key: cord-0075804-t6dm13er
authors: Choudhary, Renu; Singh, Satpal; Kumar, Devendra
title: A second-order numerical scheme for the time-fractional partial differential equations with a time delay
date: 2022-03-18
journal: Comp
DOI: 10.1007/s40314-022-01810-9
sha: 8d26674128ab68eb98239caf1c07c51f61d59753
doc_id: 75804
cord_uid: t6dm13er

This work proposes a numerical scheme for a class of time-fractional convection–reaction–diffusion problems with a time lag. Time-fractional derivative is considered in the Caputo sense. The numerical scheme comprises the discretization technique given by Crank and Nicolson in the temporal direction and the spline functions with a tension factor are used in the spatial direction. Through the von Neumann stability analysis, the scheme is shown conditionally stable. Moreover, a rigorous convergence analysis is presented through the Fourier series. Two test problems are solved numerically to verify the effectiveness of the proposed numerical scheme.

where represents the gamma function and defined for a complex number z with non-negative real part as (z) = ∞ 0 e −t t z−1 dt.

Fractional partial differential equations (FPDEs) are where a fractional derivative replaces the classical derivative. Because of the precise and powerful descriptions of a large variety of natural phenomena, mathematicians and scientists are analyzing FPDEs with delay analytically and numerically (Yan and Kou (2012) ; Ouyang (2011); Zhou et al. (2009) ; Zhang et al. (2017) ; Sakara et al. (2016) ; Rodríguez et al. (2012) ; Mohebbi (2019); Nicaise and Pignotti (2006) ). The models containing the time delay can be seen in the automatic control systems (Lazarević and Spasić (2009); Si-Ammour et al. (2009) ), random walk (Ohira and Milton (2009) ), and the modeling human immunodeficiency virus (HIV) infection of C D4 + T -cells (Culshaw et al. (2003) ; Hao et al. (2016) ; Yan and Kou (2012) ). Furthermore, for the fractional models on HIV and corona virus disease, the readers are referred to the recent articles ; Chu et al. (2021) ; Pandey et al. (2021) . An extensive study of the delay differential equations (DDEs) in the context of ordinary differential equations (ODEs) can be seen in Bellen and Zennaro (2003) ; Guglielmi (2006) ; Guglielmi and Hairer (2004, 2001) ; Rihan (2000) and the references therein. Furthermore, the analytical solutions, including the spectral methods and the integral transform methods such as Laplace transforms and Mellin transforms for FPDEs, can be seen in Langlands et al. (2011); Jiang et al. (2012) ; Kilbas et al. (2006) ; Ding and Jiang (2013) ; Nieto (2015, 2017) . On the other hand, the DPDEs and TF-DPDEs are less explored than DDEs and FPDEs.

