# Existence and Uniqueness of Generalized Solutions to a Telegraph Equation with an Integral Boundary Condition via Galerkin's Method

Authors: Assia Guezane-Lakoud; Jaydev Dabas; Dhirendra Bahuguna
Journal: International Journal of Mathematics and Mathematical Sciences
 (2011)

---

## Abstract

We consider a telegraph equation with nonlocal boundary conditions, and using the application of Galerkin's method we established the existence and uniqueness of a generalized solution.


---

## Body

## 1. Introduction

The initial work devoted to the second-order partial differential equations with nonlocal integral conditions goes back to Cannon [1]. Subsequent investigations include [1–10] and many other references therein.Nonlocal problems for some classes of partial differential equations have attracted much interest in the last few years, and several papers have been devoted to this subject. The problem of existence and uniqueness of solutions is of growing interest, as well as methods of obtaining explicit or approximate solutions for this kind of problems. The main reason of such a wide interest in this type of problems is that the non-local conditions occur in various equations of mathematical physics and in mathematical biology.Nonlocal conditions come up when values of the function on the boundary are connected to values inside the domain. There are various types of non-local integral conditions for hyperbolic, parabolic, or elliptic equations; we point out an important non-local problem (see [11, 12]), studied for a parabolic equation with the following non local conditions. (1.1)u(0,t)=∫01α1(x)u(x,t)dx+d1(t),u(1,t)=∫01α2(x)u(x,t)dx+d2(t),
connected with quasistationary thermoelasticity theory. The non-local boundary conditions of type (1.2)α(t)=∫01u(x,t)dx,u(1,t)=0,
arise in modeling the technology of integral circuits, studied in [13].Problems with nonlocal integral conditions for multidimensional  hyperbolic equations have been intensively studied in recent years. However, the nonlocal integral conditions for equations were in the form of second kind as(1.3)∂u∂η|(x,t)∈∂Ω×I+∫0T∫ΩK(x,ξ,τ)u(ξ,τ)dξdτ=0,∀x∈∂Ω,
or(1.4)u(x,t)|(x,t)∈∂Ω×I+∫ΩK(x,ξ)u(ξ,t)dξ=0,∀x∈∂Ω,
where Ω is a bounded domain in ℝnwith a smooth boundary, ∂u/∂η is the inward normal vector to ∂Ω; see [14, 15]. Non-local integral conditions of first kind as (1.5)∫ΩK(x,ξ)u(ξ,t)dξ=0,∀x∈∂Ω,
are studied in [15]. Under some assumptions on the kernel K, existence and uniqueness of the generalized solution are established.The present work deals with the application of the Galerkin method to determine a functionu=u(x,t), x∈Ω∈ℝn is a bounded domain with a smooth boundary ∂Ω, t∈I=(0,T), which satisfies, in weak sense, the telegraph equation(1.6)lu=∂2u∂t2+a∂u∂t+bu-c∂2u∂x2=f(x,t),(x,t)∈Q=Ω×I,
subject to the initial conditions(1.7)u(x,0)=φ(x),∂u∂t(x,0)=ψ(x)
and the integral condition(1.8)∂u∂η|(x,t)∈∂Ω×I+∫ΩK(x,ξ)u(ξ,t)dξ=0,∀x∈∂Ω,
where ∂u/∂η is the inward normal vector to ∂Ω. This equation arises in the analysis of propagation of electrical signals in a cable transmission line. Both the current and voltage satisfy an equation of this form. This equation arises also in the propagation of pressure waves in the study of pulsate blood flow in arteries.Cannon et al. [5] have developed and analyzed finite-difference and finite-element Galerkin’s methods for the solution of one-dimensional heat equation with the Neumann and integral conditions. Cannon and van der Hoek [4] presented numerical schemes based on the finite difference method. In [16], the authors have applied the Galerkin method to a parabolic problem with an integral boundary condition and established the existence, uniqueness, and continuous dependence upon the data of a weak solution. In [17–21], the authors have used the method of semidiscretization in time and have established the existence and uniqueness of a weak solution. In [22], the authors have discussed some existence uniqueness results for a neutral functional differential equation with a nonlocal initial condition via the Galerkin approximation.The paper is organized as follows. In Section2, we specify notations, state some inequalities and make the sense of the desired solution precise. In Section 3, we establish the uniqueness of the solution. Finally, Section 4, is devoted to the construction of the approximate solution and its existence via the Galerkin method.


