\\ r_{2}=0,\dots ,\mathcal{M} }} \varsigma _{r_{1},r_{2}}\mathcal{G}_{r_{1}}(x) \mathcal{G}^{t_{\mathrm{end}}}_{r_{2}, \mu }(t), $$

where \(\mathcal{G}^{t_{\mathrm{end}}}_{r_{2},\mu }(t)=_{0}^{c}\mathcal{D}^{\mu }_{t} \mathcal{G}^{t_{\mathrm{end}}}_{r_{2}}(t)\), see [58]. The integrated operator’s treatment provides us with

$$ \begin{aligned} \int _{0}^{1}\kappa (\mu )_{0}^{c} \mathcal{D}^{ \mu }_{t}\mathcal{U}(x,t)\,d\mu &= \int _{0}^{1} \biggl(\kappa (\mu ) \sum _{ \substack{r_{1}=0,\dots ,\mathcal{N}\\ r_{2}=0,\dots ,\mathcal{M} }} \varsigma _{r_{1},r_{2}}\mathcal{G}_{r_{1}}(x) \mathcal{G}^{t_{\mathrm{end}}}_{r_{2}, \mu }(t) \biggr)\,d\mu \\ &=\sum_{ \substack{r_{1}=0,\dots ,\mathcal{N}\\ r_{2}=0,\dots ,\mathcal{M} }} \varsigma _{r_{1},r_{2}} \int _{0}^{1}\kappa (\mu ) \mathcal{G}_{r_{1}}(x) \mathcal{G}^{t_{\mathrm{end}}}_{r_{2},\mu }(t)\,d\mu \\ &=\sum_{ \substack{r_{1}=0,\dots ,\mathcal{N}\\ r_{2}=0,\dots ,\mathcal{M}\\r_{3}=0,\dots ,\mathcal{Q}}} \varsigma _{r_{1},r_{2}} \mathcal{G}_{r_{1}}(x)\mathcal{F}^{t_{\mathrm{end}}}_{r_{2},r_{3}}(t), \end{aligned} $$

where \(\mathcal{F}^{t_{\mathrm{end}}}_{r_{2},r_{3}}(t)=\varpi _{\mathcal{Q},r_{3}} \kappa (\mu _{\mathcal{Q},r_{3}})\mathcal{G}^{t_{\mathrm{end}}}_{r_{2},\mu _{ \mathcal{Q},r_{3}}}\) and \(\mu _{\mathcal{Q},r_{3}}\), \(r_{3}=0,1,\ldots ,\mathcal{Q}\) are the shifted Legendre Gauss–Lobatto collocation points in the interval \([0,1]\), see [54, 59, 60].

At selected nodes, the preceding derivatives are calculated as follows:

$$\begin{aligned}& \bigl((-\Delta )^{\frac{\delta }{2}} \mathcal{U}(x,t) \bigr)^{x=x_{ \mathcal{N},n},}_{t=t^{t_{\mathrm{end}}}_{\mathcal{M},m}}=\sum _{ \substack{r_{1}=0,\dots ,\mathcal{N}\\ r_{2}=0,\dots ,\mathcal{M} }} \varsigma _{r_{1},r_{2}}\mathcal{G}^{\Delta }_{r_{1}}(x_{\mathcal{N},n}) \mathcal{G}^{t_{\mathrm{end}}}_{r_{2}} \bigl(t^{t_{\mathrm{end}}}_{\mathcal{M},m} \bigr), \end{aligned}$$
$$\begin{aligned}& \biggl( \int _{0}^{1}\kappa (\mu )_{0}^{c} \mathcal{D}^{\mu }_{t}\mathcal{U}(x,t)\,d\mu \biggr)^{x=x_{\mathcal{N},n},}_{t=t^{t_{\mathrm{end}}}_{ \mathcal{M},m}}=\sum_{ \substack{r_{1}=0,\dots ,\mathcal{N}\\ r_{2}=0,\dots ,\mathcal{M}\\r_{3}=0,\dots ,\mathcal{Q}}} \varsigma _{r_{1},r_{2}}\mathcal{G}_{r_{1}}(x_{\mathcal{N},n}) \mathcal{F}^{t_{\mathrm{end}}}_{r_{2},r_{3}} \bigl(t=t^{t{\mathrm{end}}}_{\mathcal{M},m} \bigr), \end{aligned}$$

where \(n=0,1,\ldots , \mathcal{N}\), \(m=0,1,\ldots , \mathcal{M}\) and \(x_{\mathcal{N},n}\), \(t^{t_{\mathrm{end}}}_{\mathcal{M},m}\) are Gauss–Lobatto Legendre collocation and Gauss–Radau shifted Legendre collocation nodes, respectively.

Equation (2.1) is coerced to zero at the \((\mathcal{N}-1)\times (\mathcal{M})\) nodes in the approach.

$$ \Omega \bigl(x_{\mathcal{N},n},t^{t{\mathrm{end}}}_{\mathcal{M},m} \bigr)= \Lambda \biggl(x_{ \mathcal{N},n},t^{t{\mathrm{end}}}_{\mathcal{M},m},\sum _{ \substack{r_{1}=0,\dots ,\mathcal{N}\\ r_{2}=0,\dots ,\mathcal{M} }} \varsigma _{r_{1},r_{2}} \mathcal{G}_{r_{1}}(x_{\mathcal{N},n}) \mathcal{G}^{t_{\mathrm{end}}}_{r_{2}} \bigl(t^{t{\mathrm{end}}}_{\mathcal{M},m} \bigr) \biggr), $$


$$\begin{aligned} \Omega \bigl(x_{\mathcal{N},n},t^{t{\mathrm{end}}}_{\mathcal{M},m} \bigr) =&\sum _{ \substack{r_{1}=0,\dots ,\mathcal{N}\\ r_{2}=0,\dots ,\mathcal{M} }} \varsigma _{r_{1},r_{2}} \mathcal{G}^{\Delta }_{r_{1}}(x_{\mathcal{N},n}) \mathcal{G}^{t_{\mathrm{end}}}_{r_{2}} \bigl(t^{t_{\mathrm{end}}}_{\mathcal{M},m} \bigr) \\ &{}+ \varepsilon \sum_{ \substack{r_{1}=0,\dots ,\mathcal{N}\\ r_{2}=0,\dots ,\mathcal{M}\\r_{3}=0,\dots ,\mathcal{Q}}} \varsigma _{r_{1},r_{2}}\mathcal{G}_{r_{1}}(x_{\mathcal{N},n}) \mathcal{F}^{t_{\mathrm{end}}}_{r_{2},r_{3}} \bigl(t^{t{\mathrm{end}}}_{\mathcal{M},m} \bigr). \end{aligned}$$

Otherwise, initial-boundary can be obtained by

$$ \begin{aligned} &\sum_{ \substack{ l,r_{1}=0,\dots ,\mathcal{N}\\ r_{2}=0,\dots ,\mathcal{M} }} \varsigma _{r_{1},r_{2}}\mathcal{G}_{r_{1}}(x)\mathcal{G}^{t_{\mathrm{end}}}_{r_{2}}(0)= \Theta _{1}(x), \\ &\sum_{ \substack{ l,r_{1}=0,\dots ,\mathcal{N}\\ r_{2}=0,\dots ,\mathcal{M} }} \varsigma _{r_{1},r_{2}} \mathcal{G}_{r_{1}}(-1)\mathcal{G}^{t_{\mathrm{end}}}_{r_{2}} \bigl(t^{t{\mathrm{end}}}_{ \mathcal{M},m} \bigr)=\Theta _{2} \bigl(t^{t{\mathrm{end}}}_{\mathcal{M},m} \bigr), \\ &\sum_{ \substack{ l,r_{1}=0,\dots ,\mathcal{N}\\ r_{2}=0,\dots ,\mathcal{M} }} \varsigma _{r_{1},r_{2}} \mathcal{G}_{r_{1}}(1)\mathcal{G}^{t_{\mathrm{end}}}_{r_{2}} \bigl(t^{t{\mathrm{end}}}_{ \mathcal{M},m} \bigr)=\Theta _{3}(t). \end{aligned} $$

