finite difference method for hyperbolic equations …for solving the one-dimensional hyperbolic...

Hindawi Publishing CorporationDiscrete Dynamics in Nature and SocietyVolume 2011, Article ID 562385, 15 pagesdoi:10.1155/2011/562385

Research ArticleFinite Difference Method for Hyperbolic Equationswith the Nonlocal Integral Condition

Allaberen Ashyralyev1, 2 and Necmettin Aggez1

1 Department of Mathematics, Fatih University, Istanbul 34500, Turkey2 Department of Mathematics, ITTU, Ashgabat 74400, Turkmenistan

Correspondence should be addressed to Necmettin Aggez, [email protected]

Received 12 April 2010; Accepted 4 January 2011

Academic Editor: Leonid Shaikhet

Copyright q 2011 A. Ashyralyev and N. Aggez. This is an open access article distributed underthe Creative Commons Attribution License, which permits unrestricted use, distribution, andreproduction in any medium, provided the original work is properly cited.

The stable difference schemes for the approximate solution of the nonlocal boundary valueproblem for multidimensional hyperbolic equations with dependent in space variable coefficientsare presented. Stability of these difference schemes and of the first- and second-order differencederivatives is obtained. The theoretical statements for the solution of these difference schemes forone-dimensional hyperbolic equations are supported by numerical examples.

1. Introduction

Nonlocal problems have been a major research area in modern physics, biology, chemistry,and engineering when it is impossible to determine the boundary values of the unknownfunction. Numerical methods and theory of solutions of the nonlocal boundary valueproblems for partial differential equations of variable type were carried out in for example,[1–10] and the references therein. Hyperbolic equations with nonlocal integral conditionsare widely used for chemical heterogeneity, plasma physics, thermoelasticity, and so forth.The solutions of hyperbolic equations with nonlocal integral conditions were investigatedin [11–15]. The method of operators as a tool for investigation of the solution to hyperbolicequations in Hilbert and Banach spaces has been studied extensively (see, e.g., [16–28]).

In the present paper, the nonlocal boundary value problem for the multidimensionalhyperbolic equation with nonlocal integral condition

∂2u(t, x)∂t2

−m∑

r=1

(ar(x)uxr )xr = f(t, x),

x = (x1, . . . , xm) ∈ Ω, 0 < t < 1,

2 Discrete Dynamics in Nature and Society

u(0, x) =∫1

0α(ρ)u(ρ, x)dρ + ϕ(x), ut(0, x) = ψ(x), x ∈ Ω,

u(t, x) = 0, 0 < t < 1, x ∈ S(1.1)

is considered. Here Ω is the unit open cube in the m-dimensional Euclidean spaceRm = {x = (x1, . . . , xm) : 0 < xj < 1, 1 ≤ j ≤ m} with boundary S, Ω = Ω ∪ S,

ar(x) (x ∈ Ω), ϕ(x), ψ(x) (x ∈ Ω), and f(t, x) (t ∈ (0, 1), x ∈ Ω) are given smooth functions,and ar(x) ≥ a > 0.

The first and second orders of approximation in t and the second order of approx-imation in space variables difference schemes for the approximate solution of nonlocalboundary value problem (1.1) are presented. Stability of these difference schemes and ofthe first- and second-order difference derivatives established. Error analysis is obtained bynumerical solutions of one-dimensional hyperbolic equations with integral condition.

2. Difference Schemes and Stability Estimates

The discretization of problem (1.1) is carried out in two steps. In the first step, let us definethe grid sets

Ωh ={x = xr = (h1r1, . . . , hmrm), r = (r1, . . . , rm), 0 ≤ rj ≤Nj, hjNj = 1, j = 1, . . . , m

},

Ωh = Ωh ∩Ω, Sh = Ωh ∩ S.(2.1)

We introduce the Hilbert space L2h = L2(Ωh) of the grid functions

ϕh(x) ={ϕ(h1r1, . . . , hmrm)

