Finite Difference Method for Hyperbolic Equations with the Nonlocal Integral Condition

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Discrete Dynamics in Nature and Society Volume 2011, Article ID 562385,15pages doi:10.1155/2011/562385

Research Article

Finite Difference Method for Hyperbolic Equations with the Nonlocal Integral Condition

Allaberen Ashyralyev

^{1, 2}

and Necmettin Aggez

¹

1Department of Mathematics, Fatih University, Istanbul 34500, Turkey

2Department 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

Copyrightq2011 A. Ashyralyev and N. Aggez. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The stable difference schemes for the approximate solution of the nonlocal boundary value problem for multidimensional hyperbolic equations with dependent in space variable coefficients are presented. Stability of these difference schemes and of the first- and second-order difference derivatives is obtained. The theoretical statements for the solution of these difference schemes for one-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 unknown function. Numerical methods and theory of solutions of the nonlocal boundary value problems for partial diﬀerential equations of variable type were carried out in for example, 1–10 and the references therein. Hyperbolic equations with nonlocal integral conditions are widely used for chemical heterogeneity, plasma physics, thermoelasticity, and so forth.

The solutions of hyperbolic equations with nonlocal integral conditions were investigated in11–15. The method of operators as a tool for investigation of the solution to hyperbolic equations in Hilbert and Banach spaces has been studied extensivelysee, e.g.,16–28.

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

∂²ut, x

∂t² −^m

r1

arxuxr_x_r ft, x, x x₁, . . . , x_m∈Ω, 0< t <1,

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u0, x ₁

0

α ρ

u ρ, x

dρϕx, ut0, x ψx, x∈Ω, ut, x 0, 0< t <1, x∈S

1.1 is considered. Here Ω is the unit open cube in the m-dimensional Euclidean space R^m {x x1, . . . , x_m : 0 < x_j < 1,1 ≤ j ≤ m} with boundary S, Ω Ω ∪ S, arx x∈Ω,ϕx,ψx x∈Ω, andft, x t∈0,1, x ∈Ωare given smooth functions, andarx≥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 nonlocal boundary value problem 1.1are presented. Stability of these difference schemes and of the first- and second-order difference derivatives established. Error analysis is obtained by numerical solutions of one-dimensional hyperbolic equations with integral condition.

2. Difference Schemes and Stability Estimates

The discretization of problem1.1is carried out in two steps. In the first step, let us define the grid sets

Ωh

xx_r h₁r₁, . . . , h_mr_m, r r₁, . . . , r_m, 0≤r_j≤N_j, h_jN_j1, j 1, . . . , m , ΩhΩh∩Ω, S_hΩh∩S.

2.1

We introduce the Hilbert spaceL2hL2Ωhof the grid functions ϕ^hx

ϕh1r₁, . . . , h_mr_m

2.2

defined onΩh, equipped with the norm

ϕ^h

L2Ωh

⎛

⎝

x∈Ωh

ϕ^hx²h1,· · · , hm

⎞

⎠

1/2

. 2.3

To the diﬀerential operatorA^xgenerated by problem1.1, we assign the diﬀerence operator A^x_hby the formula

A^x_hu^h_x−^m

r1

arxu^h_x_r

xrrj 2.4

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acting in the space of grid functionsu^hx, satisfying the conditionu^hx 0 for allx∈Sh. It is known thatA^x_his a self-adjoint positive definite operator inL₂Ωh. With the help ofA^x_hwe arrive at the nonlocal boundary value problem

d²v^ht, x

dt² A^x_hv^ht, x f^ht, x, 0< t <1, x∈Ωh, v^h0, x

₁

0

α ρ

v^h ρ, x

dρϕ^hx, x∈Ωh, dv^h0, x

dt ψ^hx, x∈Ωh

2.5

for an infinite system of ordinary diﬀerential equations.

