﻿Hyers-Ulam-Rassias Stability for the Heat Equation

Applied Mathematics
Vol.4 No.7(2013), Article ID:33958,8 pages DOI:10.4236/am.2013.47137

Hyers-Ulam-Rassias Stability for the Heat Equation

Maher Nazmi Qarawani

Department of Mathematics, AlQuds Open University, Salfit, Palestine

Email: mkerawani@qou.edu

Copyright © 2013 Maher Nazmi Qarawani. 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.

Received April 26, 2013; revised May 27, 2013; accepted June 5, 2013

Keywords: Hyers-Ulam-Rassias Stability; Heat Equation; Fourier Transform; Laplace Transform

ABSTRACT

In this paper we apply the Fourier transform to prove the Hyers-Ulam-Rassias stability for one dimensional heat equation on an infinite rod. Further, the paper investigates the stability of heat equation in with initial condition, in the sense of Hyers-Ulam-Rassias. We have also used Laplace transform to establish the modified Hyers-Ulam-Rassias stability of initial-boundary value problem for heat equation on a finite rod. Some illustrative examples are given.

1. Introduction and Preliminaries

The study of stability problems for various functional equations originated from a famous talk given by Ulam in 1940. In the talk, Ulam discussed a problem concerning the stability of homomorphisms. A significant breakthrough came in 1941, when Hyers [1] gave a partial solution to Ulam’s problem. Afterthen and during the last two decades a great number of papers have been extensively published concerning the various generalizations of Hyers result (see [2-10]).

Alsina and Ger [11] were the first mathematicians who investigated the Hyers-Ulam stability of the differential equation Theyproved that if a differentiable function satisