A Hybrided Method for Temporal Variable-Order Fractional Partial Differential Equations with Fractional Laplace Operator

: In this paper, we present a more general approach based on a Picard integral scheme for non-linear partial differential equations with a variable time-fractional derivative of order α ( x , t ) ∈ ( 1,2 ) and space-fractional order s ∈ ( 0,1 ) , where v = u ′ ( t ) is introduced as the new unknown function and u is recovered using the quadrature. In order to get rid of the constraints of traditional plans considering the half-time situation, integration by parts and the regularity process are introduced on the variable v . The convergence order can reach O ( τ 2 + h 2 ) , different from the common L 1,2 − α schemes with convergence rate O ( τ 2,3 − α ( x , t ) ) under the infinite norm. In each integer time step, the stability, solvability and convergence of this scheme are proved. Several error results and convergence rates are calculated using numerical simulations to evidence the theoretical values of the proposed method


Introduction
One of the most useful and applicable generalizations of the ordinary derivatives of integer orders and integrals is the fractional calculus.Utilizing the models based on derivatives of fractional orders in several branches of science and engineering is a major study of many mathematicians and physicists [1][2][3][4].Fractional partial differential equations (FPDEs), particularly space-and time-fractional equations, have been widely studied to demonstrate the existence of solutions and the validity of these problems [5][6][7].
In this paper, we will consider the nonlinear multi-dimensional fractional advectiondiffusion equation involving variable time-space orders.
Nonetheless, the analytical solutions for the majority of fractional partial differential equations remain elusive.Consequently, over the past two decades, a significant portion of researchers has concentrated on approximations and numerical methods for tackling these fractional-order systems.Many researchers have particularly emphasized difference schemes, given their superior stability and solvability compared to alternative approaches.For example, Adomian decomposition method (ADM) was utilized to approximate the time-space FPDE with the Caputo sense in [13].The space-time fractional advection-diffusion equations are linear partial pseudo-differential equations with Feller space-fractional differentiation derivatives and are used to model transport at the earth's surface in [14].The space-time fractional diffusion equation with Caputo time-fractional derivative α and Riesz-Feller space-fractional derivative β is studied in [15], and the convergence rate is O(τ 3−α + h 3−β ).In [16], multi-dimensional space-time variable-order fractional Schrödinger equations are introduced with a Caputo time-fractional derivative and Riesz-Feller space-fractional derivative.The time-space fractional telegraph equation with local fractional derivatives is investigated in [17].Among them, the fractional advection-diffusion equation, as an important model, has been widely studied in engineering applications and fast calculations.In [18], the fractional advection-diffusion equation model is a new approach to describe the vertical distribution of suspended sediment concentration in steady turbulent flow.Gu has successively published multiple outstanding achievements in rapid calculation in [19][20][21].
The foundational time model, tailored for discrete schemes, often requires the incorporation of half-time steps for the variable u due to their suitability in representing derivatives and integration processes.Conversely, conventional energy methods for u typically involve its coupling with the variable v.In [22], we pivot our focus to primarily address the situation of v.We meticulously establish the system's stability and solvability and demonstrate that the convergence of u is of second order.It is worth noting that the derivative operation is inherently unbounded, whereas integration acts as a refining operator.In Section 4 of this paper, we substantiate the effectiveness of the integral formula when coupled with the difference scheme, illustrating its robust stability.

Preliminaries and Some Lemmas
Consider the set {t n |n ≥ 0}, which comprises uniformly spaced time intervals with t n = nτ and τ > 0. Suppose where u n i and v n i correspond to the function values and their respective first derivative values at time t n for the point x i , while g(t) represents a smooth function defined within the interval (0, T]. Here is the difference scheme we will explore for Equation (1): Fractal Fract.2024, 8, 105 where and