In the second step, we replace problem2.5by the diﬀerence scheme

u^h_k1x−2u^h_kx u^h_k−1x

τ² A^x_hu^h_k1x f_k1^h x, f_k1^h x f^ht_k1, x, t_k1 k1τ,

1≤k≤N−1, Nτ1, x∈Ωh, u^h₀x ^N

m1

αtmu^h_mxτϕ^hx, x∈Ωh, Iτ²A^x_h

u^h₁x−u^h₀x

τ⁻¹ψ^hx, x∈Ωh

2.6

of the first order accuracy in t.

Theorem 2.1. Letτandhbe suﬃciently small numbers. Then, the solutions of the diﬀerence scheme 2.6satisfy the following stability estimates:

0≤k≤Nmax u^h_k

L2h

max

0≤k≤N

m r1

u^h_k

xrrj

L2h

≤M1

1≤k≤N−1max f_k^h

L2h ψ^h

L2h^m

r1

ϕ^h_x_r_r_j

L2h

,

1≤k≤N−1max τ⁻²

u^h_k1−2u^h_ku^h_k−1

L2h

max

0≤k≤N

m r1

u^h_k

xrxrrj

L2h

≤M₁ f₁^h

L2h

max

2≤k≤N−1

τ⁻¹

f_k^h−f_k−1^h

L2h

^m

r1

ψ_x^h

rrj

L2h

^m

r1

ϕ^h_x

rxrrj

L2h

,

2.7

whereM₁does not depend onτ,h,ϕ^hx,ψ^hx, andf_k^hx,1≤k < N.

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The proof ofTheorem 2.1is based on the symmetry property of diﬀerence operator A^x_hdefined by the formula2.4and on the following theorem on coercivity inequality of the elliptic diﬀerence problem.

Theorem 2.2. For the solutions of the elliptic diﬀerence problem

A^x_hu^hx ω^hx, x∈Ωh, u^hx 0, x∈S_h, 2.8

the following coercivity inequality holds [29]:

m r1

u^h_x_r_x_r_r_j

L2h

≤M ω^h

L2h

. 2.9

Moreover, the second order of accuracy diﬀerence schemes

τ⁻²

u^h_k1x−2u^h_kx u^h_k−1x 1

2A^x_hu^h_kx 1 4A^x_h

u^h_k1x u^h_k−1x

f_k^hx, x∈Ωh, f_k^hf_k^htk, x, t_kkτ, 1≤k≤N−1, Nτ 1,

u^h₀x ^N

j1

τ 2

α tj

u^h tj, x

α t_j−1

u^h

t_j−1, x

ϕ^hx, x∈Ωh,

Iτ²A^x_h 2

τ⁻¹

u^h₁x−u^h₀x

−τ 2

f₀^hx−A^x_hu^h₀x

ψ^hx, f₀^hf^h0, x, f_N^h f^h1, x, x∈Ωh,

2.10

and

τ⁻²

A^x_hu^h_kτ² 4

A^x_h2

u^h_k1f_k^hx, f_k^hf_k^htk, x, tkkτ, 1≤k≤N−1, Nτ 1, x∈Ωh, u^h₀x ^N

j1

τ 2

α t_j

u^h t_j, x

α t_j−1

u^h

t_j−1, x

ϕ^hx, x∈Ωh,

Iτ²A^x_h 4

τ⁻¹

u^h₁x−u^h₀x

−τ 2

f₀^hx−A^x_hu^h₀x

ψ^hx, f₀^hf^h0, x, f_N^h f^h1, x, x∈Ωh

2.11

for approximately solving the boundary value problem1.1is presented.

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We have the following theorem.

Theorem 2.3. Letτ and hbe suﬃciently small numbers. Then, for the solution of the diﬀerence schemes2.10and2.11the stability inequalities

0≤k≤Nmax u^h_k

L2h

max

0≤k≤N

m r1

u^h_k

xr,jr

L2h

≤M₂

0≤k≤Nmax f_k^h

L2h

ψ^h

L2h

^m

r1

ϕ^h_x_r_,j_r

L2h

,

1≤k≤N−1max τ⁻²

L2h

≤M₂ f₀^h

L2h

max

1≤k≤N−1

τ⁻¹

f_k^h−f_k−1^h

L2h

^m

r1

ψ_x^h

r,jr

L2h

^m

r1

ψ_x^h

r,xr,jr

L2h

2.12

hold, whereM2is independent ofτ,h, ϕ^hx,ψ^hx, andf_k^hx, 0≤k≤N.

The proof ofTheorem 2.3is based on the symmetry property of diﬀerence operator A^x_hdefined by formula2.4and onTheorem 2.2on coercivity inequality of elliptic diﬀerence problem2.8.

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 schemes for 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

∂²ut, x

∂t² −1x∂²ut, x

∂x² −u_xt, x ut, x ft, x, ft, x x2sinx−cosx

e^t−1−t

e^tsinx, 0< t <1, 0< x < π, u0, x

₁

0

e^−sus, xdsψx, ψx

1−3e⁻¹ sinx, u_t0, x 0, 0≤x≤π,

ut,0 ut, π 0, 0≤t≤1

3.1

for one dimensional hyperbolic equation is considered.

The exact solution of problem3.1is ut, x

e^t−1−t

sinx. 3.2

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Applying the formulas

uxn1−uxn−1

2h −uxn O

h² , uτ−u0

τ −u0 Oτ, ux_n1−2uxn ux_n−1

h² −uxn O

h²

3.3

and using the first order of accuracy in t implicit diﬀerence scheme 2.6, we obtain the diﬀerence scheme first order of accuracy intand second order of accuracy inx

u^k1_n −2u^k_nu^k−1_n

τ² −1xnu^k1_n1−2u^k1_n u^k1_n−1

h² −u^k1_n1−u^k1_n−1

2h u^k1_n ftk1, xn, ft_k1, x_n xn2sinx_n−cosx_n

e^t^k1−1−t_k1

e^t^k1sinx_n, Nτ1, xn nh, 1≤n≤M−1, Mhπ,

u⁰_n−^N

k1

e^−kττu^k_nψxn, t_kkτ,1≤k≤N−1, ψxn

1−3e⁻¹

sinx_n, 0≤n≤M,

u^k₀ u^k_M0, u¹_n−u⁰_n0, 0≤k≤N,1≤n≤M−1

3.4

for approximate solutions of nonlocal boundary value problem3.1. It can be written in the matrix form

A_nu_n1B_nu_nC_nu_n−1Dϕ_n, 1≤n≤M−1, u00, uM0.

3.5

Here

An

⎡

⎢⎢

⎣

0 0 0 0 · · · 0 0

0 0 a_n 0 · · · 0 0

0 0 0 a_n · · · 0 0

· · · ·

0 0 0 0 · · · an 0

0 0 0 0 · · · 0 an

0 0 0 0 · · · 0 0

⎤

⎥⎥

⎦

N1×N1

,

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B_n

⎡

⎢⎢

⎢⎣

1 −τe^−τ −τe^−2τ −τe^−3τ · · · −τe^−N−1τ −τe^−Nτ

b c d_n 0 · · · 0 0

0 b c d_n · · · 0 0

0 0 b c · · · 0 0

· · · ·

0 0 0 0 · · · d_n 0

0 0 0 0 · · · c d_n

−1 1 0 0 · · · 0 0

⎤

⎥⎥

⎥⎦

N1×N1

,

C_n

⎡

⎢⎢

⎣

0 0 0 0 · · · 0 0

0 0 en 0 · · · 0 0

0 0 0 en · · · 0 0

· · · ·

0 0 0 0 · · · e_n 0

0 0 0 0 · · · 0 e_n

0 0 0 0 · · · 0 0

⎤

⎥⎥

⎦

N1×N1

,

a_n −1x_n h² − 1

2h, b 1

τ², c −2 τ², d_n 1

τ² 21x_n

h² 1, e_n −1x_n h² 1

2h,

ϕ_n

⎡

⎢⎢

⎣ ϕ⁰_n ϕ¹_n ϕ²_n ... ϕ^N_n

⎤

⎥⎥

⎦

N1×1

,

ϕ^k1_n ftk1, xn xn2sinxn−cosxn

e^t^k−1−tk

e^t^ksinxn, x_nnh, t_kkτ, 1≤k≤N−1,

3.6 andDI_N1is the identity matrix.