}(2.2)

defined on Ωh, equipped with the norm

∥∥∥ϕh∥∥∥L2(Ωh)

=

⎛

⎝∑

x∈Ωh

∣∣∣ϕh(x)∣∣∣2h1, · · · , hm

⎞

⎠1/2

. (2.3)

To the differential operator Ax generated by problem (1.1), we assign the difference operatorAxh by the formula

Axhu

hx = −

m∑

r=1

(ar(x)uhxr

)

xrrj(2.4)

Discrete Dynamics in Nature and Society 3

acting in the space of grid functions uh(x), satisfying the condition uh(x) = 0 for all x ∈ Sh. Itis known thatAx

h is a self-adjoint positive definite operator in L2(Ωh). With the help ofAxh we

arrive at the nonlocal boundary value problem

d2vh(t, x)dt2

+Axhv

h(t, x) = fh(t, x), 0 < t < 1, x ∈ Ωh,

vh(0, x) =∫1

0α(ρ)vh(ρ, x)dρ + ϕh(x), x ∈ Ωh,

dvh(0, x)dt

= ψh(x), x ∈ Ωh

(2.5)

for an infinite system of ordinary differential equations.In the second step, we replace problem (2.5) by the difference scheme

uhk+1(x) − 2uhk(x) + uhk−1(x)

τ2+Ax

huhk+1(x) = f

hk+1(x),

fhk+1(x) = fh(tk+1, x), tk+1 = (k + 1)τ,

1 ≤ k ≤N − 1, Nτ = 1, x ∈ Ωh,

uh0(x) =N∑

m=1

α(tm)uhm(x)τ + ϕh(x), x ∈ Ωh,

(I + τ2Ax

h

)(uh1(x) − uh0(x)

)τ−1 = ψh(x), x ∈ Ωh

(2.6)

of the first order accuracy in t.

Theorem 2.1. Let τ and h be sufficiently small numbers. Then, the solutions of the difference scheme(2.6) satisfy the following stability estimates:

max0≤k≤N

∥∥∥uhk∥∥∥L2h

+ max0≤k≤N

m∑

r=1

∥∥∥∥(uhk

)

xrrj

∥∥∥∥L2h

≤M1

[max

1≤k≤N−1

∥∥∥fhk∥∥∥L2h

+∥∥∥ψh∥∥∥L2h

+m∑

r=1

∥∥∥ϕhxrrj∥∥∥L2h

],

max1≤k≤N−1

∥∥∥τ−2(uhk+1 − 2uhk + u

hk−1)∥∥∥

L2h+ max

0≤k≤N

m∑

r=1

∥∥∥∥(uhk

)

xrxrrj

∥∥∥∥L2h

≤M1

[∥∥∥fh1∥∥∥L2h

+ max2≤k≤N−1

∥∥∥τ−1(fhk − fhk−1

)∥∥∥L2h

+m∑

r=1

∥∥∥ψhxrrj∥∥∥L2h

+m∑

r=1

∥∥∥ϕhxrxrrj∥∥∥L2h

],

(2.7)

whereM1 does not depend on τ , h, ϕh(x), ψh(x), and fhk(x), 1 ≤ k < N.


The proof of Theorem 2.1 is based on the symmetry property of difference operatorAxhdefined by the formula (2.4) and on the following theorem on coercivity inequality of the

elliptic difference problem.

Theorem 2.2. For the solutions of the elliptic difference problem

Axhu

h(x) = ωh(x), x ∈ Ωh, uh(x) = 0, x ∈ Sh, (2.8)

the following coercivity inequality holds [29]:

m∑

r=1

∥∥∥uhxrxrrj∥∥∥L2h

≤M∥∥∥ωh

∥∥∥L2h. (2.9)

Moreover, the second order of accuracy difference schemes

τ−2[uhk+1(x) − 2uhk(x) + u

hk−1(x)

]+12Axhu

hk(x) +

14Axh

[uhk+1(x) + u

hk−1(x)