## 2. Notation and Definition

LetL2(Q) be the usual space of Lebesgue square integrable real functions on Q whose inner product and norm will be denoted by (,) and ∥·∥, respectively.W1,2(Q) is the Sobolev space consisting of functions such that all derivatives lower than one belong to L2(Q) equipped with the norm (2.1)‖u‖W1,2(Q)2=‖u‖2+‖∇u‖2+‖ut‖2.We define the spaceWT1,2(Q)={v(x,t)∈W1,2(Q),v(x,T)=0}.


### 2.1. Some Useful Inequalities: Gronwall Inequalities [23]

Leth(t) and y(t) be two integrable nonnegative functions on the interval I with  h(t) nondecreasing. If (2.2)y(t)≤h(t)+c∫0ty(s)ds,∀t∈I,
where c is a positive constant, then y(t)≤h(t)ect,∀t∈I.Cauchy Schwarz inequality:
(2.3)(∫f(t)g(t)dt)2≤(∫|f(t)|2dt)(∫|g(t)|2dt).ɛ-Cauchy inequality:
(2.4)|αβ|≤ɛ2α2+12ɛβ2,∀α,β∈R,∀ɛ∈R+*.Trace inequality:
(2.5)∫∂Ω|v|2dsdt≤∫Ω(ɛ|∇v|2+c(ɛ)|v|2)dxdt,
where c(ɛ) is a positive constant that depends only on ɛ and on the domain Ω.


### 2.2. Definition of Generalized Solution

We define the generalized solution of the problem (1.6)–(1.8) as a weak solution that satisfies for all v∈WT1,2(Q)(2.6)(lu,v)=(f,v).
Integrating by parts, we get(2.7)∫Qutt(x,t)v(x,t)dxdt=-∫Ωψ(x)v(x,0)dx-∫Qut(x,t)vt(x,t)dxdt,∫Qut(x,t)v(x,t)dxdt=-∫Ωφ(x)v(x,0)dx-∫Qu(x,t)vt(x,t)dxdt,∫Quxx(x,t)v(x,t)dxdt=-∫0T∫∂Ωv(∫ΩK(x,ξ)u(ξ,t)dξ)dsdt-∫Q∇u(x,t)∇v(x,t)dxdt.
Regrouping the identities (2.7), then (2.6) becomes(2.8)∫Q(c∇u∇v-utvt-auvt+buv)dxdt+c∫0T∫∂Ωv(∫ΩK(x,ξ)u(ξ,t)dξ)dsdt=∫Qfvdxdt+∫Ωψ(x)v(x,0)dx+a∫Ωφ(x)v(x,0)dx.Definition 2.1.
By a generalized solution of problem(1.6)–(1.8) one means a functionu∈W1,2(Q)such that the identity (2.8) holds for all functionv∈WT1,2(Q).


## 2.1. Some Useful Inequalities: Gronwall Inequalities [23]

Leth(t) and y(t) be two integrable nonnegative functions on the interval I with  h(t) nondecreasing. If (2.2)y(t)≤h(t)+c∫0ty(s)ds,∀t∈I,
where c is a positive constant, then y(t)≤h(t)ect,∀t∈I.Cauchy Schwarz inequality:
(2.3)(∫f(t)g(t)dt)2≤(∫|f(t)|2dt)(∫|g(t)|2dt).ɛ-Cauchy inequality:
(2.4)|αβ|≤ɛ2α2+12ɛβ2,∀α,β∈R,∀ɛ∈R+*.Trace inequality:
(2.5)∫∂Ω|v|2dsdt≤∫Ω(ɛ|∇v|2+c(ɛ)|v|2)dxdt,
where c(ɛ) is a positive constant that depends only on ɛ and on the domain Ω.


