A Reduced-Dimension Extrapolating Method of Finite Element Solution Coefficient Vectors for Fractional Tricomi-Type Equation

Abstract

We here employ a proper orthogonal decomposition (POD) to reduce the dimensionality of unknown coefficient vectors of finite element (FE) solutions for the fractional Tricomi-type equation and develop a reduced-dimension extrapolating FE (RDEFE) method for the fractional Tricomi-type equation. For this purpose, we first develop an FE method for the fractional Tricomi-type equation and provide the existence, unconditional stability, and error analysis for the FE solutions. We then develop the RDEFE method for the fractional Tricomi-type equation by means of the POD technique and analyze the existence, unconditional stability, and errors for the RDEFE solutions by using the matrix analysis. Lastly, we provide two numerical examples to verify our theoretical results and to explain the advantages of the RDEFE method.

1. Introduction

Differential equations with fractional-order derivatives are some important mathematical models and have very extensive applications, including in physics, chemistry, biology, and other fields (see [1,2,3,4,5]). Nevertheless, the concept of fractional-order derivatives was first presented in Leibniz’s letter to L’Hospital in 1695 (see [6]). A. N. Gerasimov [7] and M. Caputo [8] generalized Leibniz and L’Hospital’s earlier work to develop the Gerasimov–Caputo fractional derivative (see [9]). Owing to the complexity of differential equations with fractional-order derivatives, in general, their analytical solutions are unavailable. One has to resort to calculating approximate numerical solutions. The numerical solutions for calculating fractional partial differential equations depend mainly on the finite difference (FD) method (see [10,11,12,13,14]), finite element (FE) method (see [15,16,17,18]), collocation method (see [19,20,21]), discontinuous Galerkin (DG) method (see [22,23]), and meshless method based on thin plate radial basis functions (see [24]). However, the above classical numerical methods have usually many unknowns.

Hence, we here mainly focus on the dimensionality reduction of unknown FE solution coefficient vectors in the FE method with the following fractional Tricomi-type equation defined on a bounded and polygonal domain $\overline{\mathsf{\Omega }}\subset {\mathbb{R}}^2$ with the boundary $\partial \mathsf{\Omega }$.

Problem 1.

For the maximal time limit $t_e$, find $\varpi (x,y,t):[0,t_e]\to C^2(\overline{\mathsf{\Omega }})$ satisfying

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\displaystyle D_t^{\alpha }\varpi (x,y,t)-t^{2\gamma }\Delta \varpi (x,y,t)=f(x,y,t),}\hfill & (x,y,t)\in \mathsf{\Omega }\times (0,t_e),\hfill \\ {\displaystyle \varpi (x,y,t)=0,}\hfill & (x,y,t)\in \partial \mathsf{\Omega }\times (0,t_e),\hfill \\ {\displaystyle \varpi (x,y,0)={\varphi }_0(x,y),{\varpi }_t(x,y,0)={\varphi }_1(x,y),}\hfill & (x,y)\in \mathsf{\Omega },\hfill \end{array}\right.\end{array}$$

(1)

where $D_t^{\alpha }\varpi$ denotes the αth-order Gerasimov–Caputo fractional derivative (see [7,8]) and $1<\alpha <2$, which is defined by

$$\begin{array}{ccc}\hfill & & {\displaystyle D_t^{\alpha }\varpi =D_t^{\alpha }\varpi (x,y,t)=\frac{1}{\mathsf{\Gamma }(2-\alpha )}{\int }_0^t\frac{{\partial }^2\varpi (x,y,s)}{\partial s^2}\frac{\mathrm{d}s}{{(t-s)}^{\alpha -1}},0⩽s⩽t⩽t_e,}\hfill \end{array}$$

(2)

$\mathsf{\Gamma }(·)$ denotes the gamma function, γ is a non-negative real constant, $\Delta ={\partial }^2/\partial x^2+{\partial }^2/\partial y^2$, $f(x,y,t)$ is a known source function, ${\varpi }_t=\partial \varpi /\partial t$, and ${\varphi }_0(x,y)$ and ${\varphi }_1(x,y)$ are the given initial value functions.

Owing to Problem 1 including the Gerasimov–Caputo fractional order time derivative, especially when the source function $f(x,y,t)$ or the initial value functions ${\varphi }_0(x,y)$ and ${\varphi }_1(x,y)$ are slightly complex, we also cannot find its analytical solutions, so we can only find its numerical solutions.

The fractional Tricomi-type equations are also mainly solved with the FD scheme (see [12,25]), the spectral method (see in [26]), and the FE method (see [27,28,29,30,31]). Among these numerical methods, the FE method is more popular since it can be suitable for irregular regions and requires a lower smoothness of analytical solution than the FD scheme and spectral method. However, the classical FE method usually includes many unknowns, especially when it is applied to solving actual engineering problems; it would have tens of millions of unknowns. Therefore, the central task of this paper is to resort to the proper orthogonal decomposition (POD) to reduce the dimensionality of FE solution coefficient vectors in the FE method for the fractional Tricomi-type equation and develop a reduced-dimension extrapolated FE (RDEFE) method for the fractional Tricomi-type equation so as to reduce the unknowns of the FE method, alleviate computational load, slow down the accumulation of computing errors, save CPU running time, and improve calculating efficiency.

A lot of numerical tests (see [32,33,34,35,36,37,38,39,40,41]) have shown that the POD method is one of the most effective ways to reduce the unknowns of numerical models for the unsteady partial differential equations (PDEs). Although some reduced-dimension methods for the coefficient vectors of unknown FE solutions for the hyperbolic equation, parabolic equation, unsteady Stokes equation, Sobolev equation, and Rosenau equation have been developed in [42,43,44,45,46], the fractional Tricomi-type equation is much more complicated than the above five kinds of equations since it contains the Gerasimov–Caputo fractional order time derivative. Therefore, both the development of the RDEFE method and the theoretical analysis of the existence and stability as well as errors to the RDEFE solutions could face more challenges and require more techniques than the earlier works. However, the RDEFE method for the fractional Tricomi-type equation has very significant applications.

Although a based-POD reduced-order FE method of FE subspace for the fractional Tricomi-type equation has been proposed in [47], it is built by replacing the FE subspace with the subspace spanned by few main POD basis functions, and its POD basis functions are a continuous form so that the construction of POD basis functions and the theoretical analysis of existence, stability, and convergence (error estimates) require the use of an abstract functional analysis principle, which are not easily understood by engineers with weak mathematical ability. In contrast with the method in [47], the RDEFE method for the fractional Tricomi-type equation here is only to reduce the dimensionality of FE solution coefficient vectors in the system of FE equations and is a matrix form so that the construction of its POD basis vectors and the existence, stability, and convergence (error estimates) require only the use of the matrix analysis in linear algebra and do not require the abstract functional analysis, and the theoretical analysis is very simple and very easily accepted by the public. In other words, the RDEFE method only requires the addition of a subprogram for POD dimensionality reduction in the calculation, and its POD basis vectors are a discrete form so as to be easily obtained by matrix analysis. Especially, its FE basis functions do not change so as to keep its accuracy unchanged, but can greatly reduce the unknowns. Therefore, the RDEFE method here is absolutely distinguished from the existing reduced order/dimension methods, including the method in [47]. Hence, it is valuable to research the RDEFE method of the fractional Tricomi-type equation.

