Abstract
This paper investigates the one dimensional mixed problem with nonlocal boundary conditions, for the quasilinear parabolic equation. Under some natural regularity and consistency conditions on the input data, the existence, uniqueness, convergence of the weak generalized solution, and also continuous dependence upon the data of the solution are shown by using the generalized Fourier method. We construct an iteration algorithm for the numerical solution of this problem.
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1 Introduction
D denotes the domain
Consider the equation
with the initial condition
the nonlocal boundary condition
for a quasilinear pseudo-parabolic equation with the nonlinear source term .
The functions and are given functions on and , respectively. is a small parameter, .
Denote the solution of problem (1)-(4) by .
This problem was investigated with different boundary conditions by various researchers by using Fourier methods [1].
In this study, we consider the initial-boundary value problem (1)-(4) with nonlocal boundary conditions (2)-(3). The periodic nature of (2)-(3) type boundary conditions is demonstrated in [2]. In this study, we prove the existence, uniqueness convergence of the weak generalized solution, continuous dependent upon the data of the solution; and we construct an iteration algorithm for the numerical solution of this problem. We analyze computationally convergence of the iteration algorithm, as well as treating a test example.
We will use the weak solution approach from [3] for the considered problem (1)-(4).
According to [4, 5] we assume the following definitions.
Definition 1 The function is called test function if it satisfies the following conditions:
Definition 2 The function satisfying the integral identity
for arbitrary test function , is called a generalized (weak) solution of problem (1)-(4).
2 Reducing the problem to countable system of integral equations
Consider the following system of functions on the interval :
The system of functions (6) and (7) arise in [6] for the solution of a nonlocal boundary value problem in heat conduction.
It is easy to verify that the systems of function (6) and (7) are biorthonormal on . They are also Riesz bases in (see [7, 8]).
We will use the Fourier series representation of the weak solution to transform the initial-boundary value problem to the infinite set of nonlinear integral equations.
Any solution of (1)-(4) can be represented as
where the functions , , satisfy the following system of equations:
where
Definition 3 Denote the set
of functions continuous on satisfying the condition
by B. Let
be the norm in B. It can be shown that B is a Banach space [9].
We denote the solution of the nonlinear system (9) by .
Theorem 4
-
(1)
Assume the function is continuous with respect to all arguments in and satisfies the following condition:
where , ,
-
(2)
, ,
-
(3)
.
Then the system (9) has a unique solution in D.
Proof For , let us define an iteration for the system (9) as follows:
where, for simplicity, we let
where , , .
From the condition of the theorem we have . We will prove that the other sequential approximations satisfy this condition.
Let us write in (11).
Adding and subtracting to both sides of the last equation, we obtain
Applying the Cauchy Inequality to the last equation, we have
Applying the Lipschitzs Condition to the last equation, we have
Let us use
Taking the maximum of both sides of the last inequality yields the following:
Adding and subtracting to both sides of the last equation, we obtain
Applying the Cauchy Inequality to the last equation, we have
Taking the summation of both sides with respect to k and using the Hölder Inequality yield the following:
Using the Bessel Inequality in the last inequality, we obtain
Applying the Lipschitzs Condition to the last equation and taking the maximum of both sides of the last inequality yield the following:
Adding and subtracting
to both sides of the last equation and applying the derivative to , we obtain
Applying the Cauchy Inequality to the last equation, we have the following:
Taking the summation of both sides with respect to k and using the Hölder, Bessel, and Lipschitzs Inequalities yields the following:
Taking the maximum of both sides of the last inequality yields the following:
Finally we have the following inequality:
Hence . In the same way, for a general value of N we have
considering the induction hypothesis that , we deduce that , and by the principle of mathematical induction we obtain
Now we prove that the iterations converge in B, as . We have
Applying the Cauchy Inequality, the Hölder Inequality, the Lipschitzs Condition, and the Bessel Inequality to the right side of (11), respectively, we obtain
Let
Applying the Cauchy Inequality, the Hölder Inequality, the Lipschitzs Condition, and the Bessel Inequality to the right hand side of (9), respectively, we obtain
Applying the Cauchy Inequality, the Hölder Inequality, the Lipschitzs Condition, and the Bessel Inequality to the right hand side of (9), respectively, we obtain
In the same way, for a general value of N we have
Then the last inequality shows that the converge in B.
