Parameter-Robust Numerical Scheme for Time

Numerical Functional Analysis and Optimization

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Parameter-Robust Numerical Scheme for Time-Dependent Singularly Perturbed Reaction–Diffusion Problem with Large Delay Komal Bansal & Kapil K. Sharma To cite this article: Komal Bansal & Kapil K. Sharma (2017): Parameter-Robust Numerical Scheme for Time-Dependent Singularly Perturbed Reaction–Diffusion Problem with Large Delay, Numerical Functional Analysis and Optimization, DOI: 10.1080/01630563.2016.1277742 To link to this article: https://doi.org/10.1080/01630563.2016.1277742

Accepted author version posted online: 26 May 2017. Published online: 27 Nov 2017. Submit your article to this journal

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Date: 04 December 2017, At: 08:42

NUMERICAL FUNCTIONAL ANALYSIS AND OPTIMIZATION https://doi.org/10.1080/01630563.2016.1277742

Parameter-Robust Numerical Scheme for Time-Dependent Singularly Perturbed Reaction–Diffusion Problem with Large Delay Komal Bansala and Kapil K. Sharmab,c a

Downloaded by [122.173.119.78] at 08:42 04 December 2017

Department of Mathematics, Center for Advance Study in Mathematics, Panjab University, Chandigarh, India; b Department of Mathematics, South Asian University, New Delhi, India; c Department of Mathematics, Panjab University, Chandigarh, India ABSTRACT

ARTICLE HISTORY

This work presents the development of numerical scheme for second-order time-dependent singularly perturbed reactiondiffusion problem with large delay in the undifferentiated term. These types of problems arise frequently in many areas of science and engineering that take into consideration the effect of present situation as well as the past history of the physical system. As the characteristics of the reduced problem (ε = 0) corresponding to the original singularly perturbed problem considered here are parallel to the boundary of the domain this implies, parabolic layers exhibit in the solution. In this paper, we initiate the study of parabolic layers together with interior layers in the solution of singularly perturbed parabolic partial differential-difference equations due to propagation of singularity. Proposed numerical scheme comprised of finite difference scheme and piecewise uniform Shishkin mesh. The method is shown to be accurate of order (M−1 + N−2 (lnN)2 ), where M and N are the number of mesh elements in time and spatial direction, respectively. Proposed numerical scheme is proved to be parameter uniform convergent in the maximum norm. Numerical experiments have been performed to show the existence of interior layer due to large state-dependent delay argument in the reaction term and to confirm the predicted theory.

Received 27 January 2016 Revised 24 December 2016 Accepted 27 December 2016 KEYWORDS

Convection–diffusion parabolic problem; differential-difference equations; finite difference scheme; fitted mesh; interior layer; singular perturbation AMS SUBJECT CLASSIFICATIONS

65L11; 65M12; 35K20

1. Introduction and Problem Formulation The feasibility of recording single neuron movement induces the development of accurate mathematical models of neuronal variability. While modeling of spiking movement of neuron to any level of exactness, one has to consider special features of each kind of neuron and its input processes. In 1965, Stein [37] proposed a practical model for the stochastic movement of neuron. In 1967, the author generalized the model to handle a distribution of past synaptic potential amplitudes [38].

CONTACT Kapil K. Sharma [email protected]; [email protected] Department of Mathematics, South Asian University, New Delhi 110021, India; Department of Mathematics, Panjab University, Chandigarh 160014, India. © 2017 Taylor & Francis


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K. BANSAL AND K. K. SHARMA

In 1991, Musila and Lánsk`y [29] generalized Stein’s model and proposed the following mathematical model in terms of singularly perturbed parabolic partial differential-difference equations (SPPPDDE) to consider the time evolution trajectories of the membrane potential: ∂u σ 2 ∂ 2u x ∂u − = + µ − +λs u(x+as , t)+ws u(x+is , t)−(λs +ws )u(x, t), D ∂t 2 ∂x2 τ ∂x (1.1) where The first derivative term is due to the exponential decay between two consecutive jumps caused by the input processes. The membrane potential decays exponentially to the resting level with a membrane time constant τ . µD and σ are diffusion moments of Wiener process characterizing the influence of dendritic synapses on the cell excitability. The reaction terms correspond to the superposition of excitatory and inhibitory inputs and we can assume that they are Poissonian [29]. The excitatory input contributes to the membrane potential by an amplitude as with intensity λs and similarly the inhibitory input contributes by an amplitude is with intensity ws . This model makes available time evolution of the trajectories of the membrane potential. The model (1.1) is a differential-difference equation, one can hardly derive its exact solution. Thus, to stimulate this model, one has to land to numerical techniques. Lange and Miura [25, 26] gave asymptotic solution of a wide class of boundary value problems for singularly perturbed delay differential equations. Kadalbajoo and Sharma [14] initiated the numerical study of singularly perturbed ordinary differential-difference equations. A wide variety of literature are available on the numerical analysis of singularly perturbed ordinary differential equations [6, 15–17, 28, 35]. In 2007, Amiraliyev and Erdogan [2] considered singularly perturbed initial value problem for a linear first-order delay differential equation with fixed delay. The authors proposed a parameter uniform numerical method for this problem based on appropriate piecewise uniform mesh on each time subinterval. In 2010, Amiraliyev and Cimen [1] studied second-order singularly perturbed convection–diffusion problem with constant delay. In this continuation, Amiraliyeva et al. [3] approximated the solution of an initial value problem for the nonlinear second-order singularly perturbed delay differential equation. Subburayan and Ramanujam [39, 40] developed numerical techniques for the solution of second-order ordinary singularly perturbed boundary value delay problems with discontinuous convection–diffusion coefficient and reaction– diffusion problem, respectively. In the same year, Nicaise and Xenophontos [32] gave robust approximation to the solution of second-order singularly perturbed ordinary differential equations with large delay by the hp finite element method.


NUMERICAL FUNCTIONAL ANALYSIS AND OPTIMIZATION

3

A wide variety of literature for the last two decades are based on the numerical solution of singularly perturbed parabolic partial differential equations (SPPPDEs) with and without time delay [4, 8, 10, 13, 18, 19]. It is a challenging problem to deal with SPPPDDEs containing general shift arguments in the space variable. For the literature on the parameter uniform numerical scheme for SPPPDDE with space-dependent small shift arguments, we refer the readers to [23, 33, 34]. In [23, 33, 34] the authors developed and analyzed the numerical schemes for the approximated initial-boundary value problem rather than the original differential-difference problem with initial-interval boundary conditions. This approximation is valid till size of the shift arguments is sufficiently small and fails in the case when size of the shift arguments is bigger. Bansal et al. [5, 7] proposed numerical schemes for SPPPDDE with state-dependent general shift arguments, which resolve the two major issues, i.e., general space-dependent shift arguments and parameter uniform convergence. In addition to boundary layers, interior layers can also occur in the solution of time-dependent SPPPDEs. But, there is no literature till date on SPPPDDEs exhibiting interior layers. Motivated by the work of Lange and Miura [25], the main contribution of the present work is to study the propagation of singularity in the form of interior layers due to large state dependent delay in timedependent SPPPDEs. The solution of the problem considered in this paper has parabolic boundary layers in the neighborhood of L and R and an interior layer due to unit shift in the reaction term. We design a numerical scheme based upon condensing grid method and discretize the time variable using the implicit Euler method. We calculate the error estimates by decomposing the solution into two components, i.e., smooth and singular and deal with the point (1, 0) separately. We are concerned with an SPPPDEs on ≡ D × (0, T] ≡ (0, 2) × (0, T] in the S space–time S plane, T is some fixed positive time and ∂ = − = {(0, t) (2, t) (x, 0) : 0 ≤ x ≤ 2, 0 ≤ t ≤ T} ∂u ∂ 2u + ε 2 − a(x)u(x, t) − b(x)u(x − 1, t) = f (x, t), (1.2) ∂t ∂x where (x, t) ∈ , subject to the following interval boundary conditions and the initial condition u(x, t) = φ(x, t), ∀ (x, t) ∈ L = {(x, t) : −1 ≤ x ≤ 0 and 0 ≤ t ≤ T}, −

u(2, t) = ψ(t), u(x, 0) = u0 (x),

∀ (2, t) ∈ R = {(2, t) : 0 ≤ t ≤ T},

(1.3)

∀ x ∈ D = [0, 2].

