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Uniqueness of system integration scheme of artificial intelligence technology in fractional differential mathematical equation

Published Online: 15 Jul 2022
Volume & Issue: AHEAD OF PRINT
Page range: -
Received: 18 Feb 2022
Accepted: 24 Apr 2022
Journal Details
License
Format
Journal
eISSN
2444-8656
First Published
01 Jan 2016
Publication timeframe
2 times per year
Languages
English
Introduction

With the continuous advancement of the accounting informatization process, various financial software have been developed one after another. At present, financial software is showing a mushrooming development momentum, there are many kinds of financial software on the market, there are also different emphases for different enterprise scales and industry types, but it can be seen that these financial software have many common characteristics. The current financial software mainly includes Uyouyou, Golden Disc, Golden Abacus, etc. Accounting information, it refers to a process of effectively integrating information technology and accounting development, that is to say, focusing on accounting information as the basis of resources, using network technology and long-distance communication technology, and computer technology, effective collection, transmission and storage of accounting information resources, and more specific practical functions such as processing applications, and gradually provide strong and powerful information resources for the good operation of enterprises, rational economic management, scientific control and decision-making.

Fractional calculus in anomalous diffusion, fluid mechanics, biology, signal processing, control theory and many other fields have a wide range of applications. In control science, the fractional order transfer function can be established by applying fractional order theory, which expands the research scope of control science. The fractional transfer function is equivalent to a linear fractional differential equation, so to calculate its time domain response, is equivalent to solving the corresponding fractional differential equation, and because the initial value condition defined by Caputo fractional order has clear physical meaning, therefore, linear Caputo fractional differential equations are widely used in practical engineering. Although some simple linear Caputo fractional differential equations have analytical solutions, however, these solutions are often very complex, which is not convenient for direct application in practical engineering, therefore, how to solve the numerical solutions of such equations has received extensive attention [1]. The author mainly studies numerical algorithms for solving linear Caputo fractional differential equations. In recent years, some numerical algorithms for solving fractional differential equations have appeared, for example, fractional linear multi-step algorithm, prediction correction algorithm, matrix algorithm, exponential algorithm, block diagram-based method diagrams, etc.; In addition, the author proposes a set of test problems to verify the actual effect of numerical algorithms [2].

Hamaguchi Y studied fractional convection-the finite difference method for dispersion fluid equations, and the finite difference method for fractional partial differential equations with left and right Grinward-Letnikov type fractional operators, and proved its stability properties [3]. Combined with the definition of fractional differential in Riesz space, Kheybari S et al, the finite difference method of the Riesz fractional-order convection-dispersion equation whose time direction is an integer order is studied [4]. Wang Z C discussed the finite difference method for complex fractional differential equations [5].

On the basis of this research, a high-precision numerical algorithm for solving linear Caputo fractional differential equations is proposed, it has been experimentally verified, the algorithm fusion effect is good.

Research Methods
The second-order finite difference scheme of the time fractional diffusion wave equation

Fractional calculus as a new modeling tool, are widely used in various fields, such as viscoelastic materials, hydrogeography, finance and Control Systems, etc. Compared with the traditional integer order model, the fractional order model represented by the fractional order differential equation in the accounting information financial software, memory effects and genetic effects in multiple materials engineering can be better described [6]. The fractional derivative in modeling anomalous diffusion in porous media, has a very important role. The first or second time derivative in the classical diffusion equation or wave equation, replaced by the fractional derivative of order a>0, a time fractional differential equation can be obtained. When the order 1<a<2, the time fractional diffusion wave equation can fuse the properties of diffusion and wave. When the order is 0<a<1, it is called the time fractional order diffusion equation, it is a derivative process of the continuous-time random walk problem, usually a non-Markov process.