For this purpose, we first review the classical FE method for the fractional Tricomi-type equation and provide the unconditional stability and errors of the FE solutions in Section 2. Next, in Section 3, we resort to the POD technique to produce the POD basis vectors and build the RDEFE method for the fractional Tricomi-type equation, and we resort to matrix analysis to analyze the existence and stability as well as errors to the RDEFE solutions. Then, in Section 4, we provide two numerical examples to verify the feasibility and effectiveness of the RDEFE method and show that the numerical computing results agree well with the theoretical ones. We finally sum up the main conclusions and provide the prospect of future research in Section 5.

2. The Classical FE Method for the Fractional Tricomi-Type Equation

The Sobolev spaces as well as their norms used later are classical (see [48]). We assume that $\mathbb{W}=H_0^1(\mathsf{\Omega })$ in the following for convenience.

2.1. The FE Format in Functional Form

In order to establish the FE format in functional form, we need to approximate the time derivative by difference quotient and the spatial variables by the FE method. To this end, let $N>0$ stand for an integer, $\Delta t=t_e/N$ denote the time step, ${\Im }_h$ be the quasi-uniform triangulation on $\overline{\mathsf{\Omega }}$, ${\varpi }_h^n$ stand for the FE approximations for $\varpi$ at $t_n=n\Delta t$ $(n=0,1,2,\cdots ,N)$, and ${\mathbb{W}}_h$ be an M-dimensional FE space, which is spanned with the orthonormal bases ${\left\{{\xi }_i(x,y)\right\}}_{i=1}^M$ under the inner product $a(\varpi ,\omega )=(\varpi ,\omega )+{\gamma }_0{\alpha }_0(\nabla \varpi ,\nabla \omega )$ in $H_0^1(\mathsf{\Omega })$ (where ${\xi }_i(x,y)$ are obtained by the standard orthogonalization in Section 6.3 in [48], ${\gamma }_0={(n\Delta t)}^{2\gamma }$, ${\alpha }_0={(\Delta t)}^{\alpha }\mathsf{\Gamma }(3-\alpha )$, and $(·,·)$ is the $L^2$ inner product) and denoted by

$$\begin{array}{ccc}\hfill {\displaystyle {\mathbb{W}}_h}& =& {\displaystyle \left\{{\omega }_h\in C(\mathsf{\Omega })\cap H_0^1(\mathsf{\Omega }):{\omega }_h{|}_E\in {\mathbb{P}}_l(E),E\in {\Im }_h\right\}}\hfill \\ \hfill & =& {\displaystyle \mathrm{span}\left\{{\xi }_i(x,y):1⩽i⩽M\right\},}\hfill \end{array}$$

(3)

in which ${\mathbb{P}}_l(E)$ is made up of lth degree polynomials on $E\in {\Im }_h$.

Thereupon, the Gerasimov–Caputo fractional order time derivative for the formula (2) is approximated as follows (see [29]):

$$\begin{array}{c}{\displaystyle D_t^{\alpha }{\varpi }_h^n=\frac{1}{\Delta t\mathsf{\Gamma }(3-\alpha )}\sum _{j=1}^n{b}_{n-j}({\varpi }_h^j-2{\varpi }_h^{j-1}+{\varpi }_h^{j-2})+R(\varpi ),}\hfill \end{array}$$

(4)

where $b_{n-j}={(n-j+1)}^{2-\alpha }-{(n-j)}^{2-\alpha }$, $R(\varpi )⩽\frac{C_{\varpi }{M}_0{t}_e^{2-\alpha }\Delta t}{\mathsf{\Gamma }(3-\alpha )}+O(\Delta t^2)$, $C_{\varpi }>0$ represents the constant that is dependent on $\varpi$, and $M_0$ is the upper bound of $|{\partial }^3\varpi (x,y,t)/\partial t^3|$. It is easily verified that the coefficients $b_j$ satisfy the following properties:

$$\begin{array}{c}{\displaystyle \left\{\begin{array}{c}{\displaystyle 1=b_0>b_1>b_2>\cdots >b_n>0,b_n\to 0(n\to \infty ),}\hfill \\ {\displaystyle \sum _{j=1}^{n-1}(b_{j-1}-b_j)=b_0-b_{n-1}<1,1⩽n⩽N.}\hfill \end{array}\right.}\hfill \end{array}$$

(5)

Thereupon, the FE format in functional form can be expressed as follows:

Problem 2.

Find ${\left\{{\varpi }_h^n\right\}}_{n=1}^N\subset {\mathbb{W}}_h$ satisfying

$$\begin{array}{c}{\displaystyle a({\varpi }_h^n,w_h)=\sum _{j=1}^{n-1}(2b_j-b_{j+1}-b_{j-1})({\varpi }_h^{n-j-1},w_h)+(2b_0-b_1)({\varpi }_h^{n-1},w_h)}\hfill \\ {\displaystyle +b_0({\varpi }_h^0,w_h)-b_{n-1}({\varpi }_h^{-1},w_h)+{\alpha }_0(f^n,w_h),\forall w_h\in {\mathbb{W}}_h,1⩽n⩽N;}\hfill \end{array}$$

(6)

$$\begin{array}{c}{\displaystyle {\varpi }_h^0=P_h{\varphi }_0,{\varpi }_h^{-1}=P_h{\varphi }_0+\Delta t{P}_h{\varphi }_1,}\hfill \end{array}$$

(7)

where ${\varpi }_h^0=P_h{\varpi }^0$ and $P_h$ is the $L^2$ projection, namely, satisfies the following equation:

$$\begin{array}{ccc}& & (P_h{\varpi }^0,{\upsilon }_h)=({\varpi }^0,{\upsilon }_h),\forall {\upsilon }_h\in {\mathbb{W}}_h.\hfill \end{array}$$

Noting that $a(·,·)$ is the bounded symmetric positive definite bilinear functional and, for the given $f^n$, ${\varpi }_h^{n-1}$ $(1⩽n⩽N)$, the right-hand side in (6) is the bounded linear functional, by Lax–Milgram’s theorem and standard FE method, we can easily demonstrate that Problem 2 has a unique set of solutions ${\left\{{\varpi }_h^n\right\}}_{n=1}^N\subset {\mathbb{W}}_h$ satisfying the unconditional stability and errors, whose detailed proof can be found in [31,47].

Theorem 1.