]= fhk (x), x ∈ Ωh,

fhk = fhk (tk, x), tk = kτ, 1 ≤ k ≤N − 1, Nτ = 1,

uh0(x) =N∑

j=1

τ

2

[α(tj)uh(tj , x)+ α(tj−1)uh(tj−1, x

)]+ ϕh(x), x ∈ Ωh,

(I +

τ2Axh

2

)τ−1[uh1(x) − uh0(x)

]− τ

2

[fh0 (x) −Ax

huh0(x)

]= ψh(x),

fh0 = fh(0, x), fhN = fh(1, x), x ∈ Ωh,

(2.10)

and

τ−2(uhk+1 − 2uhk + u

hk−1)+Ax

huhk +

τ2

4(Axh

)2uhk+1 = f

hk (x),

fhk = fhk (tk, x), tk = kτ, 1 ≤ k ≤N − 1, Nτ = 1, x ∈ Ωh,

uh0(x) =N∑

j=1

τ

2

[α(tj)uh(tj , x)+ α(tj−1)uh(tj−1, x

)]+ ϕh(x), x ∈ Ωh,

(I +

τ2Axh

4

)[(I +

τ2Axh

4

)τ−1[uh1(x) − uh0(x)

]− τ

2

[fh0 (x) −Ax

huh0(x)

]]= ψh(x),

fh0 = fh(0, x), fhN = fh(1, x), x ∈ Ωh

(2.11)

for approximately solving the boundary value problem (1.1) is presented.


We have the following theorem.

Theorem 2.3. Let τ and h be sufficiently small numbers. Then, for the solution of the differenceschemes (2.10) and (2.11) the stability inequalities

max0≤k≤N

∥∥∥uhk∥∥∥L2h

+ max0≤k≤N

m∑

r=1

∥∥∥∥(uhk

)

xr ,jr

∥∥∥∥L2h

≤M2

[max0≤k≤N

∥∥∥fhk∥∥∥L2h

+∥∥∥ψh∥∥∥L2h

+m∑

r=1

∥∥∥ϕhxr ,jr∥∥∥L2h

],

max1≤k≤N−1

∥∥∥τ−2(uhk+1 − 2uhk + u

hk−1)∥∥∥

L2h

≤M2

[∥∥∥fh0∥∥∥L2h

+ max1≤k≤N−1

∥∥∥τ−1(fhk − fhk−1

)∥∥∥L2h

+m∑

r=1

∥∥∥ψhxr ,jr∥∥∥L2h

+m∑

r=1

∥∥∥ψhxr ,xr ,jr∥∥∥L2h

]

(2.12)

hold, whereM2 is independent of τ , h, ϕh(x), ψh(x), and fhk (x), 0 ≤ k ≤N.

The proof of Theorem 2.3 is based on the symmetry property of difference operatorAxhdefined by formula (2.4) and on Theorem 2.2 on coercivity inequality of elliptic difference

problem (2.8).In Theorems 2.1 and 2.3, the constants M1 and M2 cannot be obtained sharply.

Therefore, in the following section, we will study the accuracy of these difference schemesfor solving the one-dimensional hyperbolic equations with the integral condition. Moreover,the method is supported by numerical experiments.

3. Numerical Analysis

3.1. The First Order of Accuracy in Time Difference Scheme

In this section, the nonlocal boundary value problem

∂2u(t, x)∂t2

− (1 + x)∂2u(t, x)∂x2

− ux(t, x) + u(t, x) = f(t, x),

f(t, x) = [(x + 2) sinx − cosx](et − 1 − t) + et sinx, 0 < t < 1, 0 < x < π,

u(0, x) =∫1

0e−su(s, x)ds + ψ(x), ψ(x) =

(1 − 3e−1

)sinx,

ut(0, x) = 0, 0 ≤ x ≤ π,u(t, 0) = u(t, π) = 0, 0 ≤ t ≤ 1

(3.1)

for one dimensional hyperbolic equation is considered.The exact solution of problem (3.1) is

u(t, x) =(et − 1 − t) sinx. (3.2)


