numerical solution of first-order hyperbolic partial differential-difference equation with shift

Numerical Solution of First-Order HyperbolicPartial Differential-Difference Equation with ShiftParamjeet Singh, Kapil K. SharmaDepartment of Mathematics, Center for Advanced Study in Mathematics,Panjab University, Chandigarh, India

Received 24 January 2008; accepted 23 October 2008Published online 12 February 2009 in Wiley InterScience (www.interscience.wiley.com).DOI 10.1002/num.20419

In this article, we continue the numerical study of hyperbolic partial differential-difference equation that wasinitiated in (Sharma and Singh, Appl Math Comput 201(2008), 229–238). In Sharma and Singh, the authorsconsider the problem with sufficiently small shift arguments. The term negative shift and positive shift areused for delay and advance arguments, respectively. Here, we propose a numerical scheme that works nicelyirrespective of the size of shift arguments. In this article, we consider hyperbolic partial differential-differenceequation with negative or positive shift and present a numerical scheme based on the finite difference methodfor solving such type of initial and boundary value problems. The proposed numerical scheme is analyzedfor stability and convergence in L∞ norm. Finally, some test examples are given to validate convergence,the computational efficiency of the numerical scheme and the effect of shift arguments on the solution.© 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 26: 107–116, 2010

Keywords: hyperbolic partial differential equation; differential-difference equation; negative shift; positiveshift; finite difference; upwind scheme

I. INTRODUCTION

Hyperbolic partial differential-difference equations provide a tool to simulate several realisticphysical and biological phenomena. Several biological phenomena can be modeled by the time-dependent partial differential-difference equations of hyperbolic type such as in Stein’s model[1,2], and the distribution of neuronal firing intervals is modeled by a first-order hyperbolic partialdifferential-difference equation:

∂F

∂t(v, t) − (v/τ0)

∂F

∂v(v, t) = pe[F(v − 1, t) − F(v, t)] + pi[F(v + v0, t) − F(v, t)], (1.1)

where Vt equal the depolarization at time t ; F(v, t) equal the probability that Vt ≤ v at timet . Here, it is assumed that excitatory and inhibitory impulse occur randomly with a frequencype/sec and pi /sec, respectively. After each neuronal firing there is a refractory period of duration

Correspondence to: Kapil K. Sharma, Department of Mathematics, Center for Advanced Study in Mathematics, PanjabUniversity, Chandigarh, India (e-mail: [email protected].)

© 2009 Wiley Periodicals, Inc.

108 SINGH AND SHARMA

t0, during which the impulse have no effect and the membrane depolarization Vt is reset to be zero.At times t > t0, an excitatory impulse produces unit depolarization while an inhibitory impulseproduces v0 unit repolarization, and if the depolarization reaches a threshold of r units, the neuronfires. For subthreshold levels, the depolarization decays exponentially between impulses with thetime constant τ0.

Equation (1.1) is a first-order hyperbolic partial differential-difference equation that does nothave an analytical solution and cannot be solved using classical numerical method [3–8]. Thehyperbolic partial differential-difference equation with sufficiently small delay argument is con-sidered in [9], where the authors use the Taylor series approximation for the terms containingshift arguments. The Taylor series approximation may lead to a bad approximation when the sizeof the shift argument is large. Therefore, the numerical scheme presented in [9] does not work forthe problems with large shift argument. To overcome this difficulty, we generate a special typeof mesh so that the difference term lies on the nodal point in the discretized domain and presenta numerical scheme that works nicely in both the cases, i.e., when the size of shift argument issmaller as well as bigger.

This article is organized as follows. We consider two model problems in the next two sections.In section II, we consider a hyperbolic partial differential equation with negative shift, and anumerical scheme based on finite difference method is constructed for finding the approximatesolution. Stability and convergence analysis is done in L∞ norm. In section III, we consider ahyperbolic partial differential equation with positive shift and discuss the numerical solution. Insection IV, some numerical experiments are done to check the efficiency of the numerical schemeand convergence. In the last section, we conclude the analysis done in this article.

II. HYPERBOLIC PARTIAL DIFFERENTIAL-DIFFERENCE EQUATIONWITH NEGATIVE SHIFT

A. Formulation of the Problem

We consider the following hyperbolic partial differential-difference equation with negative shift,i.e., with delay,

∂u

∂t(x, t) + a(x, t)

∂u

∂x(x, t) = b(x, t)u(x − δ, t), (2.1)

where a(x, t) and b(x, t) are functions of x and t and δ is any positive constant.Let region of interest is � = [0, 1] × [0, tf ]. Since the differential Eq. (2.1) is first-order

hyperbolic, we require only one boundary condition according to the direction of characteristics[10]. If a > 0, due to the presence of delay in Eq. (2.1), we need a boundary-interval conditionin the left side of domain, i.e., in the interval [−δ, 0] and if a < 0, we need a boundary conditionat x = 1. Thus, initial and boundary-interval conditions for this equation are given by

u(x, 0) = u0(x) (2.2a)

u(s, t) = g(s, t); ∀s ∈ [−δ, 0]; for a > 0 (2.2b)

u(1, t) = f (t); for a < 0 (2.2c)

B. Numerical Scheme for Approximation to the Solution

In this section, we construct a numerical scheme for solving the problem (2.1–2.2) based on anupwind finite difference scheme.