Problem 2 has a unique set of solutions ${\left\{{\varpi }_h^n\right\}}_{n=1}^N\subset {\mathbb{W}}_h$ meeting the following unconditional stability:

$$\begin{array}{c}\parallel {\varpi }_h^n{\parallel }_0+\sqrt{{\gamma }_0{\alpha }_0}{\parallel \nabla {\varpi }_h^n\parallel }_0⩽c,1⩽n⩽N,\hfill \end{array}$$

(8)

and error estimates

$$\begin{array}{c}\parallel \varpi (t_n)-{\varpi }_h^n{\parallel }_0⩽c(\Delta t^{3-\alpha }+\Delta t^{1-\alpha }{h}^{l+1}),1⩽n⩽N,\hfill \end{array}$$

(9)

where c is a generic positive constant but may be unequal in different occurring and $\varpi (t_n)$ $(1⩽n⩽N)$ denotes the solution $\varpi \in {[H^{l+1}(\mathsf{\Omega })\cap H_0^1(\mathsf{\Omega })]}^2$ to Problem 2 at $t=t_n$.

2.2. The FE Format in Matrix Form

Let ${\mathsf{\varpi }}^n={({\varpi }_1^n,{\varpi }_2^n,\cdots ,{\varpi }_M^n)}^T$ and $\mathsf{\xi }=({\xi }_1,{\xi }_2,\cdots ,$ ${\xi }_M{)}^T$. Then, the solutions ${\varpi }_h^n$ for Problem 2 may be denoted by

$$\begin{array}{ccc}& & {\varpi }_h^n={\displaystyle \sum _{j=1}^M}{\varpi }_j^n{\xi }_j=\mathsf{\xi }·{\mathsf{\varpi }}^n.\hfill \end{array}$$

Thereupon, Problem 2 may be rewritten in the following matrix form:

Problem 3.

Find ${\mathsf{\varpi }}^n\in {\mathbb{R}}^M$ and ${\varpi }_h^n\in {\mathbb{W}}_h$ $(1⩽n⩽N)$ satisfying

$$\begin{array}{ccc}\hfill & {\displaystyle {\mathsf{\varpi }}^n}& {\displaystyle =\sum _{j=1}^{n-1}(2b_j-b_{j+1}-b_{j-1})\mathit{B}{\mathsf{\varpi }}^{n-j-1}+(2b_0-b_1)\mathit{B}{\mathsf{\varpi }}^{n-1}}\hfill \\ \hfill {\displaystyle }& & {\displaystyle +b_0\mathit{B}{\mathsf{\varpi }}^0-b_{n-1}\mathit{B}{\mathsf{\varpi }}^{-1}+{\alpha }_0{\mathit{F}}^n,\forall w_h\in {\mathbb{W}}_h,1⩽n⩽N,}\hfill \end{array}$$

(10)

where $\mathit{B}={(({\xi }_i,{\xi }_j))}_{M\times M}$, ${\mathsf{\varpi }}^{-1}=({\varpi }_h^{-1},{\xi }_1),({\varpi }_h^{-1},{\xi }_2),\cdots ,({\varpi }_h^{-1},{\xi }_M){)}^T$, ${\mathsf{\varpi }}^0=(({\varpi }_h^0,{\xi }_1),$ $({\varpi }_h^0,{\xi }_2),\cdots ,({\varpi }_h^0,{\xi }_M){)}^T$, ${\mathit{F}}^n={((f^n,{\xi }_1),(f^n,{\xi }_2),\cdots ,(f^n,{\xi }_M))}^T$, and $(·,·)$ is the $L^2$ inner product.

To discuss the stability and convergence of the FE solution vectors in Problem 3 needs the following lemma, which can be proved by using the matrix norm in [49] and Lemma 1.22 in [50].

Lemma 1.

If ${\xi }_i$ $(1⩽i⩽M)$ is a set of FE basis functions, $(·,·)$ denotes the $L^2$ inner product, and the matrix $\mathit{B}$ is made use of $({\xi }_i,{\xi }_j)$, then the following estimates hold:

$$\begin{array}{ccc}& & {\displaystyle {\parallel \mathit{B}\parallel }_2⩽ch,{\parallel {\mathit{B}}^{-1}\parallel }_2⩽c{h}^{-1},}\hfill \end{array}$$

here, ${\parallel \mathit{B}\parallel }_2={sup}_{\mathbf{\chi }\in {\mathbb{R}}^M}\parallel \mathit{B}\mathbf{\chi }\parallel /\parallel \mathbf{\chi }\parallel$ and $\parallel \mathbf{\chi }\parallel$ is the Euclidean norm of vector $\mathbf{\chi }$.

The FE solution vectors for Problem 3 have the following result:

Theorem 2.

Problem 3 has a unique set of solution vectors ${\left\{{\mathsf{\varpi }}^n\right\}}_{n=1}^N\subset {\mathbb{R}}^M$ satisfying the following unconditional stability:

$$\begin{array}{c}\parallel {\mathsf{\varpi }}^n\parallel ⩽c,1⩽n⩽N.\hfill \end{array}$$

(11)

Further, the set of solution vectors ${\left\{{\mathsf{\varpi }}^n\right\}}_{n=1}^N$ is unconditionally convergent.

Proof. 

We can conclude from the positive definiteness of $a(·,·)$ that Problem 3 has a unique set of solution vectors ${\left\{{\mathsf{\varpi }}^n\right\}}_{n=1}^N\subset {\mathbb{R}}^M$.

By (5), Lemma 1, and (10), we obtain

$$\begin{array}{c}{\displaystyle \parallel {\mathsf{\varpi }}^n\parallel ⩽ch(\parallel {\mathsf{\varpi }}^0\parallel +\parallel {\mathsf{\varpi }}^{-1}\parallel )+c\Delta t^{\alpha }\parallel {\mathit{F}}^n\parallel +ch\sum _{j=0}^{n-1}\parallel {\mathsf{\varpi }}^j\parallel ,1⩽n⩽N.}\hfill \end{array}$$

(12)

By applying the Gronwall lemma to (12), we obtain

$$\begin{array}{c}{\displaystyle \parallel {\mathsf{\varpi }}^n\parallel ⩽[ch(\parallel {\mathsf{\varpi }}^0\parallel +\parallel {\mathsf{\varpi }}^{-1}\parallel )+c\Delta t^{\alpha }\parallel {\mathit{F}}^n\parallel ]exp(chn)⩽c,1⩽n⩽N.}\hfill \end{array}$$

(13)

Therefore, the inequality (11) holds so that the solution vectors ${\left\{{\mathsf{\varpi }}^n\right\}}_{n=1}^N\subset {\mathbb{R}}^M$ are unconditionally stable. Further, by the Lax equivalence theorem (see [48], Theorem 2.3.18), we can conclude that the set of solution vectors ${\left\{{\mathsf{\varpi }}^n\right\}}_{n=1}^N$ is unconditionally convergent. □

Remark 1.