Therefore, adapting (2.1)–(2.10), we get

$$ \Omega \bigl(x_{\mathcal{N},n},t^{t{\mathrm{end}}}_{\mathcal{M},m} \bigr)= \Lambda \biggl(x_{\mathcal{N},n},t^{t{\mathrm{end}}}_{\mathcal{M},m},\sum _{ \substack{r_{1}=0,\dots ,\mathcal{N}\\ r_{2}=0,\dots ,\mathcal{M} }} \varsigma _{r_{1},r_{2}} \mathcal{G}_{r_{1}}(x_{\mathcal{N},n}) \mathcal{G}^{t_{\mathrm{end}}}_{r_{2}} \bigl(t^{t{\mathrm{end}}}_{\mathcal{M},m} \bigr)\biggr), $$

with \(n=1,\ldots ,\mathcal{N}-1\), \(m=1,\ldots ,\mathcal{M}\), additionally

$$ \begin{aligned}&\sum_{ \substack{ l,r_{1}=0,\dots ,\mathcal{N}\\ r_{2}=0,\dots ,\mathcal{M} }} \varsigma _{r_{1},r_{2}}\mathcal{G}_{r_{1}}(x_{\mathcal{N},n}) \mathcal{G}^{t_{\mathrm{end}}}_{r_{2}}(0) =\Theta _{1}(x_{\mathcal{N},n}),\quad k=1,\ldots ,\mathcal{N}-1, \\ &\sum_{ \substack{ l,r_{1}=0,\dots ,\mathcal{N}\\ r_{2}=0,\dots ,\mathcal{M} }} \varsigma _{r_{1},r_{2}} \mathcal{G}_{r_{1}}(-1)\mathcal{G}^{t_{\mathrm{end}}}_{r_{2}} \bigl(t^{t{\mathrm{end}}}_{ \mathcal{M},m} \bigr) =\Theta _{2} \bigl(t^{t{\mathrm{end}}}_{\mathcal{M},m} \bigr),\quad l=0, \ldots ,\mathcal{M}, \\ &\sum_{ \substack{ l,r_{1}=0,\dots ,\mathcal{N}\\ r_{2}=0,\dots ,\mathcal{M} }} \varsigma _{r_{1},r_{2}} \mathcal{G}_{r_{1}}(1)\mathcal{G}^{t_{\mathrm{end}}}_{r_{2}} \bigl(t^{t{\mathrm{end}}}_{ \mathcal{M},m} \bigr) =\Theta _{3} \bigl(t^{t{\mathrm{end}}}_{\mathcal{M},m} \bigr),\quad l=0, \ldots ,\mathcal{M}. \end{aligned} $$

An algebraic equations system is produced by combining Eqs. (2.11) and (2.12), and it is easy to solve.

Distributed and Riesz fractional Schrödinger equation

To address the TDSRFSP, GLLCT and GRLCT are developed

$$ \begin{aligned} & i \int _{0}^{1}\kappa (\mu ){}_{0}^{c} \mathcal{D}^{ \mu }_{t}\psi (x,t)\,d\mu +\varepsilon (-\Delta )^{\frac{\delta }{2}} \psi (x,t)+ \bigl\vert \psi (x,t) \bigr\vert ^{2}\psi (x,t) \\ &\quad =\Lambda (x,t), \quad (x,t)\in \Lambda ^{\bullet } \times \Lambda ^{\diamond }, \end{aligned} $$

where \(\Lambda ^{\bullet }\equiv [-1,1]\) and \(\Lambda ^{\diamond } \equiv [0,t_{\mathrm{end}}]\). Related to

$$ \begin{aligned} &\psi (x,0)=\Theta _{1}(x), \quad x\in \Lambda ^{\bullet }, \\ &\psi (0,t)=\Theta _{2}(t), \qquad \psi ( x_{\mathrm{end}},t)=\Theta _{3}(t),\quad t\in \Lambda ^{\diamond } . \end{aligned} $$