Liu F uses a first-order approximation for the time and space directions, the first-order finite difference scheme is obtained, and the corresponding stability conditions are obtained. Yuste is based on the weighted average method for the constant diffusion equation (non-fractional), a new differential format is proposed. Lin and Xu solved the subdiffusion equation using spectral methods. In time, the finite difference method is used, and in space, the Legendre spectral method is applied, and the stability and convergence of this method are proved. Compared with many numerical solutions for solving time fractional diffusion equations, there are not many numerical methods for solving the time fractional diffusion wave equation [7]. Sun and Wu proposed a fully discrete difference scheme, this format works by introducing two new variables, transform the original equation into a lower-order equation, and give the error analysis of this format. When LiCP solves the time fractional diffusion wave equation, in the time direction, the finite difference method is used, in the spatial direction, applying the finite element method, numerical analyses of semi-discrete and fully discrete formats are also given. Based on the equivalent integral form of the fractional diffusion wave equation, Huang J designed a first-order accuracy in the time direction, a difference scheme with second-order precision in the spatial direction, and stability and convergence analysis of the scheme is given. However, there are few higher-order difference schemes for solving fractional diffusion wave equations, based on the given equivalent integral form of the fractional diffusion wave equation, using the Crank-Ni colson method and the fractional echelon method, a higher-order difference scheme of the fractional-order diffusion wave equation is constructed; This format achieves second-order accuracy in both time and space directions [8]. Numerical examples are used to further verify the second-order convergence and efficiency of the constructed scheme.

Fractional Diffusion Wave Equation and Its Equivalent Form

Consider the following time fractional diffusion wave equation OCDtαu(x,t)=c2u(x,t)x2+f(x,t)0xL,0tT,1<α<2 \matrix{{_O^CD_t^\alpha u\left({x,t} \right) = c{{{\partial ^2}u\left({x,t} \right)} \over {\partial {x^2}}} + f\left({x,t} \right)} \hfill & {0 \le x \le L,0} \hfill \cr} \le t \le T,1 < \alpha < 2

Its initial condition is u(x,0)=ϕ(x),ut(x,0)=φ(x),0xL u\left({x,0} \right) = \phi \left(x \right),{u_t}\left({x,0} \right) = \varphi \left(x \right),0 \le x \le L

The boundary conditions are u(0,t)=u(L,t)=0,t>0 u\left({0,t} \right) = u\left({L,t} \right) = 0,t > 0

Among them, c represents the diffusion fluctuation coefficient constant, x is the spatial direction variable, t is the time direction variable, f(x, t) is the source-sink term, OCDtα _O^CD_t^\alpha is the α-order Caputo fractional derivative, defined as follows OCDtαu(x,t)=1Γ(2α)ot(ts)1α2u(x,s)s2ds _O^CD_t^\alpha u\left({x,t} \right) = {1 \over {\Gamma \left({2 - \alpha} \right)}}\int_o^t {{{\left({t - s} \right)}^{1 - \alpha}}} {{{\partial ^2}u\left({x,s} \right)} \over {\partial {s^2}}}ds

According to the splitting property of the Caputo fractional derivative operator, the Caputo fractional derivative operator can be represented by operators OCDtα1 _O^CD_t^{\alpha - 1} and Dt, namely OCDtαu(x,t)=1Γ(1(α1))Ot(ts)(α1)su(u,s)sds=OCDtα1Dtu(x,t) \matrix{{_O^CD_t^\alpha u\left({x,t} \right)} \hfill & = \hfill & {{1 \over {\Gamma \left({1 - \left({\alpha - 1} \right)} \right)}}\int_O^t {{{\left({t - s} \right)}^{- \left({\alpha - 1} \right)}}{\partial \over {\partial s}}} {{\partial u\left({u,s} \right)} \over {\partial s}}ds} \hfill \cr {} \hfill & = \hfill & {_O^CD_t^{\alpha - 1}{D_t}u\left({x,t} \right)} \hfill \cr}

Theorem 1 Equation (2) is equivalent to the following partial differential-integral equation ut(x,t)=φ(x)+cΓ(α1)ot(ts)α22u(x,s)x2ds+F(x,t) {u_t}\left({x,t} \right) = \varphi \left(x \right) + {c \over {\Gamma \left({\alpha - 1} \right)}}\int_o^t {{{\left({t - s} \right)}^{\alpha - 2}}{{{\partial ^2}u\left({x,s} \right)} \over {\partial {x^2}}}ds + F\left({x,t} \right)}

Among, F(x,t)=Jtα1f(x,t) F\left({x,t} \right) = J_t^{\alpha - 1}\,f\left({x,t} \right) , Jtβ J_t^\beta is the Riemann-Liouville fractional integral operator with exponent > 0, defined as Jtβf(x,t)=1Γ(β)ot(ts)β1f(x,s)ds J_t^\beta f\left({x,t} \right) = {1 \over {\Gamma \left(\beta \right)}}\int_o^t {{{\left({t - s} \right)}^{\beta - 1}}\,f\left({x,s} \right)ds}

Construct a numerical format for equation (4). There are two advantages to constructing a finite difference scheme for equation (4): 1. It can reduce the smoothness requirement of the solution function in the time dimension; 2. Compared with the numerical format obtained by discretizing higher-order derivatives, the numerical format obtained by discretizing the integral is more stable.