Numerical Methods for Partial Differential Equations DOI 10.1002/num

PARTIAL DIFFERENTIAL-DIFFERENCE EQUATION 109

Case (i) a(x , t ) > 0. In this case, we use forward difference in time and backward differencein space and the difference scheme for problem (2.1–2.2) is given by,

Un+1j − Un

j

�t+ an

j

Unj − Un

j−1

�x= bn

j u(xj − δ, tn), (2.3)

where (xj = j�x, tn = n�t); j = 0, 1, . . . , J ; n = 0, 1, 2 . . . .; are mesh points, where�x = 1/J , is space step and �t is the time step, Un

j ≈ u(xj , tn) and anj = a(xj , tn).

To tackle the delay term in the difference scheme (2.3), we discretize the domain in such a waythat (xj − δ) is a nodal point; ∀j = 0, 1, . . . , J ; i.e., we choose �x s.t . δ = m0�x, m0 ∈ N andwe take total number of points in x−direction s.t.

J = 1

�x= k

mantissa(δ)

δ, k ∈ N.

Thus, the delay term (∀j = 0, 1, . . . , J ) in right side of Eq. (2.3) can be written as

u(xj − δ, tn) = u(j�x − m0�x, tn)

= u((j − m0)�x, tn)

= Unj−m0

,

which reduces the difference scheme (2.3) as

Un+1j − Un

j

�t+ an

j

Unj − Un

j−1

�x= bn

j Unj−m0

, (2.4)

and the initial and boundary-interval conditions for the difference Eq. (2.4) are given by

U 0j = u0(xj ), j = 1, . . . , J − 1 (2.5a)

Un0 = u(s, tn) = g(s, tn); ∀s ∈ [−δ, 0], n = 0, 1, 2 . . . . (2.5b)

Stability Analysis. Definition—The finite difference method is called stable in the norm ‖.‖ ifthere exists constant C > 0, independent of the space and possibly depending on the time stepsuch that

‖Un‖ ≤ C‖U 0‖, ∀n = 1, 2, . . . .

Consider the finite difference scheme as given in Eq. (2.8),

Un+1j =

(1 − an

j

�t

�x

)Un

j + anj

�t

�xUn

j−1 + bnj �tUn

j−m0

∣∣Un+1j

∣∣ ≤∣∣∣∣1 − an

j

�t

�x

∣∣∣∣∣∣Un

j

∣∣ +∣∣∣∣an

j

�t

�x

∣∣∣∣∣∣Un

j−1

∣∣ + ∣∣bnj �t

∣∣∣∣Unj−m0

∣∣taking the norm, we get

‖Un+1‖L∞ = supj

∣∣Un+1j

∣∣

≤∣∣∣∣1 − an

j

�t

�x

∣∣∣∣ supj

∣∣Unj

∣∣ +∣∣∣∣an

j

�t

�x

∣∣∣∣ supj

∣∣Unj−1

∣∣ + ∣∣bnj �t

∣∣ supj

∣∣Unj−m0

∣∣



if∣∣an

j�t

�x

∣∣ ≤ 1, the earlier inequality reduces to

‖Un+1‖L∞ ≤ ‖Un‖L∞ + ∣∣bnj

∣∣|�t |‖Un‖L∞

≤ (1 + ∣∣bn

j

∣∣�t)‖Un‖L∞ .

Here, it is assumed that b(x, t) is bounded in the domain, i.e., |bnj | ≤ M , where M is any constant

and M�t is of the form O(�(t)). Therefore, the stability inequality is given by

‖Un+1‖L∞ ≤ C‖U 0‖L∞ ,

where C = 1 + O(�(t)). This proves the stability of the numerical scheme.

Convergence Analysis. Definition—Numerical scheme is said to be convergent if for any fixedpoint (x∗, t∗) in the domain �,

xj → x∗, tn → t∗ ⇒ Unj → u(x∗, t∗).