The Equation (1.2) can be rewritten as: Lε u(x, t) = F(x, t),

(1.4)

4


where

Lε u(x, t) ≡


and F(x, t) =

 ∂u ∂ 2u   − + ε − a(x)u(x, t),   ∂t ∂x2        if 0 < x ≤ 1, 0 < t ≤ T, 

  ∂u ∂ 2u    − + ε − a(x)u(x, t) − b(x)u(x − 1, t),  2  ∂t ∂x      if 1 < x < 2, 0 < t ≤ T,

 f (x, t) + b(x)φ(x − 1, t), if 0 < x ≤ 1, 0 < t ≤ T, f (x, t),

(1.5)

(1.6)

if 1 < x < 2, 0 < t ≤ T.

Here 0 < ε ≪ 1 is a small positive parameter. The functions a(x), b(x), f (x, t), φ(x, t), ψ(t), and u0 (x) are assumed to be sufficiently smooth, bounded, and independent of ε. Undifferentiated term in the problem has unit shift argument and on taking b(x) = 0 ∀ x ∈ D, the differential-difference equation reduces to differential equation. To avoid oscillation in the solution, it is assumed that a(x) + b(x) ≥ 2b∗ > 0,

∀ x ∈ D,

(1.7)

where b∗ is a positive constant and b(x) < 0. The outline of the paper is as follows: In the introductory section, the study focuses on the problem formulation. In the next section, we discuss some properties of the exact solution. Section 3 is devoted to the time semi discretization and parameter uniform error estimate in the time direction is proved. In Section 4, fitted mesh is introduced for spatial discretization and fully discrete problem is discussed. Section 5 deals with the decomposition of the discrete solution to calculate error estimates for the numerical solution and parameter uniform convergence in the spatial direction is proved. Section 6 consists of numerical experiments on some test examples. A summary of the main conclusion is also given in the last section.

1.1. Notations and terminology

Most of the notations and symbols we use are fairly standard. It is convenient to introduce the notation for jump in any function say at a point “b” with [g](b) = g(b+ ) − g(b− ). We define S = 1 ∪ 2 , where 1 = D1 × (0, T], 2 = D2 × (0, T], D1 = (0, 1), D2 = (1, 2). Further, throughout the paper, C will represent a positive constant, which may take different values in different equations (inequalities) but is independent of perturbation parameter,


5

the spatial mesh parameters and the time step. Here ||.|| stands for the standard supremum norm, which is defined by: ||g|| = sup(x,t)∈ |g(x, t)|, for a function g defined on some domain . 2. Some properties of the exact solution The existence and uniqueness of the solution of (1.2)–(1.3) can be established by assuming that the given data are Hölder continuous and imposing appropriate compatibility conditions at the points (0, 0), (2, 0), (−1, 0), and (1, 0) [24]. The required compatibility conditions are


u0 (0) = φ(0, 0),

(2.1)

u0 (2) = ψ(0), and ∂ 2 u0 (0) ∂φ(0, 0) +ε − a(0)u0 (0) − b(0)φ(−1, 0) = f (0, 0), − ∂t ∂x2 ∂ψ(0) ∂ 2 u0 (2) − +ε − a(2)u0 (2) − b(2)u0 (1) = f (2, 0). ∂t ∂x2

(2.2)

Remark 1. The solution of the problem exhibits parabolic boundary layers in the neighborhood of L and R . So by using compatibility conditions in (2.1) and (2.2), it is guaranteed that there exists a constant C independent of ε such that for all (x, t) ∈ , we have |u(x, t) − u(x, 0)| = |u(x, t) − u0 (x)| ≤ Ct.

(2.3)

For the proof of (2.3), reader can refer to [12, 35]. Lemma 2.1. The bound on the solution u(x, t) of the continuous problem is given by: |u(x, t)| ≤ C,

∀ (x, t) ∈ .

Proof. From the inequality (2.3), we have |u(x, t)| − |uo (x)| ≤ |u(x, t) − uo (x)| ≤ Ct, ⇒ |u(x, t)| ≤ Ct + |uo (x)|,

∀ (x, t) ∈ .

Since t ∈ (0, T], so it is bounded and uo (x) ∈ C2 (D). Therefore, Ct + |uo (x)| is bounded by some constant C and hence |u(x, t)| ≤ C, ∀ (x, t) ∈ . Lemma 2.2. Let ξ(x, t) ∈ C2,1 (). If ξ(x, t) ≥ 0, ∀ (x, t) ∈ ∂ and Lε ξ(x, t) ≤ 0, ∀ (x, t) ∈ , then ξ(x, t) ≥ 0, ∀ (x, t) ∈ .

6


Proof. Let (x∗ , t ∗ ) ∈ be such that ξ(x∗ , t ∗ ) = min ξ(x, t) and suppose ξ(x∗ , t ∗ ) < 0 then by the given condition, we have (x∗ , t ∗ ) ∈ / ∂ and ξx (x∗ , t ∗ ) = 0, ξt (x∗ , t ∗ ) = 0, ξxx (x∗ , t ∗ ) > 0. To prove Lε ξ(x∗ , t ∗ ) > 0, we have two following cases: Case 1: 0 ≤ x ≤ 1, Lε ξ(x∗ , t ∗ ) = −

∂ξ ∗ ∗ ∂ 2ξ (x , t ) + ε 2 (x∗ , t ∗ ) − a(x∗ )ξ(x∗ , t ∗ ) ∂t ∂x

> 0.