Because the evolution of a dependent variable of TF-DPDEs at any time t depends not only on its value at t −τ (for some time delay τ ) but also on all previous solutions, it is a difficult task to solve TF-PDEs with delay effectively and accurately. However, some analytical methods have been investigated to find the solutions to these problems. For instance, to obtain the actual solution to the TF-DPDEs, Prakash et al. (2020) proposed the invariant subspace approach. For the existence of solutions to DDEs, the readers are referred to Lakshmikantham (2008) . Moreover, the asymptotic properties of the solution can be found in Hale and Verduyn Lunel (1993) ; Küchler and Mensch (1992) . Furthermore, the stability bounds on the solution for the class of fractional delay difference equations (FDDEs) can be seen in Chen and Moore (2002) ; Deng et al. (2007) ; Hwang and Cheng (2006) . For the finite-time stability of robotic systems where a time delay appears in P D γ fractional control system, the readers are referred to Lazarević (2006) ; Lazarević and Spasić (2009) . As a consequence, we look for numerical methods for TF-DPDEs. Some numerical methods have also been developed for these problems; for instance, for a numerical scheme for space-fractional diffusion equations with a delay in time, the readers are referred to Hao et al. (2016) . They have used the Taylor series expansion to linearize the nonlinear term and weighted shifted Grünwald-Letnikov formula to approximate the space-fractional derivative. Rihan (2009) extended the θ method to solve the TF-DPDEs of parabolic type in the Caputo sense. To test the BIBO stability of the system of TF-DDEs, Hwang and Cheng (2006) presented an effective numerical algorithm based on Cauchy's integral theorem. Mohebbi (2019) constructed an efficient numerical method to solve the TF-DPDEs of convection-reaction-diffusion type with a nonlinear source term. The Chebyshev spectral collocation method and second-order finite difference approximations are used in spatial and temporal directions. Sakara et al. (2016) presented a homotopy perturbation method for nonlinear TF-DPDEs. Zhang et al. (2017) proposed a linearized compact finite difference method (FDM) for the semilinear TF-DPDEs. We use the compact finite difference approximation and L 1 approximation to discretize the spatial and temporal derivatives. Through rigorous analysis, the method is shown fourth-order convergent in space and of order 2 − γ in time. Chen et al. (2020) suggested an approach comprising the use of global radial basis functions and a collocation method to solve a class of time-delay fractional optimal control problems. proposed a direct scheme comprising the use of a set of Dickson polynomials as basis functions and a collocation method to solve a class of time-delay fractional optimal control problems.

Motivated by the initial-boundary value problems for singularly perturbed delay parabolic PDEs (Kumar and Kumari (2020) ), and the model considered by Kumar et al. (2021) for the singularly perturbed fractional delay PDEs, we analyze and develop an efficient numerical algorithm for the regular perturbation case. Furthermore, the following time-fractional diffusion model with a time-lag considered by Rihan (2009) 

motivates us to extend this model for the convection-reaction-diffusion fractional order PDEs.

In this work, we consider the following TF-DPDEs:

, with the following interval and boundary conditions

where the fractional derivative of order γ ∈ (0, 1) is estimated in the Caputo sense given by

A hybrid scheme and a compact FDM have been investigated in Marzban and Tabrizidooz (2012) ; Li et al. (2015) for the classical integer order DPDEs that is the simplified form of the Eq. (1.1) for γ = 1. Several numerical schemes have been developed when (1.1) is equivalent to the wave equation (for γ = 2) with delay, see e.g. Nicaise and Pignotti (2006) ; Garrido-Atienza and Real (2003); Rodríguez et al. (2012) . Du et al. (2010) developed a high order difference method for the fractional diffusion-wave equation associated with γ ∈ (1, 2) in (1.1) with constant coefficients and without the convective term. Li et al. (2019) investigates a linearized FDM for the problem similar to (1.1) without the reaction term and non-linear source term. First, the original problem is transformed into an equivalent semilinear fractional delay reaction-diffusion equation. Then they used the central finite difference formula for the space derivative and L 1 approximation for the Caputo derivative. Finally, the inverse exponential recovery method is used to obtain the numerical solution. This remainder article is outlined as follows. In Sect. 2, we semi-discretize the problem (1.1) in the temporal direction followed by the full-discretization in the spatial direction. The local truncation error is computed in Sect. 3, whereas the proof of the stability of the proposed scheme is given in Sect. 4. The rigorous convergence analysis is provided in Sect. 5. In Sect. 6, the numerical results of two test problems are reported to confirm the accuracy and efficiency of the proposed scheme. Finally, the paper ends with some observations, concluding remarks, and the proposed method's future scope in Sect. 7.

Remark 1 Throughout the manuscript, we take C as a positive generic constant that takes different values at different places.

To discretize the time domain, we use an equidistant mesh. 

where y represents the approximate solution to the problem (1.1). We approximate ∂ γ y(x,t n+1/2 ) ∂t γ using the following approach:

Thus, we get

Thus, we have the following approximation at (

Thus, we obtained a second-order approximation to

Let

where δ is termed as tension factor. As δ → 0 the function S(x, t, δ) is turned to be parametric cubic spline in [0, 1]. Let S n m be an approximation of y n m obtained by the segment of the functions passing through the points (

with the following interpolation conditions S π (x m , t n , δ) = y n m , S π (x m+1 , t n , δ) = y n m+1 .