## 2.2. Definition of Generalized Solution

We define the generalized solution of the problem (1.6)–(1.8) as a weak solution that satisfies for all v∈WT1,2(Q)(2.6)(lu,v)=(f,v).
Integrating by parts, we get(2.7)∫Qutt(x,t)v(x,t)dxdt=-∫Ωψ(x)v(x,0)dx-∫Qut(x,t)vt(x,t)dxdt,∫Qut(x,t)v(x,t)dxdt=-∫Ωφ(x)v(x,0)dx-∫Qu(x,t)vt(x,t)dxdt,∫Quxx(x,t)v(x,t)dxdt=-∫0T∫∂Ωv(∫ΩK(x,ξ)u(ξ,t)dξ)dsdt-∫Q∇u(x,t)∇v(x,t)dxdt.
Regrouping the identities (2.7), then (2.6) becomes(2.8)∫Q(c∇u∇v-utvt-auvt+buv)dxdt+c∫0T∫∂Ωv(∫ΩK(x,ξ)u(ξ,t)dξ)dsdt=∫Qfvdxdt+∫Ωψ(x)v(x,0)dx+a∫Ωφ(x)v(x,0)dx.Definition 2.1.
By a generalized solution of problem(1.6)–(1.8) one means a functionu∈W1,2(Q)such that the identity (2.8) holds for all functionv∈WT1,2(Q).


## 3. Uniqueness of the Generalized Solution

Now we will show that the generalized solution of problem (1.6)–(1.8), if it exists, is unique.Theorem 3.1.
Assume thatφ∈W1,2(Ω), ψ∈L2(Ω),f∈L2(Q),  K(x,y)∈C(Ω×Ω), and(3.1)max⁡Q¯|K(x,y)|≤k1,
then the generalized solution of problem  (1.6)–(1.8), if it exists, is unique.Proof.
Suppose that there exist two different generalized solutionsu1 and u2 for the problem (1.6)–(1.8); then obviously their difference u=u1-u2 is a generalized solution of the problem (1.6)–(1.8) with homogeneous equation and homogeneous initial and non local conditions, that is, f=φ=ψ=0. We will prove that u=0 in Q. Let v∈WT1,2(Q), and denote Qτ={(x,t);0<x<1,0<t≤τ≤T}. Consider the function
(3.2)v(x,t)={∫tτu(x,θ)dθ,0≤t≤τ,0,τ≤t≤T.
The identity (2.8) becomes
(3.3)∫Q(c∇u∇v-utvt-auvt+buv)dxdt+c∫0T∫∂Ωv(∫ΩK(x,ξ)u(ξ,t)dξ)dsdt=0.
Substituting v into (3.3), integrating by parts,  and then using the fact that vt(x,t)=-u(x,t), it follows that
(3.4)∫Ω(c(∇v(x,0))2+u2(x,τ)+bv2(x,0))dx=-2c∫0τ∫∂Ωv(∫ΩK(s,ξ)u(ξ,t)dξ)dsdt+2a∫0τ∫Ωu2(x,t)dxdt.
Using the assumption on the function K, we get
(3.5)∫Ω((∇v(x,0))2+u2(x,τ)+v2(x,0))dx≤C1(∫0τ∫∂Ω|v|(∫Ω|u(ξ,t)|dx)dsdt+∫0τ∫Ωu2(x,t)dxdt),
where C1=max(2ck1,2a)/min(c,1,b). Now, applying Cauchy Schwarz inequality to the first term in the right-hand side of (3.5), then ɛ-Cauchy inequality with ɛ=1, yields
(3.6)∫Ω((∇v(x,0))2+u2(x,τ)+v2(x,0))dx≤C1(∫0τ∫∂Ω|v|2dsdt+2∫0τ∫Ωu2(x,t)dxdt).
Using the trace inequality, we obtain
(3.7)∫Ω((∇v(x,0))2+u2(x,τ)+v2(x,0))dx≤C1(∫0τ∫Ω(ϵ(∇v)2+c(ɛ)v2)dxdt+2∫0τ∫Ωu2(x,t)dxdt).
Considering the function
(3.8)w(x,t)=∫0tu(x,θ)dθ,
it is easy to see that v(x,t)=w(x,τ)-w(x,t), ∇v(x,0)=∇w(x,τ), and (∇v(x,t))2≤2(∇w(x,τ))2+2(∇w(x,t))2. Consequently, substituting w in (3.7), we get
(3.9)∫Ω((∇w(x,τ))2+w2(x,τ)+u2(x,τ))dx≤2τC1∫Ω(ϵ(∇w(x,τ))2+c(ϵ)w2(x,τ))dx+2C1∫0τ∫Ω(ϵ(∇w)2+c(ϵ)w2+u2)dxdt.
Seting
(3.10)C2=2C1max⁡(ϵ,c(ϵ)),C3=2C1max⁡(ϵ,c(ϵ),1),
it follows that
(3.11)∫Ω((∇w(x,τ))2+w2(x,τ)+u2(x,τ))dx≤τC2∫Ω((∇w(x,τ))2+w2(x,τ))dx+C3∫0τ∫Ω((∇w)2+w2+u2)dxdt.
Since τ is arbitrary chosen, let 1-τC2>0; then (3.11) becomes
(3.12)(1-τC2)∫Ω((∇w(x,τ))2+w2(x,τ)+u2(x,τ))dx≤C3∫0τ∫Ω((∇w)2+w2+u2)dxdt.
Applying Gronwall Lemma, we get
(3.13)∫Ω((∇w(x,τ))2+w2(x,τ)+u2(x,τ))dx≤0,∀τ∈]0,1C2[.
Consequently, we obtain u(x,τ)=0, for all x∈Ω and τ∈]0,1/C2[.
IfT≤1/C2, then u=0 in Q. In the case where T≥1/C3, we see that ]0,T[⊂∪n=1n=n0](n-1)/C2,n/C2[, where n0=[C2T]+1,[C2T] is the entire part of C2T; then repeating the preceding reasoning for τ∈](n-1)/C2,n/C2[, we get u(x,τ)=0, for all τ∈](n-1)/C2,n/C2[, and then u(x,t)=0 in Q. Thus, the uniqueness is proved.Remark 3.2.
It should be noted that we can prove the uniqueness theorem by using a priori estimates, so we establish that the generalized solution, if it exists, satisfies the inequality(3.14)‖u‖W1,2(Q)2≤C[‖f‖2+‖ψ‖L2(Ω)2+‖φ‖W1,2(Ω)2].