Case 2: 1 < x ≤ 1, Lε ξ(x∗ , t ∗ ) = −

∂ 2ξ ∂ξ ∗ ∗ (x , t ) + ε 2 (x∗ , t ∗ ) − a(x∗ )ξ(x∗ , t ∗ ) ∂t ∂x

− b(x∗ )ξ(x∗ − 1, t ∗ ) =−

∂ξ ∗ ∗ ∂ 2ξ (x , t ) + ε 2 (x∗ , t ∗ ) − (a(x∗ ) + b(x∗ ))ξ(x∗ , t ∗ ) ∂t ∂x

− b(x∗ )(ξ(x∗ − 1, t ∗ ) − ξ(x∗ , t ∗ )) > 0. Concluding the above two cases, we get Lε ξ(x∗ , t ∗ ) > 0. Hence our supposition is wrong and ξ(x∗ , t ∗ ) ≥ 0, which implies ξ(x, t) ≥ 0, ∀ (x, t) ∈ . We follow the techniques discussed in [31] to prove the bounds on the derivatives of the solution u(x, t) of problem (1.2)–(1.3) in t direction. We may assume without loss of generality that the initial-interval boundary data given in (1.3) are identically zero, i.e., uo (x) = 0, φ(x, t) = 0, ψ(t) = 0. Lemma 2.3. Suppose the Lemmas (2.1) and (2.2) hold. Then the bound on the derivatives of u with respect to t is given by: |ut (x, t)| ≤ C, (x, t) ∈ . Proof. The bound for i = 0 is proved in Lemma 2.1. ¯ we have u ≡ 0 this implies ut = 0. On Along the sides x = 0 and x = 2 of , the side t = 0, we have u = 0, therefore uxx ≡ 0. From (1.2), we have − ut (x, 0) + εuxx (x, 0) − a(x)u(x, 0) + b(x)u(x − 1, 0) = f (x, 0).

(2.4)

On the interval [0, 2], consider the two following cases: Case 1: 0 ≤ x ≤ 1, u(x − 1, 0) = φ(x − 1, 0) = 0,

(2.5)


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Case 2: 1 < x ≤ 2, u(x − 1, 0) = uo (x − 1, 0) = 0,

(2.6)

Combining the above two cases, we get u(x − 1, 0) = 0. Now (2.4) implies − ut (x, 0) = f (x, 0),

(2.7)

i.e., |ut | ≤ C for significantly large values of C on ∂. On applying the differential operator Lε given in (1.4) on ut (x, t), we get Lε ut (x, t) = Ft (x, t),

(2.8)

which implies


|Lε ut | = |Ft (x, t)| ≤ C on . Thus an application of Lemma 2.2 yields |ut (x, t)| ≤ C on . Thus the result is proved for i = 1. By using the same methodology, we prove the result for i = 2. On the sides x = 0 and x = 2, we have utt = 0. Along x-axis, we have u ≡ 0, uxx ≡ 0. On differentiating (1.2) both sides w.r.t. t, we get − utt + εuxxt − a(x)ut (x, t) − b(x)ut (x − 1, t) = ft (x, t).

(2.9)

From (2.7), we have ut (x, 0) = f (x, 0). This implies uxxt (x, 0) = fxx (x, 0).

(2.10)

From (2.5) to (2.6), we have u(x − 1, 1t) − u(x − 1, 0) = 0. 1t→0 1t Using (2.10) and (2.11) in (2.9), we get ut (x − 1, 0) = lim

− utt (x, 0) = ft (x, 0) − εfxx (x, 0) + a(x)f (x, 0).

(2.11)

(2.12)

From (2.12), we have |utt | ≤ C along x-axis. This implies |utt | ≤ C on ∂. Consider the differential operator Lε defined in (1.4) and applying it on utt , we get Lε utt = Ftt (x, t), which implies |Lε utt (x, t)| = |Ftt (x, t)| ≤ C on . Therefore, an application of Lemma 2.2 implies ¯ |utt (x, t)| ≤ C on .

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3. Temporal discretization To obtain the totally discrete method, the first step is based on the temporal discretization of the continuous problem (1.2)–(1.3). On the time domain [0, T], we placed a uniform mesh given by: M t = {tj = j1t, j = 1, . . . , M, t0 = 0, tM = T, 1t = T/M}, where M denotes the number of mesh elements in the time direction and 1t is the time step. Then, on M t the continuous problem (1.2) is discretized by using implicit Euler method given by:


ε1tUxx (x, tj+1 )−â(x)U(x, tj+1 )−1tb(x)U(x−1, tj+1 ) = 1tf (x, tj+1 )−U(x, tj ), (3.1) subject to the following initial condition and interval boundary conditions: U(x, 0) = u0 (x), on D = {(x, 0) :

0 ≤ x ≤ 2},

U(2, tj+1 ) = ψ(tj+1 ), on SM x+ = {(2, tj+1 ) :

1 ≤ j + 1 ≤ M},

(3.2)

U(x, tj+1 ) = φ(x, tj+1 ), on SM x− = {(x, tj+1 ) : −1 ≤ x ≤ 0, 1 ≤ j + 1 ≤ M}. Here U(x, tj+1 ) is the numerical solution of continuous problem (1.2)–(1.3) at (j + 1)st time level and aˆ (x) = 1ta(x) + 1. The Equation (3.1) can be rewritten as: Lx,M U(x, tj+1 ) ≡ g(x, tj+1 ),

(3.3)

where

   ε1tUxx (x, tj+1 ) − aˆ (x)U(x, tj+1 ), if 0 < x ≤ 1, x,M L U(x, tj+1 ) ≡ ε1tUxx (x, tj+1 ) − aˆ (x)U(x, tj+1 )  if 1 < x < 2  −1tb(x)U(x − 1, tj+1 ),

and

g(x, tj+1 ) =

 1tf (x, tj+1 ) − U(x, tj ) + 1tb(x)φ(x − 1, tj+1 ), 1tf (x, t ) − U(x, t ), j+1 j

(3.4)

if 0 < x ≤ 1,

if 1 < x < 2. (3.5) Bounds for the local error are used to calculate the estimates for global error, where the following problems are introduced to define the local error. ˆ tj+1 ) ≡ gˆ (x, tj+1 ), Lx,M U(x,

(3.6)

where Lx,M is same as defined in (3.4) and  1tf (x, tj+1 ) − u(x, tj ) + 1tb(x)φ(x − 1, tj+1 ), if 0 < x ≤ 1, gˆ (x, tj+1 ) = 1tf (x, t ) − u(x, t ), if 1 < x < 2, j+1 j (3.7)


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subject to the following initial condition and interval boundary conditions: ˆ 0) = u0 (x), on D = {(x, 0) : U(x,

0 ≤ x ≤ 2},

ˆ tj+1 ) = ψ(tj+1 ), on SM U(2, x+ = {(2, tj+1 ) :

1 ≤ j + 1 ≤ M},

(3.8)


ˆ tj+1 ) = φ(x, tj+1 ), on SM U(x, x− = {(x, tj+1 ) : −1 ≤ x ≤ 0, 1 ≤ j + 1 ≤ M}. Remark 2. At x = 0, 1, 2, as ε → 0 the solution of the ordinary differential equations produced in (3.1)–(3.2) cannot be approximated by the solution of corresponding reduced problems and due to this breakdown solution has singular behavior. The sharp derivative estimates given in Lemmas 3.6–3.7 also indicate the location of layers. Thus as ε → 0, the solution approaches the reduced solution uniformly on any closed interval not containing the points where the layers are. For more details, we refer the readers to [25, 30]. Lemma 3.1 (Minimum principle). Let ξ(x, tj+1 ) be a smooth function such that ξ(0, tj+1 ) ≥ 0 and ξ(2, tj+1 ) ≥ 0, then Lx,M ξ(x, tj+1 ) ≤ 0, ∀ x ∈ D implies ξ(x, tj+1 ) ≥ 0, ∀ x ∈ D. Proof. By using the same technique as in Lemma 2.2, the minimum principle for the semi discrete operator follows. Thus, the operator Lx,M satisfies the minimum principle and consequently, we have 1 −1 . (3.9) ||(Lx,M ) || ≤ 1 + 2b∗ 1t i Lemma 3.2. Having ∂ u(x,t) ≤ C, ∀ (x, t) ∈ , i = 0, 1, 2 implies the local ∂t i truncation error in the temporal direction satisfies ||êj+1 ||∞ ≤ C(1t)2 ,

(3.10)

ˆ tj+1 ) − u(x, tj+1 ) is the local error estimate in the temporal where eˆ j+1 = U(x, direction at (j + 1)st time level. ˆ tj+1 ) satisfies Proof. Since the function U(x, ˆ tj+1 ) ≡ gˆ (x, tj+1 ) Lx,M U(x,

(3.11)

and the exact solution of (1.2)–(1.3) is smooth enough, so we have Lx,M u(x, tj+1 ) + O((1t)2 ) = gˆ (x, tj+1 ).