U_s

⎡

⎢⎢

⎣ u⁰_s u¹_s u²_s ... u^N_s

⎤

⎥⎥

⎦

N1×1

, sn−1, n, n1. 3.7

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This type system was used by Samarskii and Nikolaev30for diﬀerence equations. For the solution of the matrix equation3.5, we will use the modified Gauss elimination method. We seek a solution of the matrix equation by the following form:

unα_n1u_n1β_n1, nM−1, . . . ,2,1, 3.8

whereu_M0, α_j j 1, . . . , M−1areN1×N1square matrices,β_j j1, . . . , M−1 areN1×1 column matrices,α1, β1are zero matrices, and

α_n1−BnC_nα_n⁻¹A_n, β_n1 BnCnαn⁻¹

Dnϕn−Cnβn

, n1,2,3, . . . , M−1.

3.9

3.2. The Second Order of Accuracy in Time Difference Scheme

Applying3.3and using the second order of accuracy intimplicit diﬀerence scheme2.10, we obtain the second order of accuracy diﬀerence scheme intand inx

u^k1_n −2u^k_nu^k−1_n−1

τ² −1x_n

u^k−1_n1−2u^k−1_n u^k−1_n−1

4h² −u^k_n1−2u^k_nu^k_n−1

2h² − u^k1_n1−2u^k1_n u^k1_n−1 4h²

−

u^k−1_n1−u^k−1_n−1

8h u^k_n1−u^k_n−1

4h u^k1_n1−u^k1_n−1 8h

1

2u^k_nu^k1_n u^k−1_n 4 ϕ^k_n, ϕ^k_n x_n2sinx_n−cosx_n

e^t^k−1−t_k

e^t^ksinx_n, Mhπ, x_nnh, 1≤n≤M−1,

Nτ1, tkkτ, 1≤k≤N−1, u⁰_n^N

k1

τ 2

e^−kτu^k_n e^−k−1τu^k−1_n

ψxn, 0≤n≤M,

ψxn

1−3e⁻¹

sinx_n, 0≤n≤M, u¹_n−u⁰_n τ²

2

u⁰_n1−u⁰_n−1

2h 1xnu⁰_n1−2u⁰_nu⁰_n−1

h² −u⁰_nsinxn

, 1≤n≤M−1, u^k₀ u^k_M0, 0≤k≤N

3.10

for approximate solutions of the nonlocal boundary value problem 3.1. We have again N1×M1system of linear equations. We can write the system as a matrix equation 3.5.

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Here

A_n

⎡

⎢⎢

⎣

0 0 0 0 · · · 0 0 0

an 2an an 0 · · · 0 0 0 0 a_n 2a_n a_n · · · 0 0 0

· · · · 0 0 0 0 · · · a_n 2a_n a_n

w_n 0 0 0 · · · 0 0 0

⎤

⎥⎥

⎦

N1×N1

,

Bn

⎡

⎢⎢

⎢⎣ 1−τ

2 −τe^−τ −τe^−2τ · · · −τe^−N−1τ −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

⎤

⎥⎥

⎥⎦

N1×N1

,

C_n

⎡

⎢⎢

⎢⎣

0 0 0 0 · · · 0 0 0

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

· · · · 0 0 0 0 · · · c_n 2c_n c_n

z_n 0 0 0 · · · 0 0 0

⎤

⎥⎥

⎥⎦

N1×N1

,

a_n −1x_n 4h² − 1

8h, b_n 1

τ² 1x_n 2h² 1

4, c_n −1x_n

4h² 1

8h, d_n −2

τ² 1x_n h² 1

2, yn−11xnτ²

h² τ²

2 , wn−τ²

4h−1xnτ²

2h² , zn τ²

4h−1xnτ² 2h² ,

ϕ_n

⎡

⎢⎢

⎢⎣ ϕ⁰_n ϕ¹_n ... ϕ^N_n

⎤

⎥⎥

⎥⎦

N1×1

,

ϕ^k_nftk, xn xn2sinxn−cosxn

e^t^k−1−tk

e^t^ksinxn, 1≤k≤N−1,

DI_N1, Us

⎡

⎢⎢

⎣ u⁰_s u¹_s u²_s ... u^N_s

⎤

⎥⎥

⎦

N1×1

, sn−1, n, n1.

