Fitted Tension Spline Method for Singularly Perturbed Time Delay Reaction Diffusion Problems

A uniformly convergent numerical method is presented for solving singularly perturbed time delay reaction-diffusion problems. Properties of the continuous solution are discussed. The Crank–Nicolson method is used for discretizing the temporal derivative, and an exponentially fitted tension spline method is applied for the spatial derivative. Using the comparison principle and solution bound, the stability of the method is analyzed. The proposed numerical method is second-order uniformly convergent. The theoretical analysis is supported by numerical test examples for various values of perturbation parameters and mesh size.


Introduction
Delay di erential equations in which its highest order derivative term is multiplied by a small parameter ε are known as singularly perturbed delay di erential equations (SPDDEs). A good example of SPDDEs is the mathematical model for the control system of the furnace used to produce metal sheets which is given by where u is the temperature distribution in a metal sheet moving at an instantaneous material strip velocity v and heated by a distributed temperature source given by the function f; both v and f are dynamically changing by a controller monitoring the current temperature distribution. e controller's nite speed causes a xed delay of length τ [1].
When the perturbation parameter ε tends to zero, the smoothness of the solution of SPDDEs descends, and it forms a boundary layer [2]. Since most of the problems have no known analytical/exact solution, formulating numerical methods becomes mandatory [3]. It is of theoretical and practical interest to consider numerical methods for solving SPDDEs. Developing a numerical method whose convergence does not depend on the perturbation parameter has great importance [4]. Owning this, di erent authors have developed numerical schemes for the solution of SPDDEs. Chakravarthy et al. [5][6][7] used an adaptive mesh method and tted operator methods for solving singularly perturbed di erential-di erence equations. Kumar and Kadhalbajoo [8] computed the numerical solution of SPDDEs using a B-spline collocation method on Shishkin mesh. Kanth and Kumar in [9][10][11] used the tension spline method to solve singularly perturbed convection-dominated di erential equations. Woldaregay et al. [3,[12][13][14][15][16] developed an exponentially tted numerical method for solving di erent singularly perturbed di erential-di erence equations. In [14,17], they proposed the nonstandard nite di erence techniques and the tted mesh methods, respectively. In [18,19], an initial value technique with an exponential tting factor is applied to solve convection-dominated delay differential equations. e di erence method on Shishkin for solving a singularly perturbed linear second-order delay di erential equation is developed in [20].
Govindarao et al. [2] solved a singularly perturbed delay parabolic reaction-di usion problem using the implicit Euler scheme for time derivative on the uniform mesh, and they used the central difference scheme on the Shishkin mesh for spatial discretization. Kumar and Chandra Sekhara Rao [21] developed a numerical scheme using a suitable combination of fourth order compact difference scheme and central difference scheme on generalized Shishkin mesh. Gowrisankar and Natesan [22] proposed a numerical scheme using the backward-Euler method for temporal discretization and the upwind finite difference scheme for spatial derivative. Erdogan and Amiraliyev [23] treated singularly perturbed second-order delay differential equations using an exponentially fitted difference scheme on a uniform mesh.
In some cases, the cubic spline does not provide an accurate picture of the shape of the graph. In such cases, the tension spline method is preferable. A tension spline is a cubic spline that has a tension factor applied to it to stretch the graph closer to the given points. A tension factor δ is a number that is used to generate different conditions for the spline. When the tension factor is set to zero, it gives the usual cubic spline. e objective of this study is to formulate a uniform numerical method for solving singularly perturbed time delay reaction-diffusion problems. Furthermore, we need to establish the stability and uniform convergence of the method. To the best of our knowledge, the exponentially fitted tension spline method has not been used for treating SPDDEs. In this paper, we employ the Crank-Nicolson method to temporal discretization and the exponentially fitted tension spline method to spatial discretization.

Continuous Problem
On the domain D � Ω x × Ω t � (0, 1) × (0, T], we consider singularly perturbed time delay reaction-diffusion problems of the form with the interval condition and the boundary conditions where ε ∈ (0, 1] is the perturbation parameter, and τ is the delay parameter. e functions a, b, ψ l , ψ r , ψ b , and f are assumed to be sufficiently smooth and bounded that satisfy
Lemma 1 (see [24]). ( e maximum principle) Assume that a, b ∈ C 0 (D) and let u ∈ C 2 (D) The stability of the solution of (2)-(4) in the maximum norm is established by the following lemma: Lemma 2 (see [25]). Let u be the solution of the problem in (2)- (4). en, it satisfies the ε-uniform upper bound where the constant α � max D 0, 1 − α { } ≤ 1 and the norm ‖.‖ is denoted for the maximum norm which is defined as ‖u‖ � max D |u(x, t)|.

Numerical Scheme
We approximated the temporal derivative term of (2)-(4) using the averaged Crank-Nicolson method, which gives a system of BVPs with the boundary conditions where where C 1 is a constant independent of ε and mesh size Δt.
Proof. Using Taylor's series approximation for u(x, t j ) and u(x, t j+1 ) centering at t j+1/2 , we obtain From (11), we obtain Substituting (12) into (2), we obtain Hence, by applying the maximum principle, we obtain Next, we need to show the bound for the global error of temporal discretization. Let us denote GTE j+1 as the global error up to the (j + 1) th time step.