Firstly, we split \(\psi (x,t)\) into its real and imaginary functions \(\mathcal{U}(x,t)\) and \(\mathcal{V}(x,t)\) as \(\psi (x,t)=\mathcal{U}(x,t)+i\mathcal{U}(x,t)\). Based on this transformation, we get

$$ \begin{aligned} & \int _{0}^{1}\kappa (\mu ){}_{0}^{c} \mathcal{D}^{ \mu }_{t}\mathcal{U}(x,t)\,d\mu +\varepsilon (- \Delta )^{ \frac{\delta }{2}}\mathcal{V}(x,t)+ \bigl(\mathcal{U}^{2}(x,t)+ \mathcal{V}^{2}(x,t) \bigr) \mathcal{V}(x,t) \\ &\quad =\Lambda _{2}(x,t),\quad (x,t)\in \Lambda ^{\bullet } \times \Lambda ^{\diamond }, \\ &{-} \int _{0}^{1}\kappa (\mu ){}_{0}^{c} \mathcal{D}^{\mu }_{t} \mathcal{V}(x,t)\,d\mu +\varepsilon (- \Delta )^{\frac{\delta }{2}} \mathcal{U}(x,t)+ \bigl(\mathcal{U}^{2}(x,t)+ \mathcal{V}^{2}(x,t) \bigr) \mathcal{U}(x,t) \\ &\quad =\Lambda _{1}(x,t), \quad (x,t)\in \Lambda ^{\bullet } \times \Lambda ^{\diamond }, \end{aligned} $$

where \(\Lambda (x,t)=\Lambda _{2}(x,t)+i\Lambda _{2}(x,t)\), \(\Lambda ^{\bullet }\equiv [-1,1]\), and \(\Lambda ^{\diamond } \equiv [0,t_{\mathrm{end}}]\). Related to

$$ \begin{aligned} &\mathcal{U}(x,0)=\theta _{1}(x),\qquad \mathcal{U}(0,t)= \theta _{2}(t),\qquad \mathcal{U}( x_{\mathrm{end}},t)=\theta _{3}(t), \quad x \in \Lambda ^{\bullet }, t\in \Lambda ^{\diamond }, \\ &\mathcal{U}(x,0)=\vartheta _{1}(x), \qquad \mathcal{U}(0,t)= \vartheta _{2}(t), \qquad \mathcal{U}( x_{\mathrm{end}},t)=\vartheta _{3}(t), \quad x\in \Lambda ^{\bullet }, t\in \Lambda ^{\diamond }, \end{aligned} $$

where \(\Theta \equiv \theta +i\vartheta \).

To convert the TDSRFP into a nonlinear algebraic system, the GLLCT and GRSLCT are used. The truncated solution is written as follows:

$$ \begin{aligned}& \mathcal{U}_{\mathcal{N},\mathcal{M}}(x,t)= \sum_{ \substack{r_{1}=0,\dots ,\mathcal{N}\\ r_{2}=0,\dots ,\mathcal{M} }} \varsigma _{r_{1},r_{2}} \mathcal{G}_{r_{1}}(x)\mathcal{G}^{t_{\mathrm{end}}}_{r_{2}}(t), \\ &\mathcal{V}_{\mathcal{N},\mathcal{M}}(x,t)=\sum_{ \substack{r_{1}=0,\dots ,\mathcal{N}\\ r_{2}=0,\dots ,\mathcal{M} }} \sigma _{r_{1},r_{2}}\mathcal{G}_{r_{1}}(x)\mathcal{G}^{t_{\mathrm{end}}}_{r_{2}}(t). \end{aligned} $$

Therefore, according to the previous analysis, we get

$$ \begin{aligned} &\Upsilon _{1} \bigl(x_{\mathcal{N},n},t^{t{\mathrm{end}}}_{\mathcal{M},m} \bigr)= \Lambda _{2} \bigl(x_{\mathcal{N},n},t^{t{\mathrm{end}}}_{\mathcal{M},m} \bigr),\quad n=1, \ldots ,\mathcal{N}-1, m=1,\ldots ,\mathcal{M}, \\ &\Upsilon _{2} \bigl(x_{\mathcal{N},n},t^{t{\mathrm{end}}}_{\mathcal{M},m} \bigr)=\Lambda _{1} \bigl(x_{ \mathcal{N},n},t^{t{\mathrm{end}}}_{\mathcal{M},m} \bigr),\quad n=1,\ldots ,\mathcal{N}-1, m=1,\ldots ,\mathcal{M}, \end{aligned} $$