Applying the formulas

u(xn+1) − u(xn−1)2h

− u′(xn) = O(h2),

u(τ) − u(0)τ

− u′(0) = O(τ),

u(xn+1) − 2u(xn) + u(xn−1)h2

− u′′(xn) = O(h2)

(3.3)

and using the first order of accuracy in t implicit difference scheme (2.6), we obtain thedifference scheme first order of accuracy in t and second order of accuracy in x

uk+1n − 2ukn + uk−1n

τ2− (1 + xn)

uk+1n+1 − 2uk+1n + uk+1n−1h2

− uk+1n+1 − uk+1n−12h

+ uk+1n = f(tk+1, xn),

f(tk+1, xn) = [(xn + 2) sinxn − cosxn](etk+1 − 1 − tk+1

)+ etk+1 sinxn,

Nτ = 1, xn = nh, 1 ≤ n ≤M − 1, Mh = π,

u0n −N∑

k=1

e−kττukn = ψ(xn), tk = kτ, 1 ≤ k ≤N − 1,

ψ(xn) =(1 − 3e−1

)sinxn, 0 ≤ n ≤M,

uk0 = ukM = 0, u1n − u0n = 0, 0 ≤ k ≤N, 1 ≤ n ≤M − 1

(3.4)

for approximate solutions of nonlocal boundary value problem (3.1). It can be written in thematrix form

Anun+1 + Bnun + Cnun−1 = Dϕn, 1 ≤ n ≤M − 1,

u0 = 0, uM = 0.(3.5)

Here

An =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0 0 0 0 · · · 0 0

0 0 an 0 · · · 0 0

0 0 0 an · · · 0 0

· · · · · · · · · · · · · · · · · · · · ·0 0 0 0 · · · an 0

0 0 0 0 · · · 0 an

0 0 0 0 · · · 0 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(N+1)×(N+1)

,


Bn =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

1 −τe−τ −τe−2τ −τe−3τ · · · −τe−(N−1)τ −τe−Nτ

b c dn 0 · · · 0 0

0 b c dn · · · 0 0

0 0 b c · · · 0 0· · · · · · · · · · · · · · · · · · · · ·0 0 0 0 · · · dn 0

0 0 0 0 · · · c dn

−1 1 0 0 · · · 0 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(N+1)×(N+1)

,

Cn =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0 0 0 0 · · · 0 0

0 0 en 0 · · · 0 0

0 0 0 en · · · 0 0

· · · · · · · · · · · · · · · · · · · · ·0 0 0 0 · · · en 0

0 0 0 0 · · · 0 en

0 0 0 0 · · · 0 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(N+1)×(N+1)

,

an =−(1 + xn)

h2− 12h, b =

1τ2, c =

−2τ2,

dn =1τ2

+2(1 + xn)

h2+ 1, en =

−(1 + xn)h2

+12h,

ϕn =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

ϕ0n

ϕ1n

ϕ2n...

ϕNn

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

(N+1)×1

,

ϕk+1n = f(tk+1, xn) = [(xn + 2) sinxn − cosxn](etk − 1 − tk

)+ etk sinxn,

xn = nh, tk = kτ, 1 ≤ k ≤N − 1,

(3.6)

and D = IN+1 is the identity matrix.

Us =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

u0s

u1s

u2s...

uNs

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

(N+1)×1

, s = n − 1, n, n + 1. (3.7)


This type system was used by Samarskii and Nikolaev [30] for difference equations. For thesolution of the matrix equation (3.5), we will use the modified Gauss elimination method. Weseek a solution of the matrix equation by the following form:

un = αn+1un+1 + βn+1, n =M − 1, . . . , 2, 1, (3.8)

where uM = 0, αj (j = 1, . . . ,M−1) are (N +1)× (N +1) square matrices, βj (j = 1, . . . ,M−1)are (N + 1) × 1 column matrices, α1, β1 are zero matrices, and

αn+1 = −(Bn + Cnαn)−1An,

βn+1 = (Bn + Cnαn)−1(Dnϕn − Cnβn

), n = 1, 2, 3, . . . ,M − 1.