## 4. Existence of Generalized Solution

Theorem 4.1.
Assume that the assumptions of Theorem4.1 hold; then the non-local problem (1.6)–(1.8) has a unique solution  u∈W1,2(Q).Proof.
In order to prove the existence of the generalized solution we apply Galerkin’s method. Let{wk(x)} be a fundamental system in W1,2(Ω), such that (wk,wi)L2(Q)=δk,i. Now we will try to find an approximate solution of the problem (1.6)–(1.8) in the form
(4.1)u(n)=∑k=1nαk(t)wk(x).
The approximates of the functions φ(x) and ψ(x) are denoted, respectively, by
(4.2)φ(n)(x)=∑k=1nφkwk(x),ψ(n)(x)=∑∑k=1nψkwk(x),αk(0)=φk,αk′(0)=ψk.
Substituting the approximation solution in (1.6), multiplying both sides by wi(x), and then integrating according to x on Ω, we get
(4.3)∫Ω(c∇u(n)∇wi+utt(n)wi+aut(n)wi+bu(n)wi)dx+c∫∂Ωwi(∫ΩK(x,ξ)u(n)(ξ,t)dξ)ds=∫Ωfwidx.
Substituting (4.1) in (4.3), we obtain
(4.4)∫Ω∑k=1n(cαk∇wk∇wi+αk′′wkwi+aαk′wkwi+bαkwkwi)dx+c∫∂Ω∑k=1nwi(αk∫ΩK(x,ξ)wk(ξ)dξ)ds=∫Ωfwidx.
This implies that
(4.5)∑k=1n[αk′′(wk,wi)L2(Ω)+αk′(awk,wi)L2(Ω)+αk((c∇wk,∇wi)L2(Ω)+(bwk,wi)L2(Ω))]+∑k=1nαk∫∂Ωcwi(∫ΩK(x,ξ)wk(ξ)dξ)ds=∫Ωfwidx.
Denote
(4.6)βk,i=(c∇wk,∇wi)L2(Ω)+(bwk,wi)L2(Ω)+∫∂Ωcwi(∫ΩK(x,ξ)wk(ξ)dξ)ds,fi=∫Ωfwidx.
Then (4.5) becomes
(4.7)∑k=1n[αk′′δk,i+aαk′δk,i+αkβk,i]=fi.
We obtain a system of differential equations of second order according to the variable t with smooth coefficients and the initial conditions
(4.8)αk(0)=φk,αk′(0)=ψk.
Consequently, we get a Cauchy problem of linear differential equations with smooth coefficients that is uniquely solvable. So it has a unique solution u(n) satisfying (4.3).Lemma 4.2.
The sequence(u(n)) is bounded.Proof.
Multiplying (4.4) by αi′ and summing over i from 1 to n, we get
(4.9)∫Ω(ut(n)utt(n)+c∇u(n)∇ut(n)+aut(n)ut(n)+bu(n)ut(n))dx+c∫∂Ωut(n)(∫ΩK(x,ξ)u(n)(ξ,t)dξ)ds=∫Ωfut(n)dx.
Integrating by parts the left-hand side of (4.9) over t on (0,τ) yields
(4.10)∫0τ∫Ωut(n)utt(n)dxdt=12∫Ω[(ut(n)(x,τ))2-(ut(n)(x,0))2]dx,∫0τ∫Ω∇u(n)∇ut(n)dxdt=12∫Ω[(∇u(n)(x,τ))2-(∇u(n)(x,0))2]dx,∫0τ∫Ωu(n)ut(n)dxdt=12∫Ω[(u(n)(x,τ))2-(u(n)(x,0))2]dx,∫0τ∫∂Ωut(n)(∫ΩK(x,ξ)u(n)(ξ)dξ)dsdt=∫∂Ωu(n)(x,τ)(∫ΩK(x,ξ)u(n)(ξ,τ)dξ)ds-∫∂Ωu(n)(x,0)(∫ΩK(x,ξ)u(n)(ξ,0)dξ)ds-∫0τ∫∂Ωu(n)(∫ΩK(x,ξ)ut(n)(ξ)dξ)dsdt.