$$\begin{aligned}& \begin{aligned}\Upsilon _{1}(x,t)&=\sum _{ \substack{r_{1}=0,\dots ,\mathcal{N}\\ r_{2}=0,\dots ,\mathcal{M}\\r_{3}=0,\dots ,\mathcal{Q}}} \varsigma _{r_{1},r_{2}}\mathcal{G}_{r_{1}}(x) \mathcal{F}^{t_{\mathrm{end}}}_{r_{2},r_{3}}(t)+ \varepsilon (-\Delta )^{\frac{\delta }{2}}\sum_{ \substack{r_{1}=0,\dots ,\mathcal{N}\\ r_{2}=0,\dots ,\mathcal{M} }} \sigma _{r_{1},r_{2}}\mathcal{G}_{r_{1}}(x)\mathcal{G}^{t_{\mathrm{end}}}_{r_{2}}(t) \\ &\quad {}+ \biggl(\sum_{ \substack{r_{1}=0,\dots ,\mathcal{N}\\ r_{2}=0,\dots ,\mathcal{M} }} \varsigma _{r_{1},r_{2}} \mathcal{G}_{r_{1}}(x)\mathcal{G}^{t_{\mathrm{end}}}_{r_{2}}(t) \biggr)^{2} \biggl(\sum_{ \substack{r_{1}=0,\dots ,\mathcal{N}\\ r_{2}=0,\dots ,\mathcal{M} }} \sigma _{r_{1},r_{2}}\mathcal{G}_{r_{1}}(x)\mathcal{G}^{t_{\mathrm{end}}}_{r_{2}}(t) \biggr) \\ &\quad {}+ \biggl(\sum_{ \substack{r_{1}=0,\dots ,\mathcal{N}\\ r_{2}=0,\dots ,\mathcal{M} }} \sigma _{r_{1},r_{2}} \mathcal{G}_{r_{1}}(x)\mathcal{G}^{t_{\mathrm{end}}}_{r_{2}}(t) \biggr)^{2} \biggl(\sum_{ \substack{r_{1}=0,\dots ,\mathcal{N}\\ r_{2}=0,\dots ,\mathcal{M} }} \sigma _{r_{1},r_{2}}\mathcal{G}_{r_{1}}(x)\mathcal{G}^{t_{\mathrm{end}}}_{r_{2}}(t) \biggr), \end{aligned} \\& \begin{aligned} \Upsilon _{2}(x,t)&=-\sum_{ \substack{r_{1}=0,\dots ,\mathcal{N}\\ r_{2}=0,\dots ,\mathcal{M}\\r_{3}=0,\dots ,\mathcal{Q}}} \sigma _{r_{1},r_{2}}\mathcal{G}_{r_{1}}(x)\mathcal{F}^{t_{\mathrm{end}}}_{r_{2},r_{3}}(t)+ \varepsilon (-\Delta )^{\frac{\delta }{2}}\sum_{ \substack{r_{1}=0,\dots ,\mathcal{N}\\ r_{2}=0,\dots ,\mathcal{M} }} \varsigma _{r_{1},r_{2}}\mathcal{G}_{r_{1}}(x)\mathcal{G}^{t_{\mathrm{end}}}_{r_{2}}(t) \\ &\quad {}+ \biggl(\sum_{ \substack{r_{1}=0,\dots ,\mathcal{N}\\ r_{2}=0,\dots ,\mathcal{M} }} \varsigma _{r_{1},r_{2}} \mathcal{G}_{r_{1}}(x)\mathcal{G}^{t_{\mathrm{end}}}_{r_{2}}(t) \biggr)^{2} \biggl(\sum_{ \substack{r_{1}=0,\dots ,\mathcal{N}\\ r_{2}=0,\dots ,\mathcal{M} }} \varsigma _{r_{1},r_{2}}\mathcal{G}_{r_{1}}(x)\mathcal{G}^{t_{\mathrm{end}}}_{r_{2}}(t) \biggr) \\ &\quad {} + \biggl(\sum_{ \substack{r_{1}=0,\dots ,\mathcal{N}\\ r_{2}=0,\dots ,\mathcal{M} }} \sigma _{r_{1},r_{2}} \mathcal{G}_{r_{1}}(x)\mathcal{G}^{t_{\mathrm{end}}}_{r_{2}}(t) \biggr)^{2} \biggl(\sum_{ \substack{r_{1}=0,\dots ,\mathcal{N}\\ r_{2}=0,\dots ,\mathcal{M} }} \varsigma _{r_{1},r_{2}}\mathcal{G}_{r_{1}}(x)\mathcal{G}^{t_{\mathrm{end}}}_{r_{2}}(t) \biggr). \end{aligned} \end{aligned}$$