(3.9)

3.2. The Second Order of Accuracy in Time Difference Scheme

Applying (3.3) and using the second order of accuracy in t implicit difference scheme (2.10),we obtain the second order of accuracy difference scheme in t and in x

uk+1n − 2ukn + uk−1n−1

τ2− (1 + xn)

[uk−1n+1 − 2uk−1n + uk−1n−1

4h2− ukn+1 − 2ukn + u

kn−1

2h2− uk+1n+1 − 2uk+1n + uk+1n−1

4h2

]

−[uk−1n+1 − uk−1n−1

8h+ukn+1 − ukn−1

4h+uk+1n+1 − uk+1n−1

8h

]+12ukn +

uk+1n + uk−1n

4= ϕkn,

ϕkn = [(xn + 2) sinxn − cosxn](etk − 1 − tk

)+ etk sinxn,

Mh = π, xn = nh, 1 ≤ n ≤M − 1,

Nτ = 1, tk = kτ, 1 ≤ k ≤N − 1,

u0n =N∑

k=1

τ

2

[e−kτukn + e−(k−1)τuk−1n

]+ ψ(xn), 0 ≤ n ≤M,

ψ(xn) =(1 − 3e−1


u1n − u0n =τ2

2

[u0n+1 − u0n−1

2h+ (1 + xn)

u0n+1 − 2u0n + u0n−1

h2− u0n + sinxn

], 1 ≤ n ≤M − 1,

uk0 = ukM = 0, 0 ≤ k ≤N(3.10)

for approximate solutions of the nonlocal boundary value problem (3.1). We have again(N + 1) × (M + 1) system of linear equations. We can write the system as a matrix equation(3.5).


Here

An =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

0 0 0 0 · · · 0 0 0an 2an an 0 · · · 0 0 00 an 2an an · · · 0 0 0· · · · · · · · · · · · · · · · · · · · · · · ·0 0 0 0 · · · an 2an an

wn 0 0 0 · · · 0 0 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

(N+1)×(N+1)

,

Bn =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

1 − τ

2−τe−τ −τe−2τ · · · −τe−(N−1)τ −τ

2e−Nτ

bn dn bn · · · 0 0

0 bn dn · · · 0 0· · · · · · · · · · · · · · · · · ·0 0 0 · · · bn 0

0 0 0 · · · dn bn

yn 1 0 · · · 0 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(N+1)×(N+1)

,

Cn =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0 0 0 0 · · · 0 0 0

cn 2cn cn 0 · · · 0 0 0

0 cn 2cn cn · · · 0 0 0· · · · · · · · · · · · · · · · · · · · · · · ·0 0 0 0 · · · cn 2cn cn

zn 0 0 0 · · · 0 0 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(N+1)×(N+1)

,

an =−(1 + xn)

4h2− 18h, bn =

1τ2

+1 + xn2h2

+14,

cn =−(1 + xn)

4h2+

18h, dn =

−2τ2

+1 + xnh2

+12,

yn = −1 + (1 + xn)τ2

h2+τ2

2, wn = − τ

2

4h− (1 + xn)τ2

2h2, zn =

τ2

4h− (1 + xn)τ2

2h2,

ϕn =

⎡⎢⎢⎢⎢⎢⎣

ϕ0n

ϕ1n...

ϕNn

⎤⎥⎥⎥⎥⎥⎦

(N+1)×1

,

ϕkn = f(tk, xn) = [(xn + 2) sinxn − cosxn](etk − 1 − tk

)+ etk sinxn, 1 ≤ k ≤N − 1,

D = IN+1, Us =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

u0s

u1s

u2s...

uNs

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

(N+1)×1

, s = n − 1, n, n + 1.

(3.11)


For the solution of the matrix equation (3.5), we used the same algorithm as in the first orderof accuracy difference scheme.