Time-Discretization of the Present Scheme
To establish the time-fractional derivation in the current scheme, the following lemmas are required.
where a l (t n ) is defined in (8).
By rearranging the order of summation, we have where Obviously, α is the largest number in the series α(t n ) in the formula of a n l , and the same procedure may be easily adapted to obtain corresponding behaviours for b n l .□ where a n l is defined in (8), 1 < α(t n ) < 2 and n > 1.
Proof.Obviously, it suffices to verify Fractal Fract.2024, 8, 105 5 of 14 Combining this with the result of Lemma 1, we obtain where where ∆ c is a differential matrix format of the Laplace operator (−∆).Then the matrix representation of the operator (−∆) s can be given by ∆ s c .Thus, we use the finite-difference approximation of (−∆) s : Due to the similarity of the 2-norm, we will use the ∥.∥ ∞ format to replace the above norm for convenience in the following process.And we obtain the approximate error of the difference representation under this norm.
From the differential discretization scheme, we have the approximate property of the matrix ∆ c .Assuming that u ∈ C 4 (Ω), we obtain Theorem 1.If u ∈ C 4 (Ω), the following error estimate holds: Firstly, we estimate the first term.Using the notation The last is the estimate of Considering the boundedness of ∑ ∞ j=1 u j j Taking ( 20) into (18), we obtain the estimate Secondly, we estimate the second term, From the property of the difference matrix ∆ c , we have ∥∆ s c ∥ ∞ ≤ 4d.Then, Finally, inserting the inequalities into each other implies the following error estimate:

Analysis of the Time-Space Discretization of the Present Scheme
Based on Lemma 2 and Equations ( 6) and ( 7), we have and where Taking into account the distinct characteristics of the nonlinear part N u(x, t), When the above results are substituted into (1), the following is obtained: where and Fn i involves the original F n i and the approximate value of N u(x, t).We also define where D is the metric coefficient of Ω.Moreover, when g(Γ) = 0, it is observed that:

Main Results
In this section, the solvability, stability and convergence of this scheme are proved.The notation for the inner product discretization form is used as follows: where w i is described as the Gauss weight at the point x i .Subsequently, some lemmas are introduced.
Lemma 3. Let 0 ≤ s ≤ 1 and f , (−∆) s f ∈ L p .Then, for any arbitrary p ≥ 1, there holds Fractal Fract.2024, 8, 105 8 of 14 Lemma 4. Suppose {v n } is the solution of We have Proof.Summing i and n from 1 to m and from 1 to N by multiplying both sides of (34) with w i , the following is obtained: Using Lemma 1, we have and 1 When the boundary conditions in (34) are applied, it results in v n i (∂Ω) = 0. Consequently, Fractal Fract.2024, 8, 105 9 of 14 In addition, Substituting (37)-( 41) into (36), we obtain Replacing v with u in (43), the ensuing inequality is derived: (44) □ Theorem 2. Uniqueness in solvability is achieved by the difference scheme (6) and (7).
Proof.As (6) and ( 7) constitute the linear algebraic equations at different time t i , it is sufficient for the corresponding homogeneous equations to be demonstrated: ), Only a zero solution is attainable.Through the utilization of Lemma 4, the following is derived: Combining the boundary conditions in (2), we obtain This completes the proof.□ Theorem 3. Let u(x, t) and v(x, t) be the solution of ( 6) and (7).Then, the following inequality holds: where C, D represent constant numbers.

Numerical Experiments
In this section, some experiments showcase the effectiveness of the current scheme.