Second-Order Finite Difference Scheme of Fractional Diffusion Wave Equation

In order to obtain the second-order difference format of Equation (4), first give 1Γ(α1)t0tn+1(tn+1s)α2g(s)ds {1 \over {\Gamma \left({\alpha - 1} \right)}}\int_{{t_0}}^{{t_{n + 1}}} {{{\left({{t_{n + 1}} - s} \right)}^{\alpha - 2}}g\left(s \right)ds}

The second-order precision approximation, the fractional echelon method. Among, g(s) is a second-order smooth function. First perform linear Lagrangian interpolation on g(s) on the interval [tk, tk+1], that is, g(s)g(tk)tk+1sτ+g(tk+1)stkτ g\left(s \right) \approx g\left({{t_k}} \right){{{t_{k + 1}} - s} \over \tau} + g\left({{t_{k + 1}}} \right){{s - {t_k}} \over \tau}

And substituting into (5), we have k=0n1Γ(α1)ktk+1(tn+1s)α2[g(tk)tk+1sτ+g(tk+1)stkτ]ds=k=0ng(tk)τΓ(α1)ktk+1(tn+1s)α2(tk+1s)ds+k=0ng(tk+1)τΓ(α1)ktk+1(tn+1s)α2(stk)ds=g(t0)Γ(α+1)τα1[α(n+1)α1(n+1)α+nα]+k=1ng(tk)Γ(α+1)τα1[(nk)α+(n+2k)α2(n+1k)α]+g(tn+1)Γ(α+1)τa1=τα1Γ(α+1)k=0n+1bkn+1g(tk) \matrix{{\sum\limits_{k = 0}^n {{1 \over {\Gamma \left({\alpha - 1} \right)}}\int_k^{{t_{k + 1}}} {{{\left({{t_{n + 1}} - s} \right)}^{\alpha - 2}}\left[{g\left({{t_k}} \right){{{t_{k + 1}} - s} \over \tau} + g\left({{t_{k + 1}}} \right){{s - {t_k}} \over \tau}} \right]ds}}} \hfill \cr {= \sum\limits_{k = 0}^n {{{g\left({{t_k}} \right)} \over {\tau \Gamma \left({\alpha - 1} \right)}}\int_k^{{t_{k + 1}}} {{{\left({{t_{n + 1}} - s} \right)}^{\alpha - 2}}\left({{t_{k + 1}} - s} \right)ds + \sum\limits_{k = 0}^n {{{g\left({{t_{k + 1}}} \right)} \over {\tau \Gamma \left({\alpha - 1} \right)}}\int_k^{{t_{k + 1}}} {{{\left({{t_{n + 1}} - s} \right)}^{\alpha - 2}}\left({s - {t_k}} \right)ds}}}}} \hfill \cr {= {{g\left({{t_0}} \right)} \over {\Gamma \left({\alpha + 1} \right)}}{\tau ^{\alpha - 1}}\left[{\alpha {{\left({n + 1} \right)}^{\alpha - 1}} - {{\left({n + 1} \right)}^\alpha} + {n^\alpha}} \right]} \hfill \cr {+ \sum\limits_{k = 1}^n {{{g\left({{t_k}} \right)} \over {\Gamma \left({\alpha + 1} \right)}}{\tau ^{\alpha - 1}}\left[{{{\left({n - k} \right)}^\alpha} + {{\left({n + 2 - k} \right)}^\alpha} - 2{{\left({n + 1 - k} \right)}^\alpha}} \right] + {{g\left({{t_{n + 1}}} \right)} \over {\Gamma \left({\alpha + 1} \right)}}{\tau ^{a - 1}}}} \hfill \cr {= {{{\tau ^{\alpha - 1}}} \over {\Gamma \left({\alpha + 1} \right)}}\sum\limits_{k = 0}^{n + 1} {{b_k}^{n + 1}g\left({{t_k}} \right)}} \hfill \cr}

Among, b0n+1=α(n+1)α1(n+1)α+nα1bk(n+1)=(nk)α+(n+2k)α2(n+1k)α,k=1,2,,n,bn+1(n+1)=1 \matrix{{b_0^{n + 1} = \alpha {{\left({n + 1} \right)}^{\alpha - 1}} - {{\left({n + 1} \right)}^\alpha} + {n^{\alpha - 1}}} \hfill \cr {b_k^{\left({n + 1} \right)} = {{\left({n - k} \right)}^\alpha} + {{\left({n + 2 - k} \right)}^\alpha} - 2{{\left({n + 1 - k} \right)}^\alpha},\,k = 1,2, \ldots,n,\,b_{n + 1}^{\left({n + 1} \right)} = 1} \hfill \cr}

Lemma 1 Let g(t) ∈ C2 ([0, T]), then there is a constant C such that |1Γ(α1)t0tn+1(tn+1s)α2g(s)dsτα1Γ(α1)k=0n+1bkn+1g(tk)|Cτ2 \left| {{1 \over {\Gamma \left({\alpha - 1} \right)}}\int_{{t_0}}^{{t_{n + 1}}} {{{\left({{t_{n + 1}} - s} \right)}^{\alpha - 2}}g\left(s \right)ds - {{{\tau ^{\alpha - 1}}} \over {\Gamma \left({\alpha - 1} \right)}}\sum\limits_{k = 0}^{n + 1} {b_k^{n + 1}g\left({{t_k}} \right)}}} \right| \le C{\tau ^2}

Prove: According to the error of linear Lagrangian interpolation, we can get |1Γ(α1)t0tn+1(tn+1s)α2g(s)dsτα1Γ(α1)k=0n+1bn(n+1)g(tn)|k=0n12Γ(α1)tktk+1(tn+1s)α2|g(ξk)(stk)(stk+1)|dsCτ2 \matrix{{\left| {{1 \over {\Gamma \left({\alpha - 1} \right)}}\int_{{t_0}}^{{t_{n + 1}}} {{{\left({{t_{n + 1}} - s} \right)}^{\alpha - 2}}g\left(s \right)ds - {{{\tau ^{\alpha - 1}}} \over {\Gamma \left({\alpha - 1} \right)}}\sum\limits_{k = 0}^{n + 1} {b_n^{\left({n + 1} \right)}g\left({{t_n}} \right)}}} \right|} \hfill \cr {\le \sum\limits_{k = 0}^n {{1 \over {2\Gamma \left({\alpha - 1} \right)}}\int_{{t_k}}^{{t_{k + 1}}} {{{\left({{t_{n + 1}} - s} \right)}^{\alpha - 2}}\left| {g\left({{\xi _k}} \right)\left({s - {t_k}} \right)\left({s - {t_{k + 1}}} \right)} \right|ds \le C{\tau ^2}}}} \hfill \cr}

In order to establish the finite difference scheme, the network is first divided. In the spatial direction, make xj = jh, j = 0,1,…,M, where h = L/M is a positive integer. In the direction of time, make tk = , k = 0,1,…,N, among, τ = T/N, N is a positive integer. Let u(x, t) be a function with a certain smoothness in both time and space, and let ujk=u(xj,tk) u_j^k = u\left({{x_j},{t_k}} \right) , then u(xj,tk+1/2)=ujk+1+ujk2+O(τ2) u\left({{x_j},{t_{k + 1/2}}} \right) = {{u_j^{k + 1} + u_j^k} \over 2} + O\left({{\tau ^2}} \right)

If the function f(x, t) has second-order smoothness in the time direction, then through Taylor expansion, we can get Jtα1f(xj,t1/2)=12Jtα1f(xj,t1)+O(τ2) J_t^{\alpha - 1}f\left({{x_j},{t_{1/2}}} \right) = {1 \over 2}J_t^{\alpha - 1}f\left({{x_j},{t_1}} \right) + O\left({{\tau ^2}} \right)

Now consider the dispersion of equation (4) at point (xj, tn+1) ut(xj,tn+1)=φ(xj)+cΓ(α1)0tn+1(tn+1s)α22u(xj,s)x2ds+F(xj,tn+1) {u_t}\left({{x_j},{t_{n + 1}}} \right) = \varphi \left({{x_j}} \right) + {c \over {\Gamma \left({\alpha - 1} \right)}}\int_0^{{t_{n + 1}}} {{{\left({{t_{n + 1}} - s} \right)}^{\alpha - 2}}{{{\partial ^2}u\left({{x_j},s} \right)} \over {\partial {x^2}}}} ds + F\left({{x_j},{t_{n + 1}}} \right)

Further discrete space second derivative, we can get: ujn+1ujnτ=φ+cτα12Γ(α1)k=0n+1bk(n+1)uj+1k2ujk+uj1kh2 {{u_j^{n + 1} - u_j^n} \over \tau} = \varphi + {{c{\tau ^{\alpha - 1}}} \over {2\Gamma \left({\alpha - 1} \right)}}\sum\limits_{k = 0}^{n + 1} {{b_k}^{\left({n + 1} \right)}} {{u_{j + 1}^k - 2u_j^k + u_{j - 1}^k} \over {{h^2}}}

Ignoring the above truncation error, and suppose that Ujn+1 U_j^{n + 1} is an approximation of ujn+1 u_j^{n + 1} , then Ujn+1Ujn=τφj+τ2(Fjn+1+Fjn)+cτα2Γ(α+1)k=0n+1bkn+1Uj+1k2Ujk+Uj1kh2 U_j^{n + 1} - U_j^n = \tau {\varphi _j} + {\tau \over 2}\left({F_j^{n + 1} + F_j^n} \right) + {{c{\tau ^\alpha}} \over {2\Gamma \left({\alpha + 1} \right)}}\sum\limits_{k = 0}^{n + 1} {b_k^{n + 1}} {{U_{j + 1}^k - 2U_j^k + U_{j - 1}^k} \over {{h^2}}}

Remember d=cτα2Γ(α+1)h2 d = {{c{\tau ^\alpha}} \over {2\Gamma \left({\alpha + 1} \right){h^2}}} , and further sort out the above formula to get dUj1n+1+(2d+1)UjndUj+1n+1=dk=0n(bkn+1+bkn)(Uj+1k2Ujk+Uj1k) - dU_{j - 1}^{n + 1} + \left({2d + 1} \right)U_j^n - dU_{j + 1}^{n + 1} = d\sum\limits_{k = 0}^n {\left({b_k^{n + 1} + b_k^n} \right)\left({U_{j + 1}^k - 2U_j^k + U_{j - 1}^k} \right)}

Combined with homogeneous boundary conditions, the above equation can be written in the following matrix form (2d+1dd2d+100)(U1n+1U2n+1UM1n+1)=(U1nU2nUM1n)+k=0n(bkn+1+bkn) \left({\matrix{{2d + 1} \hfill & {- d} \hfill \cr {- d} \hfill & {2d + 1} \hfill \cr \vdots \hfill & \vdots \hfill \cr 0 \hfill & 0 \hfill \cr}} \right)\left({\matrix{{U_1^{n + 1}} \hfill \cr {U_2^{n + 1}} \hfill \cr \vdots \hfill \cr {U_{M - 1}^{n + 1}} \hfill \cr}} \right) = \left({\matrix{{U_1^n} \hfill \cr {U_2^n} \hfill \cr \vdots \hfill \cr {U_{M - 1}^n} \hfill \cr}} \right) + \sum\limits_{k = 0}^n {\left({b_k^{n + 1} + b_k^n} \right)}

Since the matrix of the linear equation system is tridiagonal, it can be solved by the chasing method to improve the solving efficiency.

High-precision algorithms for solving equations

The missing initial condition can be calculated by applying equation (20), according to these initial value conditions, the numerical solution of equation (10) can be calculated, and the high-precision numerical algorithm is directly given below:

Algorithm 1 A high-precision numerical algorithm for solving linear Caputo fractional differential equations.

(1) Select the step size h and order p of the algorithm, if the order p is greater than the number q of initial conditions of the equation, then apply equation (20) to calculate the missing numerical condition;

(2) According to the initial value condition calculated in step (1), construct the auxiliary function ŷ(t) of the form (13), decompose the solution y(t) of the equation into y¯(t)+y^(t) \bar y\left(t \right) + \hat y\left(t \right) , transform the solved equation (10) into equation (14);

(3) Apply the recursive formula (6) to calculate the coefficient ωk(α, p), substitute the result into equation (15) to calculate the numerical solution of y¯(t) \bar y\left(t \right) , then calculate the numerical solution of y(t) according to y(t)=y¯(t)+y^(t) y\left(t \right) = \bar y\left(t \right) + \hat y\left(t \right) .

Apply the basic algorithm and the high-precision algorithm, calculate the numerical solution of the following equation, and compare the calculation error. y(3)(t)+oCt2.5y(t)+y(2)(t)+4y(1)(t)+oCt0.5y(t)+4y(t)=6cos(t) {y^{\left(3 \right)}}\left(t \right) + _o^C\wp_t^{2.5}y\left(t \right) + {y^{\left(2 \right)}}\left(t \right) + 4{y^{\left(1 \right)}}\left(t \right) + _o^C\wp_t^{0.5}y\left(t \right) + 4y\left(t \right) = 6\cos \left(t \right)

The initial value condition is y(0) = 1, y(1)(0) = 1, y(2)(0) = −1. The analytical solution of the equation can be verified as y(t)=2sin(t+π/4) y\left(t \right) = \sqrt 2 \sin \left({t + \pi /4} \right)

In this way, the accuracy of the numerical solution can be checked.

First, the basic algorithm is applied to calculate the numerical solution of equation (21).

Construct the auxiliary function ŷ(t) = 1 + tt2 / 2 according to the initial value condition of the equation, equation (15) is then applied to calculate the numerical solution of the equation. Select the step size h=0.1, set the order P to 1 to 5, the calculation error is shown in Table 1, when the order p is equal to 4 and 5, the calculation error does not decrease, indicating that the order of the basic algorithm is limited by the number of initial value conditions of the equation.

Computational error of the basic algorithm

P=1 P=2 P=3 P=4
t=5 8.2×10−2 6.5×10−2 3.8×10−3 2.6×10−3
t=10 4.9×10−2 4.7×10−3 1.4×10−3 1.9×10−3
t=15 4.3×10−2 2.9×10−3 3.9×10−3 3.1×10−3

The numerical solution of equation (21) is calculated by applying the high precision algorithm below. Also set the order p of the algorithm to 1 to 5, when p≤3, the auxiliary function of the form (13) is constructed directly according to the initial value condition of the equation, when p>3, the missing initial value condition is calculated by applying equation (20), and the auxiliary function of the form (18) is constructed, and the obtained results are shown in Table 2.

Computational errors of high-precision algorithm

P=1 P=2 P=3 P=4
t=5 8.2×10−2 6.5×10−2 3.8×10−3 8.6×10−3
t=10 4.9×10−2 4.7×10−3 1.8×10−3 1.9×10−3
t=15 4.3×10−2 2.9×10−3 3.9×10−3 5.6×10−3

Equation (21) is transformed into the equation under the zero initial value condition using the auxiliary function in Table 2, the numerical solution is then calculated using equation (15). In order to compare with the basic algorithm, the same step size h = 0.1 is selected here, and the calculation error is shown in Table 3.

Take h = 0.001, for different τ, use this format to solve the error and convergence order of the fractional diffusion wave equation

α = 1.5 α = 1.8
τ error Convergence order error Convergence order
1/4 1.8921e-2 1.9625e-2 1.9654
1/8 4.9251e-3 1.9548 5.6245e-3 1.8395
1/16 1.4125e-3 1.9354 1.2584e-3 1.9578

Take τ = 0.001, for different h, errors and Convergence Orders Produced by Solving Fractional Diffusion Wave Equations Using This Format

α = 1.5 α = 1.8
h error Convergence order error Convergence order
1/4 1.8921e-2 1.9625e-2 1.9654
1/8 4.9251e-3 1.9548 5.6245e-3 1.8395
1/16 1.4125e-3 1.9354 1.2584e-3 1.9578

Comparing the calculation errors in Table 1 and Table 2, when the order p≤3, the computational error of the two algorithms is the same, when p > 3, the calculation error of the high-precision algorithm, significantly smaller than the calculation error of the basic algorithm. This result shows that, the order of the high-precision algorithm is not limited by the number of initial value conditions of the equation, which can significantly improve the calculation accuracy.

Results and Analysis
The finite element method of several types of fractional partial differential equations is studied

Firstly, the finite element method of the time-space Riesz fractional convection-diffusion equation is studied. oCDtαu(x,t)=2u(x,t)s2+f(x,t),0x1,0t1 _o^CD_t^\alpha u\left({x,t} \right) = {{{\partial ^2}u\left({x,t} \right)} \over {\partial {s^2}}} + f\left({x,t} \right),0 \le x \le 1,0 \le t \le 1

For the Caputo-type fractional derivative in time, using the Diethelm backward difference method to discretize and get the convergence order in the time direction; In the spatial direction, the discrete format of the finite element method is given for the Riesz fractional derivative, and the convergence order of the spatial direction is obtained. Further for the fully discrete form of the above equation, the unconditional stability of the method is proved, and the convergence of the method is verified by a numerical example. Then, the finite element method of fractional partial differential equations with time complex type is studied, where P(oCDt) P\left({_o^C{D_t}} \right) is the time complex Caputo fractional differential operator. P(oCDt)u(t,c)Δxu(t,x)=f(t,x) P\left({_o^C{D_t}} \right)u\left({t,c} \right) - {\Delta _x}u\left({t,x} \right) = f\left({t,x} \right)

The discretization of the complex fractional derivative in the time direction adopts the Diethelm backward difference method, applying the classical finite element method to the spatial orientation, respectively, the convergence orders for the temporal and spatial orientations are obtained, the unconditional stability of the fully discrete scheme is proved. Also for the above conclusion, an effective numerical example is given to support, and also compared with the existing fractional numerical methods to verify the effectiveness of the method. Then, the finite element method of the time-space complex Riesz fractional convection-diffusion equation is studied. P(oCDt)u(t,x)=kβRD|x|βu(t,x)+kyRD|x|yu(t,x)+f(t,x) P\left({_o^C{D_t}} \right)u\left({t,x} \right) = {k_\beta}^RD_{\left| x \right|}^\beta u\left({t,x} \right) + {k_y}^RD_{\left| x \right|}^yu\left({t,x} \right) + f\left({t,x} \right)

Here, the Diethelm backward difference method is used to give the convergence order in the time direction, the fractional-order finite element method is applied in the spatial direction and the convergence order is given, which proves the unconditional stability of the fully discrete scheme. Numerical examples are used to verify that the convergence order of the method is consistent with the theoretical results.

Consider fractional integro-differential equations with weak singular kernels
oCDtαy(t)=g(t)+p(t)y(t)+otq(t,s)y(s)ds _o^CD_t^\alpha y\left(t \right) = g\left(t \right) + p\left(t \right)y\left(t \right) + \int_o^t {q\left({t,s} \right)y\left(s \right)ds}

The slicing polynomial configuration method is used to obtain the optimal convergence order. The results show, when the exact solution is sufficiently smooth, the optimal convergence order can be obtained, and when the exact solution is not smooth enough, the numerical solution can reach the optimal convergence order by transforming the method. and then, four specific configuration methods are applied to solve the example, and the theoretically expected numerical results are obtained, that is, when the exact solution is sufficiently smooth, the convergence order of the trapezoidal configuration method reaches 2, the convergence order of the Simpson configuration method is up to 3, the convergence order of the Newton3/8 configuration method is up to 4, and the convergence order of the Cotes configuration method is up to 5. The above numerical results are consistent with the theoretical results [9].

In fact, there is still a lot of room for the study of numerical methods of fractional differential equations. The time-space fractional partial differential equation finite difference method studied by the author, it can also be applied to time-space composite fractional partial differential equations and time-space mixed fractional partial differential equations [10].

(1) The time-space complex Riesz fractional wave-diffusion equation when 1< as < … < a1 < a < 2;

(2) The time-space mixed Riesz fractional wave-diffusion equation when 0 < as < … < aso < 1 < aso−1 < … < a1 < a < 2.

Of course, this method in solving the above equation, the specific problems that may be encountered need to be further studied. For the fractional partial differential equation finite element method, because of the nonlocal nature of fractional differential operators, therefore, the finite element method of constructing irregular element meshes is very difficult. At the same time, due to the difference between the adjoint operator of the fractional differential operator and the integer order, there are also a lot of things to keep in mind when constructing a variational format. The finite element method in this paper is extended to the above-mentioned composite and hybrid time-space Riesz fractional wave-diffusion equations, it is also a subject worthy of study.