3.3. The Second Order of Accuracy in Time Difference SchemeGenerated by A2

Applying (3.3) and formulas

u(xn+2) − 4u(xn+1) + 6u(xn) − 4u(xn−1) + u(xn−2)h4

− u(iv)(xn) = O(h2),

2u(0) − 5u(h) + 4u(2h) − u(3h)h2

− u′′(0) = O(h2),

2u(π) − 5u(π − h) + 4u(π − 2h) − u(π − 3h)h2

− u′′(π) = O(h2)

(3.12)

and using difference scheme (2.11), we obtain the second order of accuracy difference scheme

uk+1n − 2ukn + uk−1n

τ2− (1 + xn)

ukn+1 − 2ukn + ukn−1

h2− ukn+1 − ukn−1

2h+ ukn

+τ2

4

[(1 + xn)2

uk+1n+2 − 4uk+1n+1 + 6uk+1n − 4uk+1n−1 + uk+1n−2

h4

+4(1 + xn)uk+1n+2 − 2uk+1n+1 + 2uk+1n−1 − uk+1n−2

2h3− 2xn

uk+1n+1 − 2uk+1n + uk+1n−1h2

−uk+1n+1 − uk+1n−1

h+ uk+1n

]= ϕkn,

ϕkn = [(xn + 2) sinxn − cosxn](etk − 1 − tk

)+ etk sinxn,

Mh = π, xn = nh, 2 ≤ n ≤M − 2,

Nτ = 1, tk = kτ, 1 ≤ k ≤N − 1,

u0n =N∑

k=1

τ

2

[e−kτukn + e

−(k−1)τuk−1n

]= ψ(xn), 0 ≤ n ≤M,

ψ(xn) =(1 − 3e−1


u1n − u0n =τ2

2

[u0n+1 − u0n−1

2h+ (1 + xn)

u0n+1 − 2u0n + u0n−1

h2− u0n + sinxn

], 2 ≤ n ≤M − 2,

uk0 = ukM = 0, 0 ≤ k ≤N,

uk1 =45uk2 −

15uk3 , 0 ≤ k ≤N,

ukM−1 =45ukM−2 −

15ukM−3, 0 ≤ k ≤N

(3.13)


for approximate solutions of problem (3.1). One can write the (N + 1) × (M + 1) system oflinear equations (3.13) as the matrix equation

Anun+2 + Bnun+1 + Cnun +Dnun−1 + Enun−2 = Rϕn,

u0 = 0, uM = 0, 2 ≤ n ≤M − 2,

u1 =45u2 − 1

5u3,

uM−1 =45uM−2 − 1

5uM−3.

(3.14)

Here, An, Bn, Cn, Dn, En, and R are (N + 1) × (N + 1) square matrices:

An =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0 0 0 0 · · · 0

0 0 an 0 · · · 0

0 0 0 an · · · 0

· · · · · · · · · · · · · · · · · ·0 0 0 0 · · · an

0 0 0 0 · · · 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

, Bn =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0 0 0 · · · 0 0

0 bn cn · · · 0 0

0 0 bn · · · 0 0

· · · · · · · · · · · · · · · · · ·0 0 0 · · · cn 0

0 0 0 · · · bn cn

yn 0 0 · · · 0 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

,

Cn =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

1 − τ

2−τe−τ −τe−2τ · · · −τe−(N−2)τ −τe−(N−1)τ −τ

2e−Nτ

d en fn · · · 0 0 0

0 d en · · · 0 0 0

· · · · · · · · · · · · · · · · · · · · ·0 0 0 · · · en fn 0

0 0 0 · · · d en fn

wn 1 0 · · · 0 0 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

,

Dn =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0 0 0 · · · 0 0 0

0 gn ln · · · 0 0 0

0 0 gn · · · 0 0 0

· · · · · · · · · · · · · · · · · · · · ·0 0 0 · · · gn ln 0

0 0 0 · · · 0 gn ln

zn 0 0 · · · 0 0 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

,


En =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

0 0 0 0 · · · 0 00 0 mn 0 · · · 0 00 0 0 mn · · · 0 0· · · · · · · · · · · · · · · · · · · · ·0 0 0 0 · · · mn 00 0 0 0 · · · 0 mn

0 0 0 0 · · · 0 0

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

.