In addition to

$$\begin{aligned}& \sum_{ \substack{ l,r_{1}=0,\dots ,\mathcal{N}\\ r_{2}=0,\dots ,\mathcal{M} }} \varsigma _{r_{1},r_{2}}\mathcal{G}_{r_{1}}(x_{\mathcal{N},n}) \mathcal{G}^{t_{\mathrm{end}}}_{r_{2}}(0) =\theta _{1}(x_{\mathcal{N},n}),\quad k=1,\ldots ,\mathcal{N}-1, \\& \sum_{ \substack{ l,r_{1}=0,\dots ,\mathcal{N}\\ r_{2}=0,\dots ,\mathcal{M} }} \varsigma _{r_{1},r_{2}} \mathcal{G}_{r_{1}}(-1)\mathcal{G}^{t_{\mathrm{end}}}_{r_{2}} \bigl(t^{t{\mathrm{end}}}_{ \mathcal{M},m} \bigr) =\theta _{2} \bigl(t^{t{\mathrm{end}}}_{\mathcal{M},m} \bigr),\quad l=0, \ldots ,\mathcal{M}, \\& \sum_{ \substack{ l,r_{1}=0,\dots ,\mathcal{N}\\ r_{2}=0,\dots ,\mathcal{M} }} \varsigma _{r_{1},r_{2}} \mathcal{G}_{r_{1}}(1)\mathcal{G}^{t_{\mathrm{end}}}_{r_{2}} \bigl(t^{t{\mathrm{end}}}_{ \mathcal{M},m} \bigr) =\theta _{3} \bigl(t^{t{\mathrm{end}}}_{\mathcal{M},m} \bigr),\quad l=0, \ldots ,\mathcal{M}, \\& \sum_{ \substack{ l,r_{1}=0,\dots ,\mathcal{N}\\ r_{2}=0,\dots ,\mathcal{M} }} \sigma _{r_{1},r_{2}} \mathcal{G}_{r_{1}}(x_{\mathcal{N},n}) \mathcal{G}^{t_{\mathrm{end}}}_{r_{2}}(0) = \vartheta _{1}(x_{\mathcal{N},n}) ,\quad k=1,\ldots ,\mathcal{N}-1, \\& \sum_{ \substack{ l,r_{1}=0,\dots ,\mathcal{N}\\ r_{2}=0,\dots ,\mathcal{M} }} \sigma _{r_{1},r_{2}} \mathcal{G}_{r_{1}}(-1)\mathcal{G}^{t_{\mathrm{end}}}_{r_{2}} \bigl(t^{t{\mathrm{end}}}_{ \mathcal{M},m} \bigr) =\vartheta _{2} \bigl(t^{t{\mathrm{end}}}_{\mathcal{M},m} \bigr),\quad l=0, \ldots ,\mathcal{M}, \\& \sum_{ \substack{ l,r_{1}=0,\dots ,\mathcal{N}\\ r_{2}=0,\dots ,\mathcal{M} }} \sigma _{r_{1},r_{2}} \mathcal{G}_{r_{1}}(1)\mathcal{G}^{t_{\mathrm{end}}}_{r_{2}} \bigl(t^{t{\mathrm{end}}}_{ \mathcal{M},m} \bigr) =\vartheta _{3} \bigl(t^{t{\mathrm{end}}}_{\mathcal{M},m} \bigr),\quad l=0, \ldots ,\mathcal{M}, \end{aligned}$$

when Eqs. (2.18) and (2.19) are combined, we have a linear system of algebraic equations that is simple to solve.

Numerical results

We demonstrate the spectral collocation scheme’s resilience and accuracy by applying the technique to three test problems.