Due to the particularity of the definition of fractional differential operators, more attention should be paid to the conversion between the definitions of the operators, for example, the Riemann-Liouville fractional derivative with respect to the constant is not zero, this is very different from the usual integer order derivatives. Moreover, the definitions of fractional order are not completely equivalent, this is where attention must be paid in theoretical and numerical solutions. Compared with integer-order differential equations, the smoothness of exact solutions of fractional-order differential equations is often lower, when some higher-order methods are applied, the desired convergence effect is often not achieved. therefore, it is a very valuable and practical subject to study numerical methods of high convergence order for fractional differential equations. In recent years, fractional differential equation models have emerged one after another in various scientific research fields, for specific numerical methods of fractional ordinary differential equations and fractional partial differential equations with strong application background, the research on its stability and convergence is also very important.

Verifying the Finite Difference Format

In order to verify that the above finite difference scheme has second-order accuracy, the initial conditions for solving the following fractional-order diffusion wave equation are: u(x,0)=ut(x,0)=0,0x1 u\left({x,0} \right) = {u_t}\left({x,0} \right) = 0,0 \le x \le 1

The boundary conditions are u(0,t)=u(1,t)=0,0<t1 u\left({0,t} \right) = u\left({1,t} \right) = 0,0 < t \le 1

The true solution to this problem is u(x,t)=t2+αsin[5x(1x)] u\left({x,t} \right) = {t^{2 + \alpha}}\sin \left[{5x\left({1 - x} \right)} \right]

When taking the time and space step h = τ = 0.05, t =1, a comparison plot giving the numerical solution and the exact solution is shown in Figure 1. Figure 1 shows that the numerical solution and the exact solution basically coincide, that is, the validity of the given difference scheme.

Figure 1

t = 1, step size h = τ = 0.05, the comparison between the numerical solution and the exact solution

Conclusion

The author starts from the equivalent integral form of the fractional diffusion wave equation, a second-order finite-difference scheme of the fractional-order diffusive wave equation is constructed, numerical experiments show that the scheme has good accuracy and efficiency. In numerical solution, discrete integral equations have better numerical stability than differential equations, therefore, the format also has better stability. The calculation results of the numerical convergence order are shown in Tables 2 and 3. In Table 1, taking a small enough space step, in this way, the error caused by the spatial direction can be ignored, and the convergence order of the format in the time direction can be observed. When taking different fractional derivative indices a=1.5 and a=1.8, it can be seen that the difference scheme constructed by the author has second-order accuracy in the time direction, which is in line with the expected results. Similarly, Table 2 shows that the difference scheme also has second-order accuracy in the spatial direction. Taking the time and space steps of h=τ=0.05 and t=1, a comparison diagram of the numerical solution and the exact solution is given, as shown in Figure 1.

In addition, using the idea of spectral delay correction, a high-precision numerical algorithm for solving the initial value problem of fractional differential equations is given. Through numerical experiments, it is found that the numerical solution obtained by this method has high precision and small amount of calculation. However, the theoretical analysis of the algorithm needs further research.

Figure 1

t = 1, step size h = τ = 0.05, the comparison between the numerical solution and the exact solution
t = 1, step size h = τ = 0.05, the comparison between the numerical solution and the exact solution

Computational error of the basic algorithm

P=1 P=2 P=3 P=4
t=5 8.2×10−2 6.5×10−2 3.8×10−3 2.6×10−3
t=10 4.9×10−2 4.7×10−3 1.4×10−3 1.9×10−3
t=15 4.3×10−2 2.9×10−3 3.9×10−3 3.1×10−3

Computational errors of high-precision algorithm

P=1 P=2 P=3 P=4
t=5 8.2×10−2 6.5×10−2 3.8×10−3 8.6×10−3
t=10 4.9×10−2 4.7×10−3 1.8×10−3 1.9×10−3
t=15 4.3×10−2 2.9×10−3 3.9×10−3 5.6×10−3

Take τ = 0.001, for different h, errors and Convergence Orders Produced by Solving Fractional Diffusion Wave Equations Using This Format

α = 1.5 α = 1.8
h error Convergence order error Convergence order
1/4 1.8921e-2 1.9625e-2 1.9654
1/8 4.9251e-3 1.9548 5.6245e-3 1.8395
1/16 1.4125e-3 1.9354 1.2584e-3 1.9578

Take h = 0.001, for different τ, use this format to solve the error and convergence order of the fractional diffusion wave equation

α = 1.5 α = 1.8
τ error Convergence order error Convergence order
1/4 1.8921e-2 1.9625e-2 1.9654
1/8 4.9251e-3 1.9548 5.6245e-3 1.8395
1/16 1.4125e-3 1.9354 1.2584e-3 1.9578

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