One-Dimensional Space-Fractional Laplace Case
Consider the following problem: The exact solution of the system is In Figure 1, the upper two figures depict the present solutions for this nonlinear system at s = 0.3 (upper left) and s = 0.7 (upper right), with h = 1/32, τ = 1/10 and k = 2.According to the exact solutions on the lower left and the values of u with different t with s = 0.3 (the circle represents the analytical solution and the asterisk represents the numerical solution), the present scheme is very accurate.Figure 2 displays the curves of the present scheme and the exact solutions at the boundary line (x = 0.5, t = 0.4) for s = 0.6 and k = 2, 3.It can be observed from these figures that the proposed methods effectively match the analytical solution.
According to the exact solutions on the lower left and the values of u with different t with s = 0.3 (the circle represents the analytical solution and the asterisk represents the numerical solution), the present scheme is very accurate.Figure 2 displays the curves of the present scheme and the exact solutions at the boundary line (x = 0.5, t = 0.4) for s = 0.6 and k = 2, 3.It can be observed from these figures that the proposed methods effectively match the analytical solution.Taking s as 0.2, 0.6, Figure 3 represents temporal convergence curves for different values of the parameter with h = 1/2056.From it, we have the conclusion that the space scheme is convergent and the slopes of the two convergence curves are close to 2. According to the exact solutions on the lower left and the values of u with different t with s = 0.3 (the circle represents the analytical solution and the asterisk represents the numerical solution), the present scheme is very accurate.Figure 2 displays the curves of the present scheme and the exact solutions at the boundary line (x = 0.5, t = 0.4) for s = 0.6 and k = 2, 3.It can be observed from these figures that the proposed methods effectively match the analytical solution.Taking s as 0.2, 0.6, Figure 3 represents temporal convergence curves for different values of the parameter with h = 1/2056.From it, we have the conclusion that the space scheme is convergent and the slopes of the two convergence curves are close to 2. Taking s as 0.2, 0.6, Figure 3 represents temporal convergence curves for different values of the parameter with h = 1/2056.From it, we have the conclusion that the space scheme is convergent and the slopes of the two convergence curves are close to 2.     To assess the numerical effectiveness, the following two-dimensional numerical experiment is considered: with the boundary conditions The equation's exact solution is as follows: with With τ = 1/1000 and s = 0.4, 0.7, the maximum errors considered for different space mesh sizes at t = 1 are provided in Table 2. From this table, the temporal convergence order is close to O(τ 2 ).This result demonstrates the robust stability of this scheme.In Figure 4, the curved surfaces of the present numerical scheme (left) and the analytical solution (right) are depicted, with h x = h y = 1/32, τ = 1/10, α = 1.4 and s = 0.7 at t = 1.It is evident that the numerical solutions closely approximate the analytical solutions.
Fractal Fract.2024, 1, 0 13 of 15 In Figure 4, the curved surfaces of the present numerical scheme (left) and the analytical solution (right) are depicted, with h x = h y = 1/32, τ = 1/10, α = 1.4 and s = 0.7 at t = 1.It is evident that the numerical solutions closely approximate the analytical solutions.

Conclusions
In this paper, a compact finite difference scheme method using the Picard integral formulation is presented for solving the multi-dimensional time-space fractional partial differential system with a variable order and different nonlinear terms.In contrast to many other schemes, the proposed method takes into account the regularity of the derivative term v. Detailed proofs establish the stability and solvability of this present scheme.To validate the practicality and accuracy of this compact scheme, three numerical experiments are computed and analyzed in different dimensional spatial domains.The numerical results show that the convergence rate aligns with the theoretical value of O τ 2 + h 2 in L ∞ norm.

Figure 1 .Figure 2 .
Figure 1.The curved surfaces for the present numerical scheme (upper), the exact solution (lower left), and the values of u with different t with s = 0.3 (lower right).

Figure 1 .
Figure 1.The curved surfaces for the present numerical scheme (upper), the exact solution (lower left), and the values of u with different t with s = 0.3 (lower right).

Figure 1 .Figure 2 .
Figure 1.The curved surfaces for the present numerical scheme (upper), the exact solution (lower left), and the values of u with different t with s = 0.3 (lower right).

Figure 2 .
Figure 2. The curves of the present numerical scheme (asterisk) and the analytical solution (circle) with x = 0.5 (left) and t = 0.4 (right).

FractalFigure 3 .
Figure 3.The curves of the present numerical scheme (asterisk) and the analytical solution (circle).

Figure 3 .
Figure 3.The curves of the present numerical scheme (asterisk) and the analytical solution (circle).

Figure 4 .
Figure 4.The surface of the numerical solution (left) and the exact solution (right) for s = 0.7 and α 0 = 1.4.

Figure 4 .
Figure 4.The surface of the numerical (left) and the exact solution (right) for s = 0.7 and α 0 = 1.4.
to the variable coefficients A(x, t) at the point (x i , t n ).

Table 1
displays the present space convergence orders at t = 1, which are close to our theoretical values O(h 2 ).

Table 1
displays the present space convergence orders at t = 1, which are close to our theoretical values O(h 2 ).

Table 2 .
The maximum errors and the corresponding spatial order at t = 1.