Example 1

The convection–diffusion equation [57] is first presented

$$ \begin{aligned} &\int _{0}^{1}\Gamma (3-\mu ){}_{0}^{c} \mathcal{D}^{\mu }_{t}\mathcal{U}(x,t)\,d\mu + \frac{\Gamma (8-\delta )}{\Gamma (8)}(-\Delta )^{\frac{\delta }{2}} \mathcal{U}(x,t) \\ &\quad =\Delta (x,t),\quad (x,t)\in [-1,1]\times [0,0.5], \end{aligned} $$

the conditions and \(\Delta (x,t)\) are given where \(\mathcal{U}(x,t)=t^{2} (x^{2}-1 )^{4}\).

Table 1 compares our results to those in [57] for various parameter values based on \(L_{2}\)-errors. Based on these findings, the recommended technique delivers superior numerical results than those reported in [57]. It is also worth mentioning that excellent predictions are made. The numerical solution and absolute errors of problem (1) are shown in Figs. 1 and 2, respectively. We showed the x-direction curves of numerical and precise solutions in Fig. 3, where \(\delta =1.8\), \(\mathcal{N}=8\), and \(\mathcal{M}=2\). Figures 4 and 5 depict the x- and t-graphs associated with absolute errors, respectively.

Figure 1
figure 1

\(\mathcal{U}_{\mathcal{N},\mathcal{M}}\) of Eq. (1) with \(\delta =1.8\), \(\mathcal{N}=8\), \(\mathcal{M}=2\)

Figure 2
figure 2

\(E(x,t)\) of Eq. (1) with \(\delta =1.8\), \(\mathcal{N}=8\), \(\mathcal{M}=2\)

Figure 3
figure 3

x-direction curves of \(\mathcal{U}\) and \(\mathcal{U}_{\mathcal{N},\mathcal{M}}\) of problem (1) with \(\delta =1.8\), \(\mathcal{N}=8\), \(\mathcal{M}=2\)

Figure 4
figure 4

\(E(0,t)\) of problem (1) with \(\delta =1.8\), \(\mathcal{N}=8\), \(\mathcal{M}=2\)

Figure 5
figure 5

\(E(x,0)\) of problem (1) with \(\delta =1.8\), \(\mathcal{N}=8\), \(\mathcal{M}=2\)

Table 1 \(L_{2}\)-comparison for problem (1)

Example 2

We introduce the temporal distributed and spatial Riesz Burgers’ equation [57]

$$ \begin{aligned} &\int _{0}^{1}\Gamma (3-\mu ){}_{0}^{c} \mathcal{D}^{\mu }_{t}\mathcal{U}(x,t)\,d\mu + \frac{\Gamma (8-\delta )}{\Gamma (8)}(-\Delta )^{\frac{\delta }{2}} \mathcal{U}(x,t)+ \frac{\partial }{\partial x} \biggl( \frac{(\mathcal{U}(x,t))^{2}}{2} \biggr) \\ &\quad =\Delta (x,t),\quad (x,t)\in [-1,1] \times [0,0.5], \end{aligned} $$

\(\Delta (x,t)\) and conditions are provided, \(\mathcal{U}(x,t)=t^{2} (x^{2}-1 )^{4}\).

Table 2 shows a comparison of our results with those in [57] at different parameter values based on \(L_{2}\)-errors. The suggested approach produces superior numerical results to those published in [57] based on these findings. It is also worth noting that great estimates are made.

Table 2 \(L_{2}\)-comparison for problem (2)

Example 3

We introduce the temporal distributed and spatial Riesz convection–diffusion and Schrödinger-type equation [57]

$$ \begin{aligned} &\int _{0}^{1}\Gamma (3-\mu ){}_{0}^{c} \mathcal{D}^{\mu }_{t}\psi (x,t)\,d\mu - \frac{\Gamma (10-\delta )}{\Gamma (10)}(-\Delta )^{\frac{\delta }{2}} \psi (x,t)+ \bigl\vert \psi (x,t) \bigr\vert ^{2}\psi (x,t) \\ &\quad =\Delta (x,t), \quad (x,t)\in [-1,1] \times [0,0.5], \end{aligned} $$

\(\Delta (x,t)\) and conditions are provided \(\psi (x,t)=t^{2}(1+i) (x^{2}-1 )^{5}\).