(3.12)

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From (3.11) and (3.12), we get Lx,M eˆ j+1 + O((1t)2 ) = 0,

(3.13)

where eˆ j+1 is the local truncation error in time direction at (j + 1)st time level. eˆ j+1 satisfies the boundary value problem of the following type Lx,M eˆ j+1 = O((1t)2 ), eˆ j+1 = (Lx,M )−1 O((1t)2 ),

(3.14)

eˆ (0, tj+1 ) = 0 = eˆ (2, tj+1 ). Downloaded by [122.173.119.78] at 08:42 04 December 2017

On using stability result (3.9) in (3.14), we get the following result: ||êj+1 || ≤ C(1t)2 . Lemma 3.3. Under the hypothesis of the Lemma (3.2), global error estimate Ej = u(x, tj ) − U(x, tj ) in the temporal direction satisfies ||Ej ||∞ ≤ C(1t),

∀ j ≤ T/1t.

Proof. Using local error estimates up to jth time step given in Lemma 3.2, we get the following global error estimates at (j + 1)st time step j X j ≤ T/1t ||Ej+1 ||∞ = eˆ l l=1 ∞

≤ ||ê1 ||∞ + ||ê2 ||∞ + ||ê3 ||∞ + · · · + ||êj ||∞

≤ C1 (j1t)1t ≤ C1 T1t

since j1t ≤ T

≤ C1t. Therefore, the proposed numerical scheme is uniformly convergent of first order in the temporal direction. Lemma 3.4. Let U(x, tj+1 ) be the solution of semidiscrete problem (3.1)–(3.2). Then U(x, tj+1 ) satisfies 1 |U(x, tj+1 )| ≤ max |U(0, tj+1 )|, ||g||, |U(2, tj+1 )| , 1 + 2b∗ 1t

∀ x ∈ D.


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Proof. Let us define two barrier functions as: 1 ± ζ (x, tj+1 ) = max |U(0, tj+1 )|, ||g||, |U(2, tj+1 )| ± U(x, tj+1 ). 1 + 2b∗ 1t

We will prove that ζ ± (x, tj+1 ) satisfies maximum principle for the semidiscrete operator. Now ζ ± (0, tj+1 ) ≥ 0, ζ ± (2, tj+1 ) ≥ 0 and we have Case 1: 0 ≤ x ≤ 1,

± Lx,M ζ ± (x, tj+1 ) = 1tεζxx (x, tj+1 ) − aˆ (x)ζ ± (x, tj+1 )


≤ 0 [by using (1.7)], Case 2: 1 < x ≤ 2,

± Lx,M ζ ± (x, tj+1 ) = 1tεζxx (x, tj+1 ) − aˆ (x)ζ ± (x, tj+1 ) − 1tb(x)ζ ± (x − 1, tj+1 )

≤ 0 [by using (1.7)]. Therefore, using Lemma 3.1, we get 1 ||g||, |U(2, tj+1 )| , |U(x, tj+1 )| ≤ max |U(0, tj+1 )|, 1 + 2b∗ 1t

∀ x ∈ D.

3.1. Decomposition of the solution

The solution U(x, tj+1 ) of the problem (3.1)–(3.2) at (j + 1)st time level can be decomposed as U(x, tj+1 ) = V(x, tj+1 ) + W(x, tj+1 ), where V(x, tj+1 ) and W(x, tj+1 ) are regular and singular components, respectively. For x ∈ (0, 1− ), V(x, tj+1 ) satisfies the following differential equation: ε1t

d2 V(x, tj+1 ) − aˆ (x)V(x, tj+1 ) = 1tf (x, tj+1 ) − U(x, tj ) dx2

+ 1tb(x)φ(x − 1, tj+1 ), V(0, tj+1 ) = V0 (0, tj+1 ), V(1− , tj+1 ) = −(â(1))−1 (1tf (1, tj+1 ) − U(1, tj ) + 1tb(1)φ(0, tj+1 )), where V0 (x, tj+1 ) is the solution of the reduced problem.

12


For x ∈ (1+ , 2), V(x, tj+1 ) satisfies the following problem: ε1t

d2 V(x, tj+1 ) − aˆ (x)V(x, tj+1 ) − 1tb(x)V(x − 1, tj+1 ) dx2 = 1tf (x, tj+1 ) − U(x, tj ),

V(1+ , tj+1 ) = −(â(1))−1 (1tf (1, tj+1 ) − U(1, tj ) + 1tb(1)V0 (0, tj+1 )), V(2, tj+1 ) = V0 (2, tj+1 ).


The singular component W(x, tj+1 ) satisfies the following differential equations: ε1t

d2 W(x, tj+1 ) − aˆ (x)W(x, tj+1 ) = 0, x ∈ (0, 1), dx2

ε1t

d2 W(x, tj+1 ) − aˆ (x)W(x, tj+1 ) − 1tb(x)W(x − 1, tj+1 ) = 0, x ∈ (1, 2), dx2

W(0, tj+1 ) = U(0, tj+1 ) − V(0, tj+1 ), W(2, tj+1 ) = U(2, tj+1 ) − V(2, tj+1 ), W(1+ , tj+1 ) − W(1− , tj+1 ) = V(1− , tj+1 ) − V(1+ , tj+1 ), Wx (1+ , tj+1 ) − Wx (1− , tj+1 ) = Vx (1− , tj+1 ) − Vx (1+ , tj+1 ). The singular component of the solution can be further decomposed as: W L (x, tj+1 ) = W1L (x, tj+1 ) + W2L (x, tj+1 ), W R (x, tj+1 ) = W1R (x, tj+1 ) + W2R (x, tj+1 ). For x ∈ (0, 1), we have ε1t

d2 W1L (x, tj+1 ) − aˆ (x)W1L (x, tj+1 ) = 0, dx2

W1L (0, tj+1 ) = W(0, tj+1 ),

(3.15)

W1L (1, tj+1 ) = 0 and W1L (x, tj+1 ) = 0, ∀ x ∈ (1, 2].

For x ∈ (0, 1), we have ε1t

d2 W1R (x, tj+1 ) − aˆ (x)W1R (x, tj+1 ) = 0, dx2

W1R (0, tj+1 ) = 0, W1R (1, tj+1 ) = A and W1R (x, tj+1 ) = 0, ∀ x ∈ (1, 2].