When the time step $\Delta t$; the spatial mesh parameter h; the constants γ and α; the source function f; and the initial value functions ${\varphi }_0$ and ${\varphi }_1$ are appointed, a set of the solution coefficient vectors ${\left\{{\mathsf{\varpi }}^n\right\}}_{n=1}^N$ can be obtained by solving Problem 3. However, when Problem 3 is used to solve the actual engineering problems, the dimensionality of the unknown solution coefficient vectors ${\left\{{\mathsf{\varpi }}^n\right\}}_{n=1}^N$ in Problem 3 is so high that we need to resort to the POD technique to reduce their dimensionality.

3. The RDEFE Method of the Fractional Tricomi-Type Equation

3.1. Generation for the POD Basis Vectors

First, we solve the L coefficient vectors ${\mathsf{\varpi }}^n$ ($1⩽n⩽L$) for Problem 3 at the initial L time nodes to make up for an $M\times L$ matrix $\mathit{A}=({\mathsf{\varpi }}^1,{\mathsf{\varpi }}^2,\cdots ,$ ${\mathsf{\varpi }}^L)$.

Next, we calculate a set of eigenvectors ${\mathsf{\psi }}_1,{\mathsf{\psi }}_2,\cdots ,$ ${\mathsf{\psi }}_r$ $(r=$ rank$(\mathit{A}))$ corresponding to the positive eigenvalues ${\lambda }_1⩾{\lambda }_2⩾\cdots ⩾{\lambda }_r>0$ for the $L\times L$ matrix ${\mathit{A}}^T\mathit{A}$.

Finally, by using the formulas ${\mathsf{\phi }}_i=\mathit{A}{\mathsf{\psi }}_i/{\lambda }_i$ $(1⩽i⩽r)$, we find the orthonormal eigenmatrix $\tilde{\mathsf{\Phi }}=({\mathsf{\phi }}_1,{\mathsf{\phi }}_2,\cdots ,$ ${\mathsf{\phi }}_r)\in {\mathbb{R}}^{M\times r}$ of $\mathit{A}{\mathit{A}}^T$ associated with the positive eigenvalues ${\lambda }_1⩾{\lambda }_2⩾\cdots ⩾{\lambda }_r>0$ to obtain a set of POD basis $\mathsf{\Phi }=({\mathsf{\phi }}_1,{\mathsf{\phi }}_2,\cdots ,$ ${\mathsf{\phi }}_d)$ ($d⩽r$), which is made up of the first d column vectors in $\tilde{\mathsf{\Phi }}$ and has the following property (see [39]):

$$\begin{array}{c}\parallel \mathit{A}-\mathsf{\Phi }{\mathsf{\Phi }}^T{\mathit{A}\parallel }_2=\sqrt{{\lambda }_{d+1}}.\hfill \end{array}$$

(14)

In addition, there are the following estimations:

$$\begin{array}{c}{\displaystyle \parallel {\mathsf{\varpi }}^n-\mathsf{\Phi }{\mathsf{\Phi }}^T{\mathsf{\varpi }}^n\parallel =\parallel (\mathit{A}-\mathsf{\Phi }{\mathsf{\Phi }}^T\mathit{A}){\mathsf{\varsigma }}^n\parallel ⩽\parallel \mathit{A}-\mathsf{\Phi }{\mathsf{\Phi }}^T{\mathit{A}\parallel }_2\parallel {\mathsf{\varsigma }}^n\parallel }\hfill \\ {\displaystyle ⩽\sqrt{{\lambda }_{d+1}},1⩽n⩽L,}\hfill \end{array}$$

(15)

where ${\mathsf{\varsigma }}^n$ $(1⩽n⩽L)$ denote the orthonormal vectors with nth component 1. Therefore, $\mathsf{\Phi }=({\mathsf{\phi }}_1,{\mathsf{\phi }}_2,\cdots ,$ ${\mathsf{\phi }}_d)$ constitutes a group of optimal POD basis.

3.2. Establishment of RDEFE Formulation

If we assume that ${\mathsf{\varpi }}_d^n={({\varpi }_{d1}^n,{\varpi }_{d2}^n,\cdots ,{\varpi }_{dM}^n)}^T$ and ${\mathsf{\beta }}^n=({\beta }_1^n,$ ${\beta }_2^n,\cdots ,$ ${\beta }_d^n{)}^T$, we can immediately obtain the first $L(L⩽N)$ coefficient vectors ${\mathsf{\varpi }}_d^n$ $=\mathsf{\Phi }{\mathsf{\Phi }}^T{\mathsf{\varpi }}^n=:\mathsf{\Phi }{\mathsf{\beta }}^n$ $(1⩽n⩽L)$ of the RDEFE solutions. If the unknown solution coefficient vectors ${\mathsf{\varpi }}^n$ of Problem 3 are replaced with ${\mathsf{\varpi }}_d^n=\mathsf{\Phi }{\mathsf{\beta }}^n$ $(L+1⩽n⩽N)$, by the orthonormality for the POD basis vectors, we can develop the RDEFE formulation as follows:

Problem 4.

Find ${\mathsf{\varpi }}_d^n\in {\mathbb{R}}^M$ and ${\varpi }_d^n\in {\mathbb{W}}_h$ $(1⩽n⩽N)$ satisfying

$$\begin{array}{c}{\displaystyle {\mathsf{\beta }}^n={\mathsf{\Phi }}^T{\mathsf{\varpi }}^n,1⩽n⩽L;}\hfill \end{array}$$

(16)

$$\begin{array}{ccc}\hfill & & {\displaystyle {\mathsf{\beta }}^n=\sum _{j=1}^{n-1}(2b_j-b_{j+1}-b_{j-1}){\mathsf{\Phi }}^T\mathit{B}\mathsf{\Phi }{\mathsf{\beta }}^{n-j-1}+(2b_0-b_1){\mathsf{\Phi }}^T\mathit{B}\mathsf{\Phi }{\mathsf{\beta }}^{n-1}}\hfill \\ \hfill {\displaystyle }& & {\displaystyle +b_0{\mathsf{\Phi }}^T\mathit{B}\mathsf{\Phi }{\mathsf{\beta }}^0-b_{n-1}{\mathsf{\Phi }}^T\mathit{B}\mathsf{\Phi }{\mathsf{\beta }}^{-1}+{\alpha }_0{\mathsf{\Phi }}^T{\mathit{F}}^n,L+1⩽n⩽N;}\hfill \end{array}$$

(17)

$$\begin{array}{c}{\displaystyle {\mathsf{\varpi }}_d^n=\mathsf{\Phi }{\mathsf{\beta }}^n,{\varpi }_d^n=\mathsf{\xi }·{\mathsf{\varpi }}_d^n,1⩽n⩽N,}\hfill \end{array}$$

(18)

in which ${\mathsf{\varpi }}^n$ $(1⩽n⩽L)$ are the initial L solution vectors to Problem 3 and the matrix $\mathit{B}$ is given in Problem 3.

Remark 2.