The local truncation error of the numerical scheme is the difference between the two sides of theEq. (2.4), when the approximation Un

j is replaced throughout by the exact solution u(xj , tn) of thedifferential equation. If u is sufficiently smooth, the truncation error T n

j of the numerical scheme(2.4) is given by

T nj = un+1

j − unj

�t+ an

j

unj − un

j−1

�x− bn

j unj−m0

=[ut + 1

2�tutt + · · ·

]n

j

+[a

(ux − 1

2�xuxx

)+ · · ·

]n

j

− bnj u

nj−m0

= [ut + aux]nj − bnj u

nj−m0

+ 1

2[�tutt − a�xuxx]nj + · · ·

As u satisfied the given differential Eq. (2.1),

Hence,

[ut + aux]nj − bnj u

nj−m0

= 0.

Thus truncation error is given by

T nj = 1

2[�tutt − a�xuxx]nj + · · ·

= 1

2[(νutt − auxx)�x]nj + · · · ,

where |ν| = |a �t

�x| ≤ 1. Hence, the numerical scheme is first-order accurate in space direction as

well as time direction.Now the error in the numerical approximation is given by

enj = Un

j − u(xj , tn).



Now Unj satisfies the Eq. (2.4) exactly, while u(xj , tn) leaves the remainder T n

j �t . Therefore, erroris given by

en+1j =

(1 − an

j

�t

�x

)en

j + anj

�t

�xen

j−1 + bnj �ten

j−m0− �tT n

j

and

en0 = 0.

Let En = max{|enj |, j = 0, 1, . . . , J }

for |anj

�t

�x| ≤ 1,

En+1 = maxj

∣∣en+1j

∣∣≤ En + ∣∣bn

j

∣∣�tEn + �t maxj

∣∣T nj

∣∣≤ En + M�tEn + �t max

j

∣∣T nj

∣∣= (1 + M�t)En + �t max

j

∣∣T nj

∣∣,since we are using the given initial value for Un

j , so E0 = 0 and if we suppose that the truncationerror is bounded i.e. |T n

j | ≤ Tmax, then by induction method

En ≤ n�tTmax ≤ tf Tmax,

which proves that the method has first-order convergence while the Courant-Friedrichs-Lewycondition (CFL) is satisfied and the solution has bounded second derivatives.

Case (ii) a(x , t ) < 0. In this case, we use forward difference in time and forward difference inspace. The difference scheme for the initial and boundary value problem (2.1) is given by

Un+1j − Un

j

�t+ an

j

Unj+1 − Un

j

�x= bn

j Unj−m0

(2.6)

Here we discretize the domain in such a way as given in case (i).Initial and boundary-interval conditions for this difference equation are given by

U 0j = u0(xj ), j = 1, . . . , J − 1 (2.7a)

UnJ = fJ (tn), n = 0, 1, 2 . . . . (2.7b)

The stability and convergence estimates for the numerical scheme (2.6) can be established viaadopting similar analysis and algebra as we did in the case (i).

After rearranging the terms in Eqs. (2.4) and (2.6), we write the following equation

Un+1j =

{(1 − an

j�t

�x

)Un

j + anj

�t

�xUn

j−1 + bnj �tUn

j−m0, if an

j > 0(1 + an

j�t

�x

)Un

j − anj

�t

�xUn

j+1 + bnj �tUn

j−m0, if an

j < 0(2.8)

with the initial and boundary-interval conditions as given by Eqs. (2.5) and (2.7).



III. HYPERBOLIC PARTIAL DIFFERENTIAL-DIFFERENCE EQUATIONWITH POSITIVE SHIFT

A. Formulation of the Problem

We consider the following hyperbolic partial differential-difference equation with positive shiftin � = [0, 1] × [0, tf ],

∂u

∂t(x, t) + c(x, t)

∂u

∂x(x, t) = d(x, t)u(x + τ , t), (3.1)

where c(x, t) and d(x, t) are functions of x and t and τ is any positive constant. Initial andboundary-interval conditions for this equation are given by

u(x, 0) = u0(x) (3.2a)

u(0, t) = φ(t); for c > 0 (3.2b)

u(s, t) = ψ(s, t); ∀s ∈ [1, 1 + τ ]; for c < 0. (3.2c)

B. Numerical Scheme for Approximation to the Solution

On the similar steps as we did in the Section IIB, we construct a numerical scheme by replacingthe negative shift with the positive shift, and the scheme is given by

Un+1j =

{(1 − cn

j�t

�x

)Un

j + cnj

�t

�xUn

j−1 + dnj �tUn

j+m0, for c > 0(

1 + cnj

�t

�x

)Un

j − cnj

�t

�xUn

j+1 + dnj �tUn

j+m0, for c < 0

(3.3)

The stability and convergence of the numerical scheme can be carried out on similar steps as wedid for the problem with negative shift in section II.