(3.16)


13

For x ∈ (1, 2), we have

d2 W2L (x, tj+1 ) − aˆ (x)W2L (x, tj+1 ) − 1tb(x)W2L (x − 1, tj+1 ) = 0, dx2 (3.17) W2L (1, tj+1 ) = B, ε1t

W2L (2, tj+1 ) = 0 and W2L (x, tj+1 ) = 0, ∀ x ∈ [0, 1).

For x ∈ (1, 2), we have

d2 W2R (x, tj+1 ) − aˆ (x)W2R (x, tj+1 ) − 1tb(x)W2R (x − 1, tj+1 ) = 0, dx2 (3.18) W2R (1, tj+1 ) = 0,


ε1t

W2R (2, tj+1 ) = W(2, tj+1 ) and W2R (x, tj+1 ) = 0, ∀ x ∈ [0, 1).

Here, A and B are constants to be chosen to satisfy the jump condition at the points (1, tj+1 ), ∀ 1 ≤ j + 1 ≤ M. To get the bounds of the derivatives of the solution U(x, tj+1 ) of (3.1) in x direction, we may assume without loss of generality that the initial and interval boundary data given in (3.2) are identically zero, i.e., U0 (x) = 0, φ(x, tj+1 ) = 0 and ψ(2, tj+1 ) = 0. Lemma 3.5. The solution U(x, tj+1 ) of the semidiscrete problem (3.1)–(3.2) satisfies, for k = 0, 1, 2, 3 dk U(x, t ) j+1 ∀ x ∈ [0, 1], ≤ C(1 + ε−k/2 e1 (x, b∗ )), dxk r

b∗ where e1 (x, b∗ ) = exp −x ε

and

!

r

b∗ + exp −(1 − x) ε

dk U(x, t ) j+1 ≤ C(1 + ε−k/2 e2 (x, b∗ )), dxk r

b∗ where e2 (x, b∗ ) = exp −(x − 1) ε

!

!

,

∀ x ∈ [1, 2], r

b∗ + exp −(2 − x) ε

!

.

Proof. For derivation of the required estimates, we refer the readers to [8, 27].

To obtain the parameter uniform error estimates, we calculate the bounds on smooth and singular components of U(x, tj+1 ) as well as on their derivatives.

14


Lemma 3.6. The singular component W(x, tj+1 ) of the solution satisfies, for 0 ≤ k≤3 dk W(x, t ) j+1 ≤ Cε−k/2 e1 (x, b∗ ), ∀ x ∈ [0, 1], dxk dk W(x, t ) j+1 ≤ Cε−k/2 e2 (x, b∗ ), ∀ x ∈ [1, 2]. dxk Proof. For proof of the lemma, please refer to [8, 9, 27].


3.2. Bounds on the regular component of the solution

Lemma 3.7. The smooth component V(x, tj+1 ) of the solution U(x, tj+1 ) at (j + 1)st time level satisfies for k = 0, 1, 2, 3, dk V(x, t ) −(k−2) j+1 ∗ 2 ≤ C 1 + ε e (x, b ) , ∀ x ∈ [0, 1), 1 dxk dk V(x, t ) −(k−2) j+1 ∗ 2 ≤ C 1 + ε e (x, b ) , ∀ x ∈ (1, 2]. 2 dxk Proof. The smooth component V(x, tj+1 ) of U(x, tj+1 ) is

V(x, tj+1 ) = V0 (x, tj+1 ) + εV1 (x, tj+1 ) + ε2 V2 (x, tj+1 ),

(3.19)

where V0 (x, tj+1 ) is the solution of reduced problem corresponding to semidiscretized problem (3.1)–(3.2) and V1 (x, tj+1 ) satisfies d2 V1 (x, tj+1 ) d2 V0 (x, tj+1 ) ε1t − aˆ (x)V1 (x, tj+1 ) = −1t , ∀ x ∈ [0, 1), dx2 dx2 2 d V1 (x, tj+1 ) ε1t − aˆ (x)V1 (x, tj+1 ) − 1tb(x)V1 (x − 1, tj+1 ) (3.20) dx2 d2 V0 (x, tj+1 ) , ∀ x ∈ (1, 2], dx2 and V1 (0, tj+1 ) = 0, V1 (1− , tj+1 ) = 0, V1 (1+ , tj+1 ) = V1 (2, tj+1 ) = 0. Further, the proof of the lemma can be easily derived following the techniques of [8, 9, 27]. = −1t

4. Spatial discretization In [0, 1], three piecewise uniform meshes are placed as follows: [0, 1] = [0, τ ] ∪ (τ , 1 − τ ] ∪ (1 − τ , 1],


15


where τ is the transition point which decides the points separating the three piecewise uniform meshes. In the interval (τ , 1 − τ ], a uniform mesh with N/4 mesh points is placed and on each of the subintervals [0, τ ] and (1 − τ , 1], a uniform mesh of N/8 mesh points is placed. Transition point τ is defined by: r 1 ε N . (4.1) τ = min , 2 ∗ ln 4 b 2 In [0, 1], the mesh elements are defined by  if i = 0,   0, xi =   xi−1 + h(i), if 1 ≤ i ≤ N 2 and

where

 N   h1 , if 0 ≤ i ≤ ,   8     N 3N h(i) = h2 , if +1≤i≤ ,  8 8        h1 , if 3N + 1 ≤ i ≤ N , 8 2

(4.2)

(4.3)

4(1 − 2τ ) 8τ , h2 = . (4.4) N N Similarly three piecewise uniform meshes are placed on the interval (1, 2] as follows: h1 =

(1, 2] = (1, 1 + τ ] ∪ (1 + τ , 2 − τ ] ∪ (2 − τ , 2], where τ is the same as defined in (4.1). To guarantee that each uniform mesh contain at least one point, we take N = 8r, r ≥ 2. In [1, 2], the mesh elements are defined by: xi = xi−1 + h(i),

∀

i=

N + 1, . . . , N, where 2

 N 5N   h1 , if +1≤i≤ ,   2 8     7N 5N h(i) = h2 , if +1≤i≤ ,  8 8        h1 , if 7N + 1 ≤ i ≤ N, 8 where h1 and h2 are same as in (4.4).

(4.5)

16


In this way, a fitted mesh is constructed to have more mesh points in the boundary layer region. The resulting discretized domain on the intervals [0, 1] and [1, 2], respectively, is given as o n = 1 , (4.6) DN = 0 = x , x , . . . , x N 0 1 1 2 n o DN (4.7) 2 = 1 + h1 = x N +1 , . . . , xN = 2 . 2

On the set of grid points DN 1

and DN 2 , the differential operator of Equation (3.3)

is now discretize by using finite differences defined as: Ui+1,j+1 − Ui,j+1 , hi+1 Ui,j+1 − Ui−1,j+1 , = hi − 2(D+ x Ui,j+1 − Dx Ui,j+1 ) = . hi + hi+1


D+ x Ui,j+1 = D− x Ui,j+1 − D+ x Dx Ui,j+1

(4.8)

The fully discrete scheme corresponding to the semi discretized problem (3.1)–(3.2) is given by: For i = 1, 2, . . . , N2 − 1, j + 1 = 1, 2, . . . , M, For i =

N 2

LN,M 1 Ui,j+1 = gî,j+1 ,

(4.9)