It follows by contrasting Problem 4 with Problem 3 that the usual FE method (Problem 3) during each time iteration contains M unknowns, while the RDEFE method (Problem 4) during the same time iteration contains only d unknowns $(d\ll M)$, but both hold the same basic functions ${\left\{{\xi }_j(x,y)\right\}}_{j=1}^M$ so that the RDEFE method holds the same precision as the usual FE method under the situation that $\sqrt{{\lambda }_{d+1}}$ is small enough. In other words, though the dimensionality to Problem 4 is highly reduced, the precision of RDEFE solutions does not change. Therefore, the RDEFE method (Problem 4) is obviously superior to the FE method (Problem 3).

3.3. The Existence, Stability, and Convergence of the RDEFE Solutions

For the RDEFE solutions, we obtain the following results of existence, unconditional stability, and unconditional convergence.

Theorem 3.

When the source function f and the initial value functions ${\varphi }_0$ and ${\varphi }_1$ are sufficiently smooth, Problem 4 has a unique set of RDEFE solutions ${\left\{{\varpi }_d^n\right\}}_{n=1}^N$ meeting the following unconditional boundedness (i.e., unconditional stability):

$$\begin{array}{c}{\displaystyle \parallel {\varpi }_d^n{\parallel }_0⩽c,1⩽n⩽N.}\hfill \end{array}$$

(19)

Further, the set of solutions ${\left\{{\varpi }_d^n\right\}}_{n=1}^N$ is unconditionally convergent. Additionally, when $\varpi \in [H^{l+1}(\mathsf{\Omega })\cap H_0^1(\mathsf{\Omega })]$, the following error estimates hold:

$$\begin{array}{c}{\displaystyle \parallel \varpi (t_n)-{\varpi }_d^n{\parallel }_0⩽c\left(\Delta t^{3-\alpha }+\Delta t^{1-\alpha }{h}^{l+1}+\sqrt{{\lambda }_{d+1}}\right),1⩽n⩽N,}\hfill \end{array}$$

(20)

in which $\varpi (t_n)$ $(1⩽n⩽N)$ denotes the solution ϖ for Problem 2 at $t=t_n$.

Proof. 

(1) Analyze the existence of solutions for Problem 4.

By ${\mathsf{\varpi }}_d^n=\mathsf{\Phi }{\mathsf{\beta }}^n$, we can rewrite Problem 4 as the following system of equations:

$$\begin{array}{c}{\displaystyle {\mathsf{\varpi }}_d^n=\mathsf{\Phi }{\mathsf{\Phi }}^T{\mathsf{\varpi }}^n,{\varpi }_d^n={\mathsf{\varpi }}_d^n·\mathsf{\xi },1⩽n⩽L;}\hfill \end{array}$$

(21)

$$\begin{array}{c}{\displaystyle {\mathsf{\varpi }}_d^n=\sum _{j=1}^{n-1}(2b_j-b_{j+1}-b_{j-1})\mathit{B}{\mathsf{\varpi }}^{n-j-1}+(2b_0-b_1)\mathit{B}{\mathsf{\varpi }}^{n-1}}\hfill \\ {\displaystyle +b_0\mathit{B}{\mathsf{\varpi }}^0-b_{n-1}\mathit{B}{\mathsf{\varpi }}^{-1}+{\alpha }_0{\mathit{F}}^n,L+1⩽n⩽N;}\hfill \end{array}$$

(22)

$$\begin{array}{c}{\displaystyle {\varpi }_d^n=\mathsf{\xi }·{\mathsf{\varpi }}_d^n,L+1⩽n⩽N.}\hfill \end{array}$$

(23)

When $(1⩽n⩽L)$, for the known ${\mathsf{\varpi }}^n$ $(1⩽n⩽L)$ provided by Problem 3, from the Equation (21), we obtain a unique set of solutions ${\left\{{\varpi }_d^n\right\}}_{n=1}^L$.

When $(L+1⩽n⩽N)$, owing to the positive definiteness of the mass matrix $\mathit{B}$, we claim that Equation (22) has a unique set of solution vectors ${\left\{{\mathsf{\varpi }}_d^n\right\}}_{n=L+1}^N$. It follows by Equation (23) that there is a unique set of solutions ${\left\{{\varpi }_d^n\right\}}_{n=L+1}^N$. Thereupon, we conclude that Problem 4 has a unique set of solutions ${\left\{{\varpi }_d^n\right\}}_{n=1}^N$.

(2) Analyze the unconditional stability and unconditional convergence of RDEFE solutions to Problem 4.

When $1⩽n⩽L$, noting that $\parallel \mathsf{\Phi }{\mathsf{\Phi }}^T{\parallel }_2⩽c$, by Theorem 1, we obtain

$$\begin{array}{c}{\displaystyle \parallel {\varpi }_d^n{\parallel }_0=\parallel {\mathsf{\varpi }}_d^n{·\mathsf{\xi }\parallel }_0={\parallel \mathsf{\Phi }{\mathsf{\Phi }}^T{\mathsf{\varpi }}^n·\mathsf{\xi }\parallel }_0}\hfill \\ {\displaystyle ⩽\parallel \mathsf{\Phi }{\mathsf{\Phi }}^T{\parallel }_2\parallel {\mathsf{\varpi }}^n{·\mathsf{\xi }\parallel }_0⩽\parallel \mathsf{\Phi }{\mathsf{\Phi }}^T{\parallel }_2{\parallel {\varpi }_h^n\parallel }_0⩽c.}\hfill \end{array}$$

(24)

When $L+1⩽n⩽N$, using Lemma 1 and (5), from (22), we obtain

$$\begin{array}{c}{\displaystyle \parallel {\mathsf{\varpi }}_d^n\parallel ⩽ch\sum _{j=1}^{n-1}\parallel {\mathsf{\varpi }}_d^j\parallel +ch\parallel {\mathsf{\varpi }}_d^0\parallel +ch\parallel {\mathsf{\varpi }}_d^{-1}\parallel +c\Delta t^{\alpha }\parallel {\mathit{F}}^n\parallel .}\hfill \end{array}$$

(25)

By applying the Gronwall lemma to (25), we obtain

$$\begin{array}{c}\hfill {\displaystyle \parallel {\mathsf{\varpi }}_d^n\parallel ⩽(ch\parallel {\mathsf{\varpi }}_d^0\parallel +ch\parallel {\mathsf{\varpi }}_d^{-1}\parallel +c\Delta t^{\alpha }\parallel {\mathit{F}}^n\parallel )exp(chn)⩽c,L+1⩽n⩽N.}\end{array}$$

(26)

Thereupon, by ${\parallel \mathsf{\xi }\parallel }_0⩽c$, we obtain

$$\begin{array}{ccc}\hfill & & {\displaystyle \parallel {\varpi }_d^n{\parallel }_0=\parallel {\mathsf{\varpi }}_d^n{·\mathsf{\xi }\parallel }_0⩽\parallel {\mathsf{\varpi }}_d^n{\parallel ·\parallel \mathsf{\xi }\parallel }_0⩽c,L+1⩽n⩽N.}\hfill \end{array}$$

(27)

Therefore, the set of solutions ${\left\{{\varpi }_d^n\right\}}_{n=1}^N$ to Problem 4 is unconditionally bounded (i.e., unconditionally stable). Thus, by the Lax equivalence theorem (see [48], Theorem 2.3.18), we can conclude that the set of solutions ${\left\{{\varpi }_d^n\right\}}_{n=1}^N$ to Problem 4 is unconditionally convergent.