(3.15)

We denote

an =τ2

4

((1 + xn)2

h4+2(1 + xn)

h3

), bn = − (1 + xn)

h2− 12h,

d =1τ2, en = − 2

τ2+2(1 + xn)

h2+ 1, gn =

12h

− 1 + xnh2

,

cn =τ2

4

(−4(1 + xn)2

h4− 4(1 + xn)

h3− 2xnh2

− 1h

),

ln =τ2

4

(−4(1 + xn)2

h4− 4(1 + xn)

h3− 2xnh2

+1h

),

fn =1τ2

+τ2

4

(6(1 + xn)2

h4+4xnh2

+ 1

),

mn =τ2

4

((1 + xn)2

h4− 2(1 + xn)

h3

),

zn =τ2

4h− (1 + xn)τ2

2h2,

wn = −1 + (1 + xn)τ2

h2+τ2

2, yn = − τ

2

4h− (1 + xn)τ2

2h2,

ϕn =

⎡⎢⎢⎢⎢⎢⎣

ϕ0n

ϕ1n...

ϕNn

⎤⎥⎥⎥⎥⎥⎦

(N+1)×1

,

ϕkn = f(tk, xn) = [(xn + 2) sinxn − cosxn](etk − 1 − tk

)+ etk sinxn, 1 ≤ k ≤N − 1,

R = IN+1 is the identity matrix, Us =

⎡⎢⎢⎢⎢⎣

U0s

U1s...

UNs

⎤⎥⎥⎥⎥⎦

(N+1)×1

,

(3.16)

where s = n ± 2, n ± 1, n.


Table 1

Difference schemes N =M = 20 N =M = 40 N =M = 80Difference schemes (3.4) 0.0743 0.0357 0.0175Difference schemes (3.10) 0.0012 0.0003009 0.0000756Difference schemes (3.13) 0.0005453 0.0001231 0.00002923

For the solution of matrix equation (3.14), we will use the modified Gauss eliminationmethod. We seek a solution of the matrix equation by the following formula:

Un = αn+1un+1 + βn+1un+2 + γn+1, n =M − 2, . . . , 2, 1, (3.17)

where αj , βj (j = 1, . . . ,M − 1) are (N + 1)× (N + 1) square matrices, γj (j = 1, . . . ,M − 1) are(N + 1) × 1 column matrices, and α1, β1, γ1, γ2 are zero matrices. α2, β2 are

α2 =45IN+1, β2 = −1

5IN+1,

Fn = Cn +Dnαn + Enαn−1αn + Enβn−1

αn+1 = F−1n

[−Bn −Dnβn − Enαn−1βn], βn+1 = −F−1

n An,

γn+1 = F−1n

[Rϕn −Dnγn + Enαn−1γn + Enγn−1

].

(3.18)

For solution of the last difference equation, we need to find uM, uM−1

uM = 0,

uM−1 =((βM−2 + 5I

) − (4I − αM−2)αM−1)−1((4I − αM−2))γM−1 − γM−2.

(3.19)

3.4. Error Analysis

The errors are computed by

ENM = max1≤k≤N−1,1≤n≤M−1

∣∣∣u(tk, xn) − ukn∣∣∣ (3.20)

of the numerical solutions, where u(tk, xn) represents the exact solution and ukn represents thenumerical solution at (tk, xn) and the results are given in Table 1.

Thus, the results show that the second order of accuracy difference schemes (3.10) and(3.13) are more accurate comparing with the first order of accuracy difference scheme (3.4).

Acknowledgment

The authors would like to thank Professor P. E. Sobolevskii (Jerusalem, Israel), for the helpfulsuggestions to the improvement of this paper.


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