+ 1, . . . , N − 1, j + 1 = 1, 2, . . . , M,

LN,M 2 Ui,j+1 = gi,j+1 ,

(4.10)

subject to the following boundary and initial conditions: Ui,0 = Uo (xi ), Ui,j+1 = φi,j+1 , UN,j+1 = ψj+1 ,

i = 0, 1, 2, . . . , N, −N −N , + 1, . . . , 0, j + 1 = 1, 2, . . . , M, (4.11) 2 2 j + 1 = 1, 2, . . . , M. i=

Also, we have − D+ x UN/2,j+1 = Dx UN/2,j+1 ,

where + − ˆ i Ui,j+1 , LN,M 1 Ui,j+1 ≡ ε1tDx Dx Ui,j+1 − a

LN,M 2 Ui,j+1 Ui,j+1 φi,j+1 aˆ i gî,j+1

− ˆ i Ui,j+1 − 1tbi Ui−N/2,j+1 , ≡ ε1tD+ x Dx Ui,j+1 − a = U(xi , tj+1 ), gi,j+1 = g(xi , tj+1 ), = φ(xi , tj+1 ), ψj+1 = ψ(tj+1 ), = aˆ (xi ), bi = b(xi ), = gi,j+1 + 1t bi φi−N/2,j+1 .

(4.12)


17

Lemma 4.1 (Discrete minimum principle). Let ψi,j+1 be any mesh function such N that ψ0,j+1 ≥ 0 and ψN,j+1 ≥ 0. Then LN,M 1 ψi,j+1 ≤ 0, ∀ i = 1, 2, . . . , 2 − 1, N + − LN,M 2 ψi,j+1 ≤ 0, ∀ i = 2 + 1, . . . , N − 1 and Dx ψN/2,j+1 − Dx ψN/2,j+1 ≤ 0 implies that ψi,j+1 ≥ 0, ∀ i = 0, 1, . . . , N − 1, N. Proof. Let j∗ ∈ {0, 1, . . . , N} be such that


ψj∗ ,j+1 =

min ψi,j+1

D1 N ∪D2 N

and suppose that ψj∗ ,j+1 < 0. This implies that j∗ ∈ / {0, N} as it is given that ψ0,j+1 ≥ 0 and ψN,j+1 ≥ 0. Also, we have N N,M ∗ ∗ L1 ψj ,j+1 > 0, ∀ j ∈ 1, 2, 3, . . . , − 1 , 2 N N,M ∗ ∗ L2 ψj ,j+1 > 0, ∀ j ∈ + 1, . . . , N − 1 , 2 − D+ x ψN/2,j+1 − Dx ψN/2,j+1 > 0,

which is a contradiction to the given conditions. This implies that our supposition is wrong, so ψj∗ ,j+1 ≥ 0. Hence ψi,j+1 ≥ 0, ∀ i = 0, 1, . . . , N. Lemma 4.2. Let ψi,j+1 be any mesh function then for 0 ≤ i ≤ N it satisfies ( ) ||LN,M ||LN,M 1 ψ|| 2 ψ|| |ψi,j+1 | ≤ max |ψ0,j+1 |, |ψN,j+1 |, , . (21tb∗ + 1) (21tb∗ + 1) Proof. Let us consider two barrier functions for 0 ≤ i ≤ N of the type ) ( N,M N,M ||L ψ|| ||L ψ|| ± 1 2 , ± ψi,j+1 . (4.13) πi,j+1 = max |ψ0,j+1 |, |ψN,j+1 |, (21tb∗ + 1) (21tb∗ + 1) ± ± From Equation (4.13), it is clear that π0,j+1 ≥ 0 and πN,j+1 ≥ 0. N N,M ± L1 πi,j+1 ≤ 0, ∀ i ∈ 1, 2, 3, . . . , − 1 , 2 N N,M ± L2 πi,j+1 ≤ 0, ∀ i ∈ + 1, . . . , N − 1 , 2 − ± (D+ x − Dx )πi,j+1 = 0, i = N/2. ± Hence by Lemma 4.1, πi,j+1 ≥ 0, ∀ 0 ≤ i ≤ N, which leads to the required result.

18


5. Error estimates To calculate the error estimate for the numerical solution, we decompose the discrete solution into regular and singular components as we earlier decomposed U(x, tj+1 ). The decomposition of Ui,j+1 is as follows: Ui,j+1 = Vi,j+1 + Wi,j+1 , where Vi,j+1 and Wi,j+1 satisfies the following differential equations: LN,M 1 Vi,j+1 = gî,j+1 ,

i = 1, 2, 3, . . . ,

N − 1, 2


V0,j+1 = V(0, tj+1 ), V N −1,j+1 = V(1− , tj+1 ), 2

LN,M 2 Vi,j+1 = gi,j+1 ,

i=

N + 1, . . . , N − 1, 2

V N +1,j+1 = V(1+ , tj+1 ), 2

VN,j+1 = V(2, tj+1 ), LN,M 1 Wi,j+1 = 0,

i = 1, 2, 3, . . . ,

N − 1, 2

W0,j+1 = W(0, tj+1 ), LN,M 2 Wi,j+1 = 0,

i=

N + 1, . . . , N − 1, 2

WN,j+1 = W(2, tj+1 ), V N +1,j+1 + W N +1,j+1 = V N −1,j+1 + W N −1,j+1 , 2

2

2

2

− + + D− x V N ,j+1 + Dx W N ,j+1 = Dx V N ,j+1 + Dx W N ,j+1 . 2

2

2

2

The nodal error ei,j+1 is given by: ei,j+1 = U(xi , tj+1 ) − Ui,j+1 = (V + W)(xi , tj+1 ) − (Vi,j+1 + Wi,j+1 ) = (V(xi , tj+1 ) − Vi,j+1 ) + (W(xi , tj+1 ) − Wi,j+1 ). The following theorem contains the error estimates for the smooth and singular components. Theorem 5.1. The smooth components V(xi , tj+1 ) and Vi,j+1 of the solution of semidiscrete problem (3.1)–(3.2) and discrete problem (4.9)–(4.12), respectively,


19

satisfy the following estimates for i 6= N/2: −2 |LN,M 1 (V(xi , tj+1 ) − Vi,j+1 )| ≤ CN ln N, 0 ≤ i ≤ N/2 − 1, −2 |LN,M 2 (V(xi , tj+1 ) − Vi,j+1 )| ≤ CN ln N, N/2 + 1 ≤ i ≤ N,

and the singular components W(xi , tj+1 ) and Wi,j+1 of semidiscrete problem (3.1)–(3.2) and discrete problem (4.9)–(4.12), respectively, satisfy the following estimates for i 6= N/2: −2 |LN,M 1 (W(xi , tj+1 ) − Wi,j+1 )| ≤ C(NlnN) , 0 ≤ i ≤ N/2 − 1,


−2 |LN,M 2 (W(xi , tj+1 ) − Wi,j+1 )| ≤ C(NlnN) , N/2 + 1 ≤ i ≤ N,

Proof. The proof can be easily derived by following the steps given in Chapter 6 of [28].