(3) Analyze the errors of RDEFE solutions for Problem 4.

Owing to ${\parallel \mathsf{\xi }\parallel }_{0,\infty }⩽c$, when $1⩽n⩽L$, by (15), we obtain

$$\begin{array}{c}{\displaystyle \parallel {\varpi }_h^n-{\varpi }_d^n{\parallel }_0={\parallel ({\mathsf{\varpi }}^n-{\mathsf{\varpi }}_d^n)·\mathsf{\xi }\parallel }_0}\hfill \\ {\displaystyle ⩽\parallel {\mathsf{\varpi }}^n-\mathsf{\Phi }{\mathsf{\Phi }}^T{\mathsf{\varpi }}^n{\parallel ·\parallel \mathsf{\xi }\parallel }_{0,\infty }⩽c\sqrt{{\lambda }_{d+1}},1⩽n⩽L.}\hfill \end{array}$$

(28)

While $L+1⩽n⩽N$, setting ${\mathit{E}}^n={\mathsf{\varpi }}^n-{\mathsf{\varpi }}_d^n$, using Lemma 1 and (5) as well as (28), from (10) and (22), we obtain

$$\begin{array}{c}{\displaystyle \parallel {\mathit{E}}^n\parallel ⩽\sum _{j=1}^{n-1}(2b_j-b_{j+1}-b_{j-1})\parallel \mathit{B}{\mathit{E}}^{n-j-1}\parallel +(2b_0-b_1)\parallel \mathit{B}{\mathit{E}}^{n-1}\parallel }\hfill \\ {\displaystyle ⩽ch\sum _{j=1}^L\parallel {\mathit{E}}^j\parallel +ch\sum _{j=L+1}^{n-1}\parallel {\mathit{E}}^j\parallel }\hfill \\ {\displaystyle ⩽chL\sqrt{{\lambda }_{d+1}}+ch\sum _{j=L+1}^{n-1}\parallel {\mathit{E}}^j\parallel ,L+1⩽n⩽N.}\hfill \end{array}$$

(29)

Applying Gronwall’s inequality to (29) yields that

$$\begin{array}{c}{\displaystyle \parallel {\mathsf{\varpi }}^n-{\mathsf{\varpi }}_d^n\parallel ⩽c\sqrt{{\lambda }_{d+1}}exp(c\Delta t(n-L-1))⩽c\sqrt{{\lambda }_{d+1}},L+1⩽n⩽N.}\hfill \end{array}$$

(30)

Owing to ${\parallel \mathsf{\xi }\parallel }_{0,\infty }⩽c$, by (30), we obtain

$$\begin{array}{c}{\displaystyle \parallel {\varpi }_h^n-{\varpi }_d^n{\parallel }_0={\parallel ({\mathsf{\varpi }}^n-{\mathsf{\varpi }}_d^n)·\mathsf{\xi }\parallel }_0}\hfill \\ {\displaystyle ⩽\parallel {\mathsf{\varpi }}^n-{\mathsf{\varpi }}_d^n{\parallel ·\parallel \mathsf{\xi }\parallel }_{0,\infty }}\hfill \\ {\displaystyle ⩽c\sqrt{{\lambda }_{d+1}},L+1⩽n⩽N.}\hfill \end{array}$$

(31)

Combining (9) in Theorem 1 with (28) and (31) yields (20). Theorem 3 is proved. □

Remark 3.

Owing to adopting the POD method to lower the dimension of the FE method, the errors of Theorem 3 have more one term $\sqrt{{\lambda }_{d+1}}$ than those of Theorem 1, which can be acted as a suggestion to elect the number of POD basis vectors. In order to ensure the RDEFE solutions to satisfy the accuracy requirement, we need only to elect d satisfying $\sqrt{{\lambda }_{d+1}}⩽max\{\Delta t^{3-\alpha },\Delta t^{1-\alpha }{h}^{l+1}\}$. A lot of numerical experiments (see, e.g., [32,33,34,35,36,37,38,39,40,41,42,43,44,45,46]) have shown that the eigenvalues ${\lambda }_j$ ($1⩽j⩽L$) to the matrix $\mathit{A}{\mathit{A}}^T$ would decrease to zero rapidly. Generally, when $d⩾6$, $\sqrt{{\lambda }_{d+1}}⩽max\{\Delta t^{3-\alpha },\Delta t^{1-\alpha }{h}^{l+1}\}$ so that it does not affect the total errors. Particularly, if the RDEFE solutions ${\varpi }_d^{n_0+1}$ obtained by Problem 4 at the time node $t_{n_0+1}$ cannot meet the desired precision, but ${\varpi }_d^n$ at the time nodes $t_n⩽t_{n_0}$ still meet the precision requirement, then we can retake a set of RDEFE solution vectors ${\mathsf{\varpi }}_d^{n_0-L+1},{\mathsf{\varpi }}_d^{n_0-L+2},\cdots ,{\mathsf{\varpi }}_d^{n_0}$ to combine a new snapshot matrix $\mathit{A}=({\mathsf{\varpi }}_d^{n_0-L+1},{\mathsf{\varpi }}_d^{n_0-L+2},\cdots ,{\mathsf{\varpi }}_d^{n_0})$, to calculate a new set of POD basis Φ, and to build a new RDEFE method to calculate out the RDEFE solutions meeting the precision requirement. Thus, we can calculate out the RDEFE solutions meeting the precision requirement at all time nodes. This is something that the classical FE method cannot do.

4. Two Numerical Examples

Here, in order to verify the correctness of theoretical results and to explain the superiority of the RDEFE method, we provide two numerical examples that the fractional Tricomi-type equation has an analytical solution, but it has no analytical solution when the source function or the initial value functions are complex.

4.1. Example 1

Assume that $\overline{\mathsf{\Omega }}=[0,1]\times [0,1]$, $f(x,y,t)=\left[\frac{2t^{2-\alpha }}{\mathsf{\Gamma }(3-\alpha )}+8{\pi }^2{t}^3\right]sin(2\pi x)sin(2\pi y)$, $t_e=1.0$, and ${\varphi }_0(x,y)={\varphi }_1(x,y)=0$. Thus, Problem 1 has an analytical solution denoted by $\varpi (x,y,t)=t^{2}sin(2\pi x)sin(2\pi y)$.