IV. NUMERICAL EXPERIMENTS

In this section, we present some numerical examples to validate the predicted results establishedin the article and to see the effect of shift arguments on the solution behavior. The maximumabsolute error for the considered examples is calculated using the half mesh principle [11] as theexact solution for the considered examples are not available.

E(�x, �t) = max0≤j≤J ,0≤n≤N

∣∣U�t�x(j , n) − U

�t/2�x/2(2j , 2n)

∣∣In the following examples, the domain of consideration is � = [0, 1] × [0, 0.6].

TABLE I. The maximum absolute error for Example 1 for δ = 0.05.

�x

�t 1/100 1/200 1/400 1/800

�x/2 0.051995 0.026089 0.013032 0.006514�x/4 0.025997 0.013045 0.006516 0.003257�x/8 0.012999 0.006522 0.003258 0.001629�x/16 0.006499 0.003261 0.001629 0.000814



TABLE II. The maximum absolute error for Example 1 for δ = 0.2.

�x

�t 1/100 1/200 1/400 1/800

�x/2 0.052318 0.026159 0.013071 0.006536�x/4 0.026159 0.013079 0.006536 0.003268�x/8 0.013079 0.006540 0.003268 0.001634�x/16 0.006540 0.003270 0.001634 0.000817

TABLE III. The maximum absolute error for Example 2 for δ = 0.05.

�x

�t 1/100 1/200 1/400 1/800

�x/2 0.054495 0.027334 0.013654 0.006825�x/4 0.027247 0.013667 0.006827 0.003413�x/8 0.013624 0.006834 0.003414 0.001706�x/16 0.006812 0.003417 0.001707 0.000853

Example 1. Consider the problem (2.1) with the following coefficients:

a(x, t) = 1 + x2

1 + 2xt + 2x2 + x4; b(x, t) = 0.5;

and under the following initial and boundary-interval conditions:

u(x, 0) = exp[−10(4x − 1)2]; u(s, t) = 0, ∀s ∈ [−δ, 0]


a(x, t) = 1 + x2

1 + 2xt + 2x2 + x4; b(x, t) = 1

1 + x2t2;

TABLE IV. The maximum absolute error for Example 3 for τ = 0.05.

�x

�t 1/100 1/200 1/400 1/800

�x/2 0.050000 0.025059 0.012516 0.006257�x/4 0.025000 0.012529 0.006258 0.003128�x/8 0.012500 0.006265 0.003129 0.001564�x/16 0.006250 0.003132 0.001565 0.000782

TABLE V. The maximum absolute error for Example 3 for τ = 0.2.

�x

�t 1/100 1/200 1/400 1/800

�x/2 0.052209 0.026115 0.013047 0.006525�x/4 0.026105 0.013058 0.006524 0.003262�x/8 0.013052 0.006529 0.003262 0.001631�x/16 0.006526 0.003264 0.001631 0.000816



FIG. 1. The approximate solution of Example 1 at t = 0.6.

and under the following initial and boundary-interval conditions:

u(x, 0) = exp[−10(4x − 1)2]; u(s, t) = 0, ∀s ∈ [−δ, 0]


c(x, t) = 1 + x2

1 + 2xt + 2x2 + x4; d(x, t) = 0.5;

FIG. 2. The approximate solution of Example 1 for different value of t .




and under the following initial and boundary conditions:

u(x, 0) = exp[−10(4x − 1)2]; u(0, t) = 0

V. CONCLUSIONS

A numerical method based on finite difference approach is proposed to find the approximatesolution of the hyperbolic partial differential-difference equation with negative or positive shift.




The method works nicely irrespective the size of shift argument, i.e., for the both cases when shiftargument is smaller and when the shift argument is the bigger one. The stability and convergenceanalysis carried out in sections (2.3–2.4) prove that the proposed numerical scheme is first-orderconvergent in space and time.

For model problem with negative shift, the maximum absolute errors are tabulated in the formof Tables I and III for the considered examples with δ = 0.05 and in Table II with δ = 0.2. Formodel problem with positive shift, the maximum absolute error are tabulated in Tables IV and Vwith τ = 0.05 and τ = 0.2, resepectively. The error table illustrate that the method has first-orderconvergence in space and time direction.

The graphs of the solution of the considered examples for different values of shift argumentsare plotted in Figs. 1–4 to study the effect of the shifts on the solution behavior. If we increasethe value of δ, height of the impulse reduces, impulse moves to the right side and also width ofthe graph is reduced. Results are shown in Fig. 1 and Fig. 3. As we increase the value of τ , heightof the impulse reduces and shifting toward the right (Fig. 4). If we fix the value of negative shift,the impulse moves toward right with time (Fig. 2).

The authors are thankful to the anonymous referees for their invaluable suggestions.

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numerical solution of first-order hyperbolic partial differential-difference equation with shift

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