5.1. Some preliminary results for the main error estimates

Consider the point (x N , tj+1 ) and let h∗ = h−N = h+N , where h−N = x N − x N −1 2

2

2

2

2

2

− and h+N = x N +1 − x N , as (D+ x − Dx )U N ,j+1 = 0 so error estimates at (x N , tj+1 ) 2

2

reduces to |(D+ x

− D− x )e N2 ,j+1 |

2

2

2

d d + − − Dx − U(x N , tj+1 ) = Dx − 2 dx dx + − d d ≤ Dx − U(x N , tj+1 ) + Dx − U(x N , tj+1 ) 2 2 dx dx d2 U(η , t ) 1 + 1 j+1 ≤ hN/2 max 2 dx2 η1 ∈(1,2) d2 U(η , t ) 1 − 2 j+1 + hN/2 max 2 dx2 η2 ∈(0b,1)

d2 U(x, t ) j+1 ≤ Ch∗ max 2 dx x∈(0,1)∪(1,2) ≤

Ch∗ . ε

20


Now for i = 0, 1, 2, . . . , N, define a set of discrete barrier functions by  Q p i  (1 + b∗ /εhk ) N  k=1  , 0 ≤ i ≤  N p  Q2 2   (1 + b∗ /εhk )  k=1 wi,j+1 = p QN−1    (1 + b∗ /εhk+1 ) N  k=i  , ≤ i ≤ N. p Q   2  N−1N (1 + b∗ /εhk+1 )

(5.1)

k= 2

From (5.1) it is clear that


w0,j+1 > 0, wN/2,j+1 = 1, wN,j+1 > 0

(5.2)

and for 0 ≤ i ≤ N, we have 0 ≤ wi,j+1 ≤ 1.

(5.3)

For i = 0, 1, . . . , N2 ,

wi+1,j+1 − wi,j+1 hi+1 p Qi p (1 + b∗ /εhk ) 1 k=1 ∗ /ε h 1 + b − 1 = i+1 p hi+1 Q N2 ∗ k=1 (1 + b /εhk p = b∗ /εwi,j+1 .

D+ x wi,j+1 =

For i = 0, 1, . . . , N2 , D− x wi,j+1

Q i

p ! ∗ /εh ) b k 1 1 1− = N p p hi Q 2 1 + b∗ /ε hi ∗ k=1 (1 + b /εhk k=1 (1 +

= For i = 0, 1, . . . , N2 , δ 2 wi,j+1

p b∗ /ε

wi,j+1 . p 1 + b∗ /ε hi

p p 1 1 = b∗ /ε − b∗ /ε p (hi + hi+1 )/2 1 + b∗ /εhi ! 2b∗ hi 1 = wi,j+1 p ε hi + hi+1 1 + b∗ /εhi ≤

(5.4)

2b∗ wi,j+1 . ε

(5.5)

!

wi,j+1

(5.6)


Similarly, for i =

N 2 , . . . , N,

p ∗ D+ x wi,j+1 = − b /ε

wi,j+1 , p (1 + b∗ /ε hi+1 )

(5.7)

p ∗ D− x wi,j+1 = − b /ε wi,j+1 , 2

and δ wi,j+1 In particular at i = − (D+ x − Dx )wi,j+1

(5.8)

2b∗ ≤ wi,j+1 . ε

(5.9)

N 2

using (5.2), (5.5), (5.7), we get  r  r ∗ ∗ b 1 b 1  wN/2,j+1 = − − p p + ∗ ε 1 + b /ε h N ε 1 + b∗ /ε h−N 2


21

2

C ≤ −√ . ε

(5.10)

Consider 2 ˆ i wi,j+1 LN,M 1 wi,j+1 = ε1tδ wi,j+1 − a

≤ (1t2b∗ − aˆ i )wi,j+1 [using (5.6)].

(5.11)

Similarly, we have ∗ ˆ i )wi,j+1 − 1tbi wi−N/2,j+1 . LN,M 2 wi,j+1 ≤ (1t2b − a

(5.12)

Theorem 5.2. Let U(xi , tj+1 ) be the solution of the problem (3.1)–(3.2) and Ui,j+1 be the solution of (4.9)–(4.12) then for 0 ≤ i ≤ N at (j + 1)st time level |U(xi , tj+1 ) − Ui,j+1 | ≤ C(N −1 lnN)2 .

Proof. Consider two mesh functions given by p ψ ± i,j+1 = C1 (N −1 lnN)2 + C2 b∗ /εh∗ wi,j+1 ± ei,j+1 ,

0 ≤ i ≤ N,

(5.13)

where C1 and C2 are constants.

p ± ˆ i (N −1 lnN)2 + C2 b∗ /εh∗ L1N,M wi,j+1 ± LN,M LN,M 1 ψ i,j+1 = −C1 a 1 ei,j+1 . (5.14) Using (5.11) in (5.14) and Theorem 5.1 p ± ˆ i (N −1 lnN)2 + C2 b∗ /εh∗ (1t2b∗ − aˆ i )wi,j+1 LN,M 1 ψ i,j+1 ≤ −C1 a ± C(N −1 lnN)2 . ≤0

± LN,M ai − 1tbi )(N −1 lnN)2 2 ψ i,j+1 = C1 (−ˆ p N,M + C2 b∗ /εh∗ LN,M 2 wi,j+1 ± L2 ei,j+1 .

(5.15)

22


Using (5.12) in (5.15) and Theorem 5.1 and by appropriate choice of C1 and C2 , we get ± LN,M ai − 1tbi )(N −1 lnN)2 2 ψ i,j+1 ≤ C1 (−ˆ p + C2 b∗ /ε h∗ ((1t2b∗ − aˆ i )wi,j+1 − 1tbi wi−N/2,j+1 )

± C(N −1 lnN)2

≤ 0.


(D+ x

± − D− x )ψ N2 ,j+1

p Ch∗ −C ∗ ± ≤ C2 h b /ε √ ε ε ∗

≤ 0, for proper choice of C2 .

(5.16)

Also, we have ψ ± 0,j+1 ≥ 0, ψ ± N,j+1 ≥ 0. Now applying the discrete minimum principle given in Lemma 4.1 on the mesh functions ψ ± i,j+1 and using (5.3), we get the required result. Theorem 5.3. Let u(x, t) be the solution of problem (1.2)–(1.3) and Ui,j+1 be the solution of (4.9)–(4.12) then, we have |u(xi , tj+1 ) − Ui,j+1 | ≤ C(1t + (N −1 lnN)2 ),

i = 0, 1, . . . , N; j = 0, 1, . . . , M.

Proof. The proof follows from Lemma 3.3 and Theorem 5.2.

6. Numerical experiments To verify the proposed theory, we do some experiments through solving some examples. The definition of maximum pointwise error EεN,1t is as follows [11]: EεN,1t =

1t

max |U N,1t (xi , tj ) − U 2N, 4 (xi , tj )|,

0≤i,j≤N,M

(6.1)

1t

where U N,1t (xi , tj ) and U 2N, 4 (xi , tj ) are numerical solutions obtained on a fitted mesh in the spatial direction and uniform mesh in the time direction. To calculate U N,1t (xi , tj ), N and M mesh intervals are considered on space and time intervals, respectively. To calculate the spatial order of convergence ζεN,1t , we used the following formula [11] given by: 2N,1t/4

ζεN,1t

log(EεN,1t /Eε = log 2

)

.