The triangulation ${\Im }_h$ on $\overline{\mathsf{\Omega }}$ is formed with the equilateral right triangles with all hypotenuses $h=\sqrt{2}/100$ and all right angles parallel to the axis, $\Delta t=1/100$, $\sqrt{{\lambda }_{d+1}}⩽{10}^{-4}$ and $l=1$. Thus, the $L^2$-morn error estimates for the classical FE solutions ${\varpi }_h^n$ and the RDEFE solutions ${\varpi }_d^n$ for the fractional Tricomi-type equation all can reach $O({10}^{-4})$ according to Theorems 1 and 3, theoretically.

When $\alpha =1.2$, $1.4$, $1.6$, and $1.8$, we find the RDEFE solutions by the following four steps:

(1) By solving Problem 3, based on experience, we find out four sets of the first $L=20$ classical FE solution coefficient vectors ${\mathsf{\varpi }}_{\alpha }^n$ $(1⩽n⩽20$ and $\alpha =1.2$, $1.4$, $1.6$, $1.8$) to construct the snapshot matrix ${\mathit{A}}_{\alpha }=({\mathsf{\varpi }}_{\alpha }^1,{\mathsf{\varpi }}_{\alpha }^2,\cdots ,{\mathsf{\varpi }}_{\alpha }^{20})$ ($\alpha =1.2$, $1.4$, $1.6$, $1.8$).

(2) By using the technique in Section 3.1, we compute four sets of eigenvalues ${\lambda }_{\alpha 1}⩾{\lambda }_{\alpha 2}⩾\cdots ⩾{\lambda }_{\alpha 20}⩾0$ for the matrix ${\mathit{A}}_{\alpha }^T{\mathit{A}}_{\alpha }$ as well as the corresponding set of orthonormal eigenvectors ${\mathsf{\phi }}_{\alpha i}$ $(1⩽i⩽20$ and $\alpha =1.2$, $1.4$, $1.6$, $1.8)$.

(3) By estimating, we deduce that $\sqrt{{\lambda }_{\alpha 7}}⩽3.23\times {10}^{-4}$ so that by using the formulas ${\mathsf{\phi }}_i=\mathit{A}{\mathsf{\psi }}_i/\sqrt{{\lambda }_i}$ $(1⩽n⩽6)$, we obtain four sets of POD basis vectors ${\mathsf{\Phi }}_{\alpha }=({\mathsf{\phi }}_{\alpha 1},{\mathsf{\phi }}_{\alpha 2},\cdots ,$ ${\mathsf{\phi }}_{\alpha 6})$ ($\alpha =1.2$, $1.4$, $1.6$, $1.8$).

(4) By substituting $\mathsf{\Phi }$ and the above data into Problem 4, we calculate four RDEFE solutions ${\omega }_{\alpha d}^{100}$ at $t=1.0$ and document the CPU runtime and errors $\parallel \omega (t_{100})-{\omega }_{\alpha d}^{100}{\parallel }_0$ ($\alpha =1.2$, $1.4$, $1.6$, $1.8$), listed on the third and fifth columns in

Table 1. The errors of the FE and RDEFE solutions and CPU runtime at $t=1.0$.

$\mathit{\alpha }$	Errors of FE	Errors of RDEFE	FE CPU	RDEFE CPU
	$\parallel \mathit{u}({\mathit{t}}_{\mathbf{100}})-{\mathit{u}}_{\mathit{h}}^{\mathbf{100}}{\parallel }_{\mathbf{0}}$	$\parallel \mathit{u}({\mathit{t}}_{\mathbf{100}})-{\mathit{u}}_{\mathit{d}}^{\mathbf{100}}{\parallel }_{\mathbf{0}}$	Runtime	Runtime
1.2	2.5118$\times {10}^{-4}$	4.2367$\times {10}^{-4}$	126.212 s	2.181 s
1.4	6.3096$\times {10}^{-4}$	4.4672$\times {10}^{-4}$	127.457 s	2.231 s
1.6	1.5849$\times {10}^{-3}$	1.6884$\times {10}^{-3}$	127.332 s	2.256 s
1.8	3.9811$\times {10}^{-3}$	3.8616$\times {10}^{-3}$	127.409 s	2.363 s

, where the contours of RDEFE solution at $t=1.0$ and $\alpha =1.2$ are shown in Figure 1a.

In order to show that the RDEFE method is superior to the classical FE method, we also use Problem 3 to calculate four classical FE solutions ${\omega }_{\alpha h}^{100}$ at $t=1.0$ and $\alpha =1.2$, $1.4$, $1.6$, $1.8$ and document the CPU runtime and errors $\parallel \omega (t_{100})-{\omega }_{\alpha h}^{100}{\parallel }_0$ ($\alpha =1.2$, $1.4$, $1.6$, $1.8$), listed on the second and fourth columns in Table 1, where the contours of the FE solution at $t=1.0$ and $\alpha =1.2$ are shown in Figure 1b. In the actual engineering calculations, we do not need to find all the classical FE solutions.

By comparing the two photos in Figure 1, we visibly see that Figure 1a and Figure 1b are highly similar, but computing the FDEFE solution saves more computer resource than computing the FE solution.

The data of Table 1 manifest that when $t=1$ (i.e., $n=100$) and $\alpha =1.2$, $1.4$, $1.6$, $1.8$, the numerical calculating errors to the RDEFE and FE solutions in Table 1 are coincided with the theoretical errors $({0.01}^{3-\alpha }+{0.01}^{1-\alpha }\times {0.01}^2)$, but the RDEFE method can greatly reduce unknowns and save the CPU runtime because the RDEFE method contains only 6 unknowns during each time iteration, while the FE method contains ${10}^4$ unknowns during the same time iteration. The data of Table 1 also show that the CPU runtime for computing the RDEFE solutions is far less than that for computing the FE solutions, and it can save time by about a factor of 54. Therefore, the RDEFE method is visibly superior to the FE method.

4.2. Example 2

Assume that $\overline{\mathsf{\Omega }}=[0,1]\times [0,1]$, the source function

$$\begin{array}{c}{\displaystyle f(x,y,t)=\left[\frac{\mathsf{\Gamma }(1+2\alpha )}{\mathsf{\Gamma }(1+\alpha )}{t}^{\alpha }+t^{2+\alpha }(400-{200}^2{(x-0.5)}^2-{200}^2{(y-0.5)}^2\right]}\hfill \\ {\displaystyle \times exp(-100({(x-0.5)}^2+{(y-0.5)}^2)),}\hfill \end{array}$$

$t_e=2.0$, and ${\varphi }_0(x,y)={\varphi }_1(x,y)=0$. Thus, Problem 1 has an analytical solution denoted by $\varpi (x,y,t)=t^{2\alpha }exp(-100({(x-0.5)}^2+{(y-0.5)}^2))$.