Substituting the four last identities into (4.9), we get
(4.11)∫Ω[(ut(n)(x,τ))2+c(∇u(n)(x,τ))2+b(u(n)(x,τ))2]dx=∫Ω[(ut(n)(x,0))2+c(∇u(n)(x,0))2+b(u(n)(x,0))2]dx-2c∫∂Ωu(n)(x,τ)(∫ΩK(x,ξ)u(n)(ξ,τ)dξ)ds+2c∫∂Ωu(n)(x,0)(∫ΩK(x,ξ)u(n)(ξ,0)dξ)ds+2c∫0τ∫∂Ωu(n)(x,t)(∫ΩK(x,ξ)ut(n)(ξ,t)dξ)dsdt-2a∫0τ∫Ω(ut(n))2dxdt+2∫0τ∫Ωfut(n)dxdt.
Applying Cauchy Schwarz inequality, ϵ-Cauchy inequality, the hypothesis on the operator K, and the fact that -2a∫0τ∫Ω(ut(n))2dxdt≤0 to the second term in the right hand side of (4.11), we get
(4.12)|∫∂Ωu(n)(x,τ)(∫ΩK(x,ξ)u(n)(ξ,τ)dξ)ds|≤12(c(ɛ)+k12|∂Ω|)∫Ω(u(n)(x,τ))2dx+ɛ2∫Ω(∇u(n)(x,τ))2dx.
We may obtain similar inequalities for the third and the fourth terms in the right-hand side of (4.11). Now if we apply Cauchy Schwarz inequality to the last term in (4.11) we get
(4.13)|∫0τ∫Ωfut(n)dxdt|≤12∫0τ∫Ω|f(x,t)|2dxdt+12∫0τ∫Ω|ut(n)(x,t)|2dxdt.
Using the result inequalities in (4.11) and then regrouping the same terms yield
(4.14)∫Ω[(ut(n)(x,τ))2+c(1-ɛ)(∇u(n)(x,τ))2+(b-c(c(ɛ)+k12|∂Ω|))(u(n)(x,τ))2]dx≤∫Ω[(ut(n)(x,0))2+(c+cɛ)(∇u(n)(x,0))2+(b+c(c(ɛ)+k12|∂Ω|))(u(n)(x,0))2]dx+c(c(ɛ)+k12|∂Ω|)∫0τ∫Ω(u(n)(x,t))2dxdt+cɛ∫0τ∫Ω(∇u(n)(x,t))2dxdt+∫0τ∫Ω|f(x,t)|2dxdt+∫0τ∫Ω|ut(n)(x,t)|2dxdt.
Choose ϵ such that (1-ɛ)>0 and b-c(c(ɛ)+k12|∂Ω|)>0 and let
(4.15)M=max⁡{1,(c+cɛ),(b+c(c(ɛ)+k12|∂Ω|))},m=min⁡{1-ɛ,b-c(c(ɛ)+k12|∂Ω|)},K2=Mm,
then (4.14) becomes
(4.16)∫Ω[(ut(n)(x,τ))2+(∇u(n)(x,τ))2+(u(n)(x,τ))2]dx≤K2[∫Ω((ψ(n)(x))2+(∇φ(n)(x))2+(φ(n)(x))2)dx+∫0τ∫Ω(|ut(n)(x,t)|2+(u(n)(x,t))2(∇u(n)(x,t))2)dxdt+∫0T∫Ω|f(x,t)|2dxdt].
Now, we apply Gronwall Lemma to get
(4.17)∫Ω[(ut(n)(x,τ))2+(∇u(n)(x,τ))2+(u(n)(x,τ))2]dx≤eK2T(∫Ω[(ψ(n)(x))2+(∇φ(n)(x))2+(φ(n)(x))2]dx+‖f‖2).
Since φ(n) and ψ(n) are the orthogonal projections of φ and ψ, respectively, from the Pythagorean relation we have that  ∥φ(n)∥W1,2(Ω)2≤∥φ∥W1,2(Ω)2 and ∥ψ(n)∥L2(Ω)2≤∥ψ∥L2(Ω)2. Integrating (4.17) according to τ on [0,T] yields
(4.18)‖u(n)‖W1,2(Q)2≤TeK2T[‖f‖2+‖ψ‖L2(Ω)2+‖φ‖W1,2(Ω)2].