(2.5)

The function derivative y (x m , t n ) has the spline derivative approximation given by

The solution of (2.4) with the interpolatory conditions (2.5) can be written as

Rewrite the Eq. (2.7) as follows

(2.9)

(2.10)

Now on replacing the second-order derivative in (2.12) by the tension spline function defined as

(2.14)

Rewrite (2.11) at (n + 1)-th time-level as

On adding (2.11) and (2.15), we get

(2.16)

A use of (2.14) in (2.16) yields

We assume that Y n m is the approximate solution of y n m and omitting the truncation error from (2.17), we find the following numerical scheme for (1.1)

with the interval and boundary conditions

then we can write the system (2.18) in matrix form as

with the coefficient matrix 

.

To find the local truncation error e n m of the proposed scheme, we replace M n m−1 , M n m , and 

Using Taylor series expansion for y n m−1 and y n m+1 in term of y n m and its derivatives, we get

(3.1)

It is clear from (3.1) that for suitable arbitrary values of α (α = 1/12) and β such that 2α + β = 1, the value of p is 2. Now, from (2.12) the local truncation error R 

In this section, we will discuss the stability of the numerical scheme (2.18) using the von Neumann stability method. We write the numerical scheme in the following form

LetŶ n m be the approximate solution of the system (2.18) and ξ n m = Y n m −Ŷ n m . The error equation of (4.1) is

along with the conditions

The grid functions

have the following Fourier series expansion

Using Parseval's identity in L 2 −norm, we find

According to above analysis, we can assume that Dividing both sides by e iθ m x , we obtain

(4.4)

After simplification for n = 0, 1, . . . , k − 1, we get the following form

and for n = k, k + 1, . . . , M t − 1, we have

where U = σ (2α cos(θ x) + β),

For stability of numerical scheme (2.18), we prove that |ζ n+1 | ≤ |ζ 0 | by use of mathematical induction. For n = 0, (4.5a) yields

.

To show that R ≤ 1, we consider two cases.

which is true for all θ and x.

which conclude that w 1 ≤ 2 2 1−γ , true for all values of γ . Now we assume that |ζ j | ≤ |ζ 0 |, for j = 2, 3, . . . , n.

(4.6)

We will prove that (4.6) holds for j = n + 1. The equation (4.5a) yields

Again, there arise two cases.

Last inequality holds true when 1 2 1−γ ≥ w 1 or 3 γ ≥ 3 2 . Now for n = k, k + 1, . . . , M t − 1, the Eq. (4.5b) becomes

It is clear from previous analysis and mathematical induction that as x, t → 0, |ζ n+1 | ≤ |ζ 0 | for each n = 0, 1, . . . , M t − 1. Hence, the numerical scheme (2.18) is conditionally stable with the condition 3 γ ≥ 3 2 .

To prove the convergence of the numerical scheme (2.18), we assume E n m = y n m − Y n m , m = 0, 1, . . . , N x , n = 0, 1, . . . , M t . (5.1) From (2.17) and (2.18), we obtain

along with the conditions

The grid functions Using Parseval's identity in L 2 −norm, we find (5.4) and

Now, we assume that where θ = 2π j. Using Eqs. (5.2) and (5.6) in (5.7) for n = 0, 1, . . . , k − 1, we have

(5.8a) and for n = k, k + 1, . . . , M t − 1, we have

(5.8b) Proposition 5.1 There exist a positive constant C such that |χ n+1 | ≤ (1 + C t) n+1 |r 1/2 |, n = 0, 1, . . . , M t − 1.