(6.2)


23

It is shown experimentally that the proposed numerical scheme is parameter uniform convergent. We take T = 2 and consider the following problems: Example 6.1. ∂u ∂ 2u + ε 2 − 3u(x, t) + u(x − 1, t) = −1, ∂t ∂x subject to the following interval boundary conditions and the initial condition


−

u(x, t) = 0,

∀ (x, t) ∈ L = {(x, t) : −1 ≤ x ≤ 0 and 0 ≤ t ≤ T},

u(2, t) = 0,

∀ (2, t) ∈ R = {(2, t) : 0 ≤ t ≤ T},

u(x, 0) = 0,

∀ x ∈ D = [0, 2].

Example 6.2. ∂u ∂ 2u + ε 2 − 5u(x, t) + 2u(x − 1, t) = −2, ∂t ∂x subject to the following interval boundary conditions and the initial condition −

u(x, t) = 0,

∀ (x, t) ∈ L = {(x, t) : −1 ≤ x ≤ 0 and 0 ≤ t ≤ T},

u(2, t) = 0,

∀ (2, t) ∈ R = {(2, t) : 0 ≤ t ≤ T},

u(x, 0) = sin(π x),

∀ x ∈ D = [0, 2].

Example 6.3. ∂u ∂ 2u + ε 2 − (x + 6)u(x, t) + (x2 + 1)u(x − 1, t) = −3, ∂t ∂x subject to the following interval boundary conditions and the initial condition −

u(x, t) = 0,

∀ (x, t) ∈ L = {(x, t) : −1 ≤ x ≤ 0 and 0 ≤ t ≤ T},

u(2, t) = 0,

∀ (2, t) ∈ R = {(2, t) : 0 ≤ t ≤ T},

u(x, 0) = sin(π x),

∀ x ∈ D = [0, 2].

In Tables 1–3, spatial order of convergence for Examples 6.1–6.3 is given, respectively. Graphs are plotted for numerical solutions of the considered problems. It is shown that the numerical solutions of the Examples 6.1–6.3 possess singular behavior at the point x = 0, 1, 2. It is observed that singularity at point x = 0 on the spatial domain is propagated at the point x = 1 due to the presence of unit shift in the reaction term. Because of this singularity, numerical solution exhibits an interior layer at x = 1. The numerical solutions are plotted in Figures 1–3 for different values of ε. It is shown that as ε decreases, an interior

24


Table 1. Numerical order of convergence ζεN,1t for Example 6.1 when 2b∗ = 0.9. ε


20 2−1 2−2 2−3 2−4 2−5 2−6 2−7 2−8 2−12 2−16 2−20

N = 64 M = 32 1.7926 1.8425 1.8681 1.8787 1.8909 1.8901 1.8870 1.8856 1.8550 1.3369 1.3369 1.3369

N = 128 M = 128 1.9408 1.9556 1.9645 1.9674 1.9712 1.9706 1.9704 1.9695 1.9664 1.5402 1.5402 1.5402

N = 256 M = 512 1.9846 1.9886 1.9909 1.9916 1.9927 1.9924 1.9923 1.9921 1.9899 1.6333 1.6333 1.6333

N = 512 M = 2048 1.9961 1.9971 1.9977 1.9979 1.9981 1.9981 1.9981 1.9980 1.9979 1.6722 1.6722 1.6722


20 2−2 2−4 2−6 2−8 2−10 2−12 2−14 2−16 2−18 2−20

N = 64 M = 32 1.4946 1.7085 1.7790 1.7887 1.7906 1.7908 1.7908 1.7908 1.7908 1.7908 1.7908

N = 128 M = 128 1.8454 1.9177 1.9369 1.9400 1.9409 1.9414 1.8148 1.8260 1.8314 1.8341 1.8354

N = 256 M = 512 1.9599 1.9787 1.9838 1.9846 1.9849 1.9849 1.5245 1.5162 1.5121 1.5101 1.5091

N = 512 M = 2048 1.9897 1.9946 1.9959 1.9961 1.9962 1.9962 1.6277 1.6266 1.6261 1.6258 1.6257


20 2−2 2−4 2−6 2−8 2−10 2−12 2−14 2−16 2−18 2−20 2−22 2−24 2−26 2−28 2−30

N = 64 M = 32 1.4730 1.6473 1.6975 1.6997 1.7005 1.7008 1.7002 1.6991 1.6979 1.6973 1.6970 1.6968 1.6967 1.6967 1.6967 1.6967

N = 128 M = 128 1.8278 1.8917 1.9076 1.9142 1.9152 1.9154 1.4175 1.4190 1.4200 1.4205 1.4208 1.4209 1.4210 1.4210 1.4210 1.4210

N = 256 M = 512 1.9524 1.9716 1.9765 1.9774 1.9778 1.9779 1.5284 1.5277 1.5273 1.5271 1.5270 1.5270 1.5270 1.5270 1.5270 1.5270

N = 512 M = 2048 1.9878 1.9927 1.9940 1.9943 1.9944 1.9944 1.6294 1.6291 1.6290 1.6289 1.6289 1.6289 1.6289 1.6289 1.6289 1.6289

N = 1024 M = 8192 1.9990 1.9992 1.9994 1.9994 1.9995 1.9995 1.9995 1.9994 1.9993 1.7052 1.7052 1.7052

N = 1024 M = 8192 1.9974 1.9986 1.9989 1.9990 1.9990 1.9990 1.6889 1.6881 1.6878 1.6876 1.6875

N = 1024 M = 8192 1.9969 1.9981 1.9985 1.9985 1.9985 1.9986 1.6836 1.6835 1.6834 1.6834 1.6834 1.6834 1.6834 1.6834 1.6834 1.6834


25


Figure 1. The numerical solution Ui,j+1 for Example 6.1 at final time level when 2b∗ = 0.8 and ε = 2−1 , 2−3 , 2−5 , 2−20 .



layer appear at x = 1 and steepness of the layers on both sides of the boundary, i.e., L and R increase and this implies width of the boundary layer region decreases.

26



7. Conclusion and discussion We analyzed the proposed numerical scheme for stability, consistency, and parameter uniform convergence. Numerical experiments have been performed to confirm the predicted theory. Numerical order of convergence is calculated for the problems considered. Delay differential equations are very important from the point of view of applications in many areas of science and engineering [20–22, 36]. We have identified several novel features in the solution due to shift argument, which are not present in the solution of differential equations without delay term. In case of large finite time, i.e., T ≫ 1 a substitution like t ′ = t/T reduces the time interval to [0, 1]. In return, ∂u/∂t is replaced in the differential equation with (1/T)∂u/∂t ′ , which means that the time derivative is now multiplied by a small parameter. So, this is an interesting topic to investigate in the future. In case T tends to ∞, the problem is reduced to the steady-state case and it may be expected that the solution of the original problem tends to the steadystate solution. In future work, we try to extend the method for other meaningful mathematical models with large time. Acknowledgments The authors express their sincere thanks to Prof. Relja Vulanović, Professor and Coordinator of Mathematics, Department of Mathematical Sciences, Kent State University at Stark, North Canton, Ohio 44720, U.S.A., for the valuable suggestions. The authors express their sincere thanks to Dr. Pratima Rai, Assistant Professor, Department of Mathematics, University of Delhi, Delhi, India for the discussion.

Funding The research work of the first author is supported by U.G.C. (letter no. F.17-7(J)/08(SA-1) dated 01-Feb-2012) New Delhi, India’.

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