Lemma 5.
e global error in temporal discretization at t j+1 time step is given by

Mathematical Problems in Engineering
Proof. Using the local truncation error up to the (j + 1) th time step given in the above lemma, we obtain the global error up to the (j + 1) th time step as where C is a constant independent of ε and Δt. □ Lemma 6. Let U j+1 (x) be a solution of the semidiscrete problem in (7), (8) at the j + 1 time level. en, its derivatives satisfy the bound where Proof. It follows from Lemma 3.
, which interpolates u(x) at the mesh points x i depending on the parameter δ, reduces to cubic spline in Ω x as δ ⟶ 0, and it is termed as a parametric cubic spline function [27,28]. e spatial domain is discretized into N equal number of subintervals, each of length h � 1/N. Let 0 � x 0 , x N � 1 and x i � ih, i � 0, 1, 2, . . . , N be the mesh points. For spatial discretization, we apply an exponentially fitted tension spline method which helps us control the influence of the singular perturbation parameter. In [x i , x i+1 ], the spline function S(x) satisfies the differential equation where S(x i ) � u(x i ) and δ > 0 is termed as a cubic spline in compression. Solving the linear second-order differential equation in (19) and determining the arbitrary constants from the interpolation conditions where λ � hδ 1/2 and M k � u ″ (x k ) for k � i, i ± 1. Now, differentiating (19) and letting x ⟶ x i , on the interval Similarly, on the interval Equating the left and right hand derivatives at Rearranging, we obtain the tridiagonal system for i � 1, 2, 3, . . . , N − 1 where λ 1 � 1/λ 2 (λ/sinhλ − 1) and . e condition of continuity in (24) ensures the continuity of the first derivatives of the spline S(x) at interior nodes. Using into (24), (8) gives To handle the effect of the perturbation parameter ε, we use the exponential fitting factor that stabilizes the discrete scheme. Substituting the fitting factor into (26), (25) we obtain the difference equation as where , forj � 0, 1, 2, . . . , m, Proof. e matrix (1 + (Δt/2)L Δt,h ε )U j+1 i is a size of (N + 1) × (N + 1) with its entries for i � 1, 2, . . . , N − 1 which are v − i < 0,v c i > 0, and v + i < 0. As a result, the coefficient matrix satisfies the M matrix property. So, there is an inverse matrix that is non-negative. is proves that the discrete solution exists and is unique.
where α > 0 is lower bound of a i � a(x i ).

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Using the discrete comparison principle, we obtain μ ± i,j+1 ≥ 0, i � 0, 1, 2 . . . , N □ Lemma 9 (see [29,30]). For a fixed number of mesh N and for positive integer k as ε ⟶ 0 Proof. Consider two barrier functions of the form Hence, using the discrete comparison principles gives |Q e discrete solution satisfies the following error bound: Proof. e truncation error of the scheme (27) is given by where θ � λ 1 (1 + Δt/2a i ) and 2λ 2 � 1 − 2λ 1 . Using Taylor's series expansion of the terms U j+1 i+1 and U j+1 i− 1 , we have Using the truncated Taylor's series expansions of the terms U j+1 i+1 and U j+1 i− 1 yields Next, we use a truncated Taylor's series expansion of the exponential fitting factor (σ) of order five Now, substituting (38) into (37), we obtain Using Lemma 6 together with Lemma 9 and the relation N − 2 > N − 4 > N − 6 . . ., the discrete scheme satisfies the bound Using the bound in Lemma 10, we obtain □ Theorem 2. Let u and U be the solution of (2)- (4) and (27), respectively. en, the following uniform error bound holds Proof. e combination of temporal and spatial error bounds gives the required result. i,2N be the computed solution on the double number of mesh points 2N and 2M by including the midpoints x i+1/2 � x i+1 + x i /2 and t j+1/2 � t j+1 + t j /2 into the mesh points. e maximum absolute error is calculated by using the formula and the uniform error is calculated by using the formula ε | e rate of convergence is calculated as and the uniform rate of convergence is calculated as Example 1. [24] Consider the following problem with the retarded argument Example 2. [25] Consider the following problem with the retarded argument Example 3. [31] Consider the following problem with the retarded argument In Tables 1, 2, and 3, respectively, we show the maximum point-wise absolute error and the rate of convergence of Examples 1, 2, and 3. In each column of the tables of 1-3 (or for each N and M as ε ⟶ 0, the maximum absolute error becomes uniform; this shows that the convergence of the method is independent of the perturbation parameter. One can observe in these tables that the method converges with an order of convergence two for each values of the perturbation parameter. e comparison in Table 4 shows that the proposed method is more accurate than the scheme in    Mathematical Problems in Engineering [24]. In Figures 1 and 2, the boundary layer formation on the solution is given as ε goes small.

Conclusion
In this paper, the exponentially fitted tension spline method is developed for solving the singularly perturbed time delay reaction-diffusion problems. e solution of considered problems exhibits exponential boundary layers on the left and right side of the spatial domain. We employed the Crank-Nicolson method in temporal discretization and an exponentially fitted tension spline method in spatial discretization to set up the numerical method. e bounds and properties of the analytical solution are explored. An exponential fitting parameter is induced to stabilize the influence of the perturbation parameter on the discrete solution.
e stability and convergence analysis of the method are discussed and proved. e performance of the method is illustrated through the numerical examples. It is proved that the method gives second-order convergence in space and in time.

Data Availability
No additional data are used in this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest. Mathematical Problems in Engineering 9