Proof Clearly χ 0 = χ 0 ( j) = 0 (as E 0 = 0). From the convergence of the series in right hand side of the Eq. (5.5), there exist a positive constant C such that |r n+1/2 | ≤ C t|r 1/2 |, n = 0, 1, . . . , M t − 1. From (5.3) and (5.5), we get

by the use of mathematical induction, for n = 0, we have cos(θ ( x) ))X 1 + X 2 + i X 3 sin(θ ( x)) ≤ C t|r 1/2 | ≤ (1 + C t)|r 1/2 |. Now assuming that |χ ℘ | ≤ (1 + C t) ℘ |r 1/2 |, ℘ = 2, 3, . . . , n, (5.9) we will prove that (5.9) is true for ℘ = n + 1. Equation (5.8a) yields

, on the basis of proof of stability it can be easily prove that |χ n+1 | ≤ (1 + C t) n+1 |r 1/2 | for n = 0, 1, . . . , k − 1.

For n = k, k + 1, . . . , M t − 1, Eq. (5.8b) yields

.

Similarly, as in the stability, we can prove that

Hence, by the mathematical induction we can say that |χ n | ≤ (1 + C t) n |r 1/2 | for n = 1, 2, . . . , M t .

where C is a positive constant independent of x and t.

Proof From Eqs. (5.4) and (5.5) and proposition (5.1), we obtain

Thus, the numerical scheme (2.18) is second-order convergent.

In this section, the efficiency and effectiveness of the proposed technique are demonstrated by implementing our numerical method on two test examples for the TF-DPDEs. The exact solutions to these problems are not available, so we employ the double mesh principle to determine the errors in L ∞ and L 2 −norms as follows: 

where y j i and y 2 j 2i are the numerical solutions obtained using (N x + 1,M t + 1) and (2N x + 1,2M t + 1) points, respectively. We also compute the corresponding orders of convergence using the formula

We consider the following TF-DPDEs with τ = 1 and suitable interval and boundary conditions. Replacing the γ order fractional derivative by first-order integer derivative, we define the following problem t) , and the interval and boundary conditions are the same as defined in Examples 6.1 and 6.2 .

We have shown the errors in L 2 and L ∞ norms for different values of γ , in Tables 1 and  3 , for Examples 6.1 and 6.2 , respectively. From these tables, it can be observed that as we increase N x and M t , the errors decrease and are in good agreement with the theoretical results. It confirms our theoretical estimates that the suggested scheme is second-order convergent. From Tables 2 and 4 , one can observe that the errors in L 2 and L ∞ norms for Examples 6.1 and 6.2 approach to the errors for the problem (6.1) as γ approaches 1. In Figures 1 and 4 , we portray the numerical solution profiles of Examples 6.1 and 6.2 , respectively for different values of γ . Figures 2 and 5 exhibit the graphs of the numerical solutions of Examples 6.1 and 6.2 at different time-levels. Keeping γ fixed, Figures 3 (γ = 0.9) and 6 (γ = 0.8) show that the error decrease when we increase the number of points in both directions.

To draw the Figs. 1, 2, 4, and 5 , we have used N x = M t = 64.

This paper suggests a numerical method comprising Crank-Nicolson scheme with tension spline for TF-DPDEs. The Caputo fractional derivative is used to discretize the fractionalorder time derivative. Using the Fourier series analysis, the method is proved conditionally stable. Moreover, through rigorous analysis, the method is shown second-order convergent for arbitrary suitable choices of α (α = 1/12) and β with 2α + β = 1. The method is easy to apply to the problem (1.1) and can easily be implemented through MATLAB software. Numerical examples demonstrate the effectiveness and adaptability of the suggested method. The suggested technique can be extended to the nonlinear TF-DPDEs and system of FPDEs.

Comparison of L 2 and L ∞ errors between Example 6.1 and problem (6.1) γ Table 4 Comparison of L 2 and L ∞ errors between Example 6.2 and problem (6.1)

γ 

N x =16 M t =16 N x =32 M t =32 N x =64 M t =64 N x =128 M t =128 N x =256 M t =2560

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The authors would like to thank the unknown reviewers for many helpful comments and corrections. The second author is grateful to UGC, New Delhi, India (award letter No. 1078/(CSIR-UGC NET JUNE 2019)) for providing the financial support.