Table 3 shows a comparison of our results with those in [57] at different parameter values based on \(L_{2}\)-errors. The suggested approach produces superior numerical results to those published in [57] based on these findings. It is also worth noting that great estimates are made.

Table 3 \(L_{2}\)-comparison for problem (3)

We see in Figs. 6 and 7 the absolute errors of problem (3) for both real and imaginary parts, respectively. In Figs. 8 and 9, we plotted the x-direction curves of numerical and exact solutions for both real and imaginary parts, respectively, where \(\delta =1.8\), \(\mathcal{N}=10\), \(\mathcal{M}=2\); while the x-graphs related to the absolute errors are sketched in Figs. 10 and 11 for both real and imaginary parts, respectively.

Figure 6
figure 6

\(E_{\mathcal{U}}\) of problem (3), with \(\delta =1.8\), \(\mathcal{N}=10\), \(\mathcal{M}=2\)

Figure 7
figure 7

\(E_{\mathcal{V}}(x,t)\) of problem (3) with \(\delta =1.8\), \(\mathcal{N}=10\), \(\mathcal{M}=2\)

Figure 8
figure 8

x-direction curves of \(\mathcal{U}\) and \(\mathcal{U}_{\mathcal{N},\mathcal{M}}\) of problem (3), where \(\delta =1.8\), \(\mathcal{N}=10\), \(\mathcal{M}=2\)

Figure 9
figure 9

x-direction curves of \(\mathcal{V}\) and \(\mathcal{V}_{\mathcal{N},\mathcal{M}}\) of problem (3), where \(\delta =1.8\), \(\mathcal{N}=10\), \(\mathcal{M}=2\)

Figure 10
figure 10

\(E_{\mathcal{U}}(x,0)\) of problem of problem (3), where \(\delta =1.8\), \(\mathcal{N}=10\), \(\mathcal{M}=2\)

Figure 11
figure 11

\(E_{\mathcal{V}}(x,0)\) of problem (3), where \(\delta =1.8\), \(\mathcal{N}=10\), \(\mathcal{M}=2\)


We present an extraordinarily accurate collocation approach for convection–diffusion and Schrodinger-type equations for mixed Riesz and distributed fractional order. A comprehensive theoretical description as well as a series of numerical tests to demonstrate the technique’s execution and eligibility are provided. We can see that our approach is highly accurate and reliable based on the findings. More fractional order issues can be included in the current theoretical debate. The current figures are completely consistent with the predicted outcomes of the spectral collocation technique, and there is clear evidence of exponential convergence. In this study, the GLLCT and GRSLCT approaches are employed to solve the given models. The shifted Legendre nodes are used as interpolation points for the independent variables, and the solution is represented as a series of shifted Legendre polynomials. After that, the residuals at the shifted Legendre quadrature points are estimated. As a consequence, we have an algebraic system that can be solved using a suitable method. A variety of numerical problems are used to illustrate the precision of the proposed technique. Due to their adaptability to both linear and nonlinear equations, the spectral collocation approach has become widely used to estimate many types of differential and integral equations. Since their global character fits well with the nonlocal notion of fractional operators, spectral techniques are excellent candidates for solving fractional differential equations. Spectral techniques provide a high degree of precision and exponential convergence rates. In the tables above, the cheap costs and excellent accuracy are readily visible. The suggested approach is efficient and accurate according to simulation results.

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The authors extend their appreciation to the Deanship of Scientific Research at Imam Mohammad Ibn Saud Islamic University for funding this work through Research Group no. RG-21-09-07.


The authors extend their appreciation to the Deanship of Scientific Research at Imam Mohammad Ibn Saud Islamic University for funding this work through Research Group no. RG-21-09-07.

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Abdelkawy, M.A., Jeelani, M.B., Alnahdi, A.S. et al. Legendre spectral collocation method for distributed and Riesz fractional convection–diffusion and Schrödinger-type equation. Bound Value Probl 2022, 13 (2022).

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  • Spectral collocation method
  • Distributed fractional
  • Riesz fractional