3.11

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For the solution of the matrix equation3.5, we used the same algorithm as in the first order of accuracy diﬀerence scheme.

3.3. The Second Order of Accuracy in Time Difference Scheme Generated byA²

Applying3.3and formulas

uxn2−4uxn1 6uxn−4uxn−1 uxn−2

h⁴ −u^ivxn O

h² , 2u0−5uh 4u2h−u3h

h² −u0 O

h² , 2uπ−5uπ−h 4uπ−2h−uπ−3h

h² −uπ O

h²

3.12

and using diﬀerence scheme2.11, we obtain the second order of accuracy diﬀerence scheme

u^k1_n −2u^k_nu^k−1_n

τ² −1x_nu^k_n1−2u^k_nu^k_n−1

h² −u^k_n1−u^k_n−1 2h u^k_n τ²

4

1x_n²u^k1_n2−4u^k1_n16u^k1_n −4u^k1_n−1u^k1_n−2 h⁴

41x_nu^k1_n2−2u^k1_n12u^k1_n−1−u^k1_n−2

2h³ −2x_nu^k1_n1−2u^k1_n u^k1_n−1 h²

−u^k1_n1−u^k1_n−1 h u^k1_n

ϕ^k_n, ϕ^k_n x_n2sinx_n−cosx_n

e^t^k−1−t_k

e^t^ksinx_n, Mhπ, xnnh, 2≤n≤M−2,

Nτ1, t_kkτ, 1≤k≤N−1, u⁰_n^N

k1

τ 2

e^−kτu^k_ne^−k−1τu^k−1_n

ψxn, 0≤n≤M, ψxn

1−3e⁻¹

sinxn, 0≤n≤M, u¹_n−u⁰_n τ²

2

u⁰_n1−u⁰_n−1

2h 1x_nu⁰_n1−2u⁰_nu⁰_n−1

h² −u⁰_nsinx_n

, 2≤n≤M−2, u^k₀ u^k_M0, 0≤k≤N,

u^k₁ 4 5u^k₂−1

5u^k₃, 0≤k≤N, u^k_M−1 4

5u^k_M−2−1

5u^k_M−3, 0≤k≤N

3.13

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for approximate solutions of problem3.1. One can write theN1×M1system of linear equations3.13as the matrix equation

A_nu_n2B_nu_n1C_nu_nD_nu_n−1E_nu_n−2Rϕ_n, u00, uM0, 2≤n≤M−2,

u₁ 4 5u₂−1

5u₃, u_M−1 4

5u_M−2−1 5u_M−3.

3.14

Here,A_n,B_n,C_n,D_n,E_n, andRareN1×N1square matrices:

An

⎡

⎢⎢

⎢⎣

0 0 0 0 · · · 0

0 0 a_n 0 · · · 0

0 0 0 a_n · · · 0

· · · ·

0 0 0 0 · · · an

0 0 0 0 · · · 0

⎤

⎥⎥

⎥⎦

, Bn

⎡

⎢⎢

⎣

0 0 0 · · · 0 0

0 b_n c_n · · · 0 0

0 0 b_n · · · 0 0

· · · ·

0 0 0 · · · c_n 0

0 0 0 · · · bn cn

yn 0 0 · · · 0 0

⎤

⎥⎥

⎦ ,

Cn

⎡

⎢⎢

⎣ 1−τ

2 −τe^−τ −τe^−2τ · · · −τe^−N−2τ −τe^−N−1τ −τ 2e^−Nτ

d e_n f_n · · · 0 0 0

0 d e_n · · · 0 0 0

· · · ·

0 0 0 · · · en fn 0

0 0 0 · · · d en fn

wn 1 0 · · · 0 0 0

⎤

⎥⎥

⎦ ,

D_n

⎡

⎢⎢

⎣

0 0 0 · · · 0 0 0

0 g_n l_n · · · 0 0 0

0 0 g_n · · · 0 0 0

· · · ·

0 0 0 · · · gn ln 0

0 0 0 · · · 0 gn ln

zn 0 0 · · · 0 0 0

⎤

⎥⎥

⎦ ,

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E_n

⎡

⎢⎢

⎣

0 0 0 0 · · · 0 0

0 0 mn 0 · · · 0 0

0 0 0 m_n · · · 0 0

· · · ·

0 0 0 0 · · · mn 0

0 0 0 0 · · · 0 m_n

0 0 0 0 · · · 0 0

⎤

⎥⎥

⎦ .

3.15

We denote

an τ² 4

1xn²

h⁴ 21xn h³

, bn−1xn h² − 1

2h, d 1

τ², e_n−2

τ² 21x_n

h² 1, g_n 1

2h−1x_n h² , c_n τ²

4

−41xn²

h⁴ − 41xn h³ −2xn

h² − 1 h

,

ln τ² 4

−41x_n²

h⁴ −41x_n h³ −2x_n

h² 1 h

,

f_n 1 τ² τ²

4

61x_n² h⁴ 4x_n

h² 1

,

mn τ² 4

1xn²

h⁴ −21xn h³

, zn τ²

4h−1xnτ² 2h² , wn−11x_nτ²

h² τ²

2, yn−τ²

4h−1x_nτ² 2h² ,

ϕn

⎡

⎢⎢

⎢⎣ ϕ⁰_n ϕ¹_n ... ϕ^N_n

⎤

⎥⎥

⎥⎦

N1×1

,

ϕ^k_nftk, x_n xn2sinx_n−cosx_n

e^t^k−1−t_k

e^t^ksinx_n, 1≤k≤N−1,

RI_N1 is the identity matrix, Us

⎡

⎢⎢

⎣ U_s⁰ U_s¹ ... U^N_s

⎤

⎥⎥

⎦

N1×1

,

3.16 wheresn±2,n±1,n.

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Table 1

Diﬀerence schemes NM20 NM40 NM80

Diﬀerence schemes3.4 0.0743 0.0357 0.0175

For the solution of matrix equation3.14, we will use the modified Gauss elimination method. We seek a solution of the matrix equation by the following formula:

Unα_n1u_n1β_n1u_n2γ_n1, nM−2, . . . ,2,1, 3.17 whereαj,βj j1, . . . , M−1areN1×N1square matrices,γj j1, . . . , M−1are N1×1 column matrices, andα₁, β₁, γ₁, γ₂are zero matrices.α₂, β₂ are

α₂ 4

5I_N1, β₂−1 5I_N1, F_nC_nD_nα_nE_nα_n−1α_nE_nβ_n−1 α_n1F_n⁻¹

−Bn−Dnβn−Enα_n−1βn

, β_n1 −F_n⁻¹An, γ_n1F_n⁻¹

Rϕ_n−D_nγ_nE_nα_n−1γ_nE_nγ_n−1 .

3.18

For solution of the last diﬀerence equation, we need to finduM,u_M−1 u_M0,

u_M−1

β_M−25I

−4I−α_M−2αM−1₋₁

4I−α_M−2γM−1−γ_M−2.

3.19

3.4. Error Analysis The errors are computed by

E^N_M max

1≤k≤N−1,1≤n≤M−1

utk, xn−u^k_n 3.20

of the numerical solutions, whereutk, xnrepresents the exact solution andu^k_nrepresents the numerical solution attk, xnand the results are given inTable 1.

Thus, the results show that the second order of accuracy diﬀerence schemes3.10and 3.13are more accurate comparing with the first order of accuracy diﬀerence scheme3.4.

Acknowledgment

The authors would like to thank Professor P. E. SobolevskiiJerusalem, Israel, for the helpful suggestions to the improvement of this paper.

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