Consequently, the sequence {u(n)} is bounded.Remark 4.3.
We have proved that the sequence{u(n)} is bounded, so we can extract a subsequence which we denote by {u(nk)} that is weakly convergent; then we prove that its limit is the desired solution of the problem (1.6)–(1.8).Lemma 4.4.
The limit of the subsequence{u(nk)} is the solution of the problem (1.6)–(1.8).Proof.
For this we prove that the limit of the subsequence{u(nk)} satisfies the identity (2.8) for all functions v=∑i=1nvi(t)wi(x)∈WT1,2(Q). Since the set Sn={v(x,t)=∑k=1nvk(t)wk(x),vk(t)∈C2(0,T),vk(T)=0} is such that ∪n=1∞Sn is dense in WT1,2(Q), it suffices to prove (2.8) for v∈Sn. Multiplying (4.3) by the function vi(t)∈W1,2(0,T),  vi(T)=0, and then taking the sum from i=0 to n, we obtain
(4.19)∫Ω(c∇u(nk)∇v+utt(nk)v+aut(nk)v+bu(nk)v)dx+c∫∂Ωv(∫ΩK(x,ξ)u(nk)(ξ,t)dξ)ds=∫Ωfvdx.
Integrating by parts on [0,T], we get
(4.20)∫Q(c∇u(nk)∇v-ut(nk)v-au(nk)v+bu(nk)v)dxdt+c∫0T∫∂Ωv(∫ΩK(x,ξ)u(nk)(ξ,t)dξ)dsdt=∫Ωψ(nk)(x)v(x,0)dx+a∫Ωφ(nknk)(x)v(x,0)dx+∫Qfvdxdt.
Denote the weak limit of the subsequence {u(nk)} by u. When k tends to infinity, we get
(4.21)∫0T∫∂Ωv(∫ΩK(x,ξ)u(nk)(ξ,t)dξ)dsdt⟶∫0T∫∂Ωv(∫ΩK(x,ξ)u(ξ,t)dξ)dsdt.
Indeed, using Cauchy Schwarz inequality then ɛ-Cauchy inequality, we obtain
(4.22)|∫0T∫∂Ωv(∫ΩK(x,ξ)u(nk)(ξ,t)dξ)dsdt-∫0T∫∂Ωv(∫ΩK(x,ξ)u(ξ,t)dξ)dsdt|≤k1|∂Ω|(∫0T∫∂Ω|v|2dsdt)1/2×(∫0T∫Ω|u(nk)(ξ,t)-u(ξ,t)|2dξdsdt)1/2⟶0,
and then by passing to the limit in (4.20) we get that the limit u satisfies(2.8).Example 4.5.
Now we present an example to demonstrate the applications of the results established in the earlier sections. We consider the following:(4.23)∂2u∂t2+∂u∂t+u-∂2u∂x2=sinx(sint+cos⁡t),0<t≤T,x∈(0,π),
with the initial conditions
(4.24)u(x,0)=0,∂u∂t(x,0)=sinx,x∈(0,π),
and the nonlocal boundary condition
(4.25)∂u∂x(π,t)-∂u∂x(0,t)-∫0πu(s,t)ds=0.
The results of the earlier sections guarantee the existence and uniqueness of a solution. It may be noted thatu(x,t)=sinxsint is the unique solution of (4.23)–(4.25).



---

*Source: 451492-2011-05-09.xml*