The triangulation ${\Im }_h$ on $\overline{\mathsf{\Omega }}$ is still formed with the equilateral right triangles with all hypotenuses $h=\sqrt{2}/100$ and all right angles parallel to the axis, $\Delta t=1/100$, $\sqrt{{\lambda }_{d+1}}⩽{10}^{-4}$, and $l=1$. Thus, both the $L^2$-morn error estimates for the classical FE solutions ${\varpi }_h^n$ and the RDEFE solutions ${\varpi }_d^n$ for the fractional Tricomi-type equation can also reach $O({10}^{-4})$ according to Theorems 1 and 3, theoretically.

When $\alpha =1.2$, $1.4$, $1.6$, and $1.8$, we still find the RDEFE solutions by the following four steps:

(1) By solving Problem 3, based on experience, we also solve out four sets of the first $L=20$ classical FE solution coefficient vectors ${\mathsf{\varpi }}_{\alpha }^n$ $(1⩽n⩽20$ and $\alpha =1.2$, $1.4$, $1.6$, $1.8$) to construct the snapshot matrix ${\mathit{A}}_{\alpha }=({\mathsf{\varpi }}_{\alpha }^1,{\mathsf{\varpi }}_{\alpha }^2,\cdots ,{\mathsf{\varpi }}_{\alpha }^{20})$ ($\alpha =1.2$, $1.4$, $1.6$, $1.8$).

(2) By resorting to the technique in Section 3.1, we also solve four sets of eigenvalues ${\lambda }_{\alpha 1}⩾{\lambda }_{\alpha 2}⩾\cdots ⩾{\lambda }_{\alpha 20}⩾0$ for the matrix ${\mathit{A}}_{\alpha }^T{\mathit{A}}_{\alpha }$ as well as the corresponding set of orthonormal eigenvectors ${\mathsf{\phi }}_{\alpha i}$ $(1⩽i⩽20$ and $\alpha =1.2$, $1.4$, $1.6$, $1.8)$.

(3) By estimating, we obtain that $\sqrt{{\lambda }_{\alpha 7}}⩽4.31\times {10}^{-4}$ so that by using the formulas ${\mathsf{\phi }}_i=\mathit{A}{\mathsf{\psi }}_i/\sqrt{{\lambda }_i}$ $(1⩽n⩽6)$, we also obtain four sets of POD basis vectors ${\mathsf{\Phi }}_{\alpha }=({\mathsf{\phi }}_{\alpha 1},{\mathsf{\phi }}_{\alpha 2},\cdots ,$ ${\mathsf{\phi }}_{\alpha 6})$ ($\alpha =1.2$, $1.4$, $1.6$, $1.8$).

(4) By substituting $\mathsf{\Phi }$ and the above data into Problem 4, we also calculate four RDEFE solutions ${\omega }_{\alpha d}^{200}$ at $t=2.0$, and document the CPU runtime and errors $\parallel \omega (t_{200})-{\omega }_{\alpha d}^{200}{\parallel }_0$ ($\alpha =1.2$, $1.4$, $1.6$, $1.8$), listed on the third and fifth columns in

Table 2. The errors of the FE and RDEFE solutions and CPU runtime at $t=2.0$.

$\mathit{\alpha }$	Errors of FE	Errors of RDEFE	FE CPU	RDEFE CPU
	$\parallel \mathit{u}({\mathit{t}}_{\mathbf{200}})-{\mathit{u}}_{\mathit{h}}^{\mathbf{200}}{\parallel }_{\mathbf{0}}$	$\parallel \mathit{u}({\mathit{t}}_{\mathbf{200}})-{\mathit{u}}_{\mathit{d}}^{\mathbf{200}}{\parallel }_{\mathbf{0}}$	Runtime	Runtime
1.2	3.1224$\times {10}^{-4}$	6.2367$\times {10}^{-4}$	252.424 s	4.263 s
1.4	8.4243$\times {10}^{-4}$	6.4672$\times {10}^{-4}$	254.864 s	4.471 s
1.6	2.8426$\times {10}^{-3}$	3.3442$\times {10}^{-3}$	254.671 s	4.523 s
1.8	6.4232$\times {10}^{-3}$	5.4323$\times {10}^{-3}$	254.816 s	4.725 s

.

In order to show that the RDEFE method is superior to the classical FE method, we also use Problem 3 to calculate four classical FE solutions ${\omega }_{\alpha h}^{200}$ at $t=2.0$ and $\alpha =1.2$, $1.4$, $1.6$, $1.8$, and document the CPU runtime and errors $\parallel \omega (t_{200})-{\omega }_{\alpha h}^{200}{\parallel }_0$ ($\alpha =1.2$, $1.4$, $1.6$, $1.8$), listed on the second and fourth columns in Table 2. In the actual engineering calculations, we do not need to find all the classical FE solutions.

The data of Table 2 also manifest that when $t=2$ (i.e., $n=200$) and $\alpha =1.2$, $1.4$, $1.6$, $1.8$, the numerical calculating errors to the RDEFE and FE solutions are also coincided with the theoretical errors $({0.01}^{3-\alpha }+{1.01}^{1-\alpha }\times {0.01}^2)$, but the RDEFE method can also lightly reduce unknowns and save the CPU runtime because the RDEFE method also contains only 6 unknowns during each time iteration, while the FE method also contains ${10}^4$ unknowns during the same time iteration. The data of Table 2 also show that the CPU runtime for finding the RDEFE solutions is much less than that for finding the FE solutions, and it can save time by about a factor of 54. Hence, the RDEFE method is feasible and effective to solve the the fractional Tricomi-type equation.

5. Conclusions and Prospect

Here, we have employed the POD technique to research the dimensionality reduction of the FE method of the fractional Tricomi-type equation. We have established the RDEFE method for the fractional Tricomi-type equation by means of the main POD basis vectors generated by the initial few FE solution vectors. We have also resorted to the matrix analysis to analyze the stability and convergence (errors) of the RDEFE solutions and employed two numerical examples to exhibit the advantages of the RDEFE method. The unknowns for the RDEFE method are far fewer than those for the FE method, so it can not only greatly lighten the calculated burden and lessen the computing error cumulation but also greatly save the CPU runtime in the calculation process. Especially, the dimensionality reduction of the solution vectors to the FE method for the fractional Tricomi-type equation is established for the first time here. Therefore, the RDEFE method for the fractional Tricomi-type equation is brand-new and distinguished from the existing reduced-order/dimension methods, for example, those mentioned in Section 1. This shows that the RDEFE method for the fractional Tricomi-type equation here is a completely new development.

Although only the RDEFE method for the fractional Tricomi-type equation is developed in this paper, the method here can be applied to other unsteady PDEs, even to more complicated real-world engineering problems. Moreover, the RDEFE method in this paper can also generalize to other numerical methods, such as the FD method, collocation method, and meshless method based on thin plate radial basis functions as mentioned in Section 1. Especially, alongside the FE and RDEFE methods, the DG method (see [22,23]) stands out as an efficient alternative for addressing the fractional Tricomi-type problem. This method demonstrates notable efficiency, and its synergy with reduced-dimension extrapolation techniques holds the promise of yielding even more enhanced outcomes. Therefore, the RDEFE method will have a broad application prospect.