Now let us show . Noting that
it follows that if we prove , then we may deduce that satisfies (9).
To this aim we estimate the difference ; after some transformation we obtain
Adding and subtracting under the appropriate integrals to the right hand side of the inequality we obtain
Applying the Gronwall Inequality to the last inequality and using inequality (11), we have
For the uniqueness, we assume that problem (1)-(4) has two solutions u, v. Applying the Cauchy Inequality, the Hölder Inequality, the Lipschitzs Condition, and the Bessel Inequality to the right hand side of , respectively, we obtain
applying the Gronwall Inequality to the last inequality we have
The theorem is proved. □
3 Solution of problem (1)-(4)
Using the solution of the system (9) we compose the series
It is evident that the series converges uniformly on D. Therefore the sum
is continuous on D. We have
From the conditions of Theorem 4 and from
it follows that
Using
and
on the left hand side of (5) we denote the obtained expression by :
Applying the integration by parts the formula on the right hand side, the last equation, and using the conditions of Theorem 4, we can show that
This shows that the function is a generalized (weak) solution of problem (1)-(4).
The following theorem shows the existence and uniqueness results for the generalized solutions of problem (1)-(4).
Theorem 5 Under the assumptions of Theorem 4, problem (1)-(4) possesses a unique generalized solution in the following form:
4 Continuous dependence upon the data
In this section, we shall prove the continuous dependence of the solution using an iteration method.
Theorem 6 Under the conditions of Theorem 4, the solution depends continuously upon the data.
Proof Let and be two sets of data which satisfy the conditions of Theorem 4. Let and be the solutions of problem (1)-(4) corresponding to the data ϕ and , respectively, and
The solution is defined by the following forms, respectively:
where
For simplicity, let us write
where , , . From the condition of the theorem we have . We will prove that the other sequential approximations satisfy this condition. First of all, we consider :
Adding and subtracting
to both sides of the last equation.
Applying the Cauchy Inequality, the Hölder Inequality, the Lipschitzs Condition, and the Bessel Inequality to the right side of (11), respectively, we obtain
Applying the Cauchy Inequality, the Hölder Inequality, the Lipschitzs Condition, and the Bessel Inequality to , respectively, we obtain
Applying the Cauchy Inequality, the Hölder Inequality, the Lipschitzs Condition, and the Bessel Inequality to , respectively, we obtain
In the same way, for a general value of N we have
where
and
(The sequence is convergent; then we can write , ∀N.)
It follows from the estimation [1] that ; then let for the last equation,
where . If and , then . □
5 Numerical procedure for the nonlinear problem (1)-(4)
We construct an iteration algorithm for the linearization of problem (1)-(4) for :
Let and . Then problem (16)-(19) can be written as a linear problem:
We use the finite difference method to solve (20)-(23).
We subdivide the intervals and into M and N subintervals of equal lengths, and , respectively.
Then we add the line to generate the fictitious point needed for the second boundary condition.
We choose the implicit scheme, which is absolutely stable and has a second order accuracy in h and a first order accuracy in τ.
The implicit monotone difference scheme for (20)-(23) is as follows:
where and are the indices for the spatial and time steps, respectively, is the approximation to , , , , [10].
At the level, adjustment should be made according to the initial condition and the compatibility requirements.
6 Numerical examples
In this section, we will consider an example of numerical solution of the nonlinear problem (1)-(4).
These problems were solved by applying the iteration scheme and the finite difference scheme which were explained in Section 4. The condition
with was used as a stopping criterion for the iteration process.
Example 1 Consider the problem
It is easy to check that the analytical solution of this problem is
The comparisons between the analytical solution and the numerical finite difference solution for different ε values when are shown in Figures 1 and 2.
We show in Figure 3 the analytical solution for and the numerical solution for , , .
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Baglan, I., Kanca, F. Weak generalized and numerical solution for a quasilinear pseudo-parabolic equation with nonlocal boundary condition. Adv Differ Equ 2014, 277 (2014). https://doi.org/10.1186/1687-1847-2014-277
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DOI: https://doi.org/10.1186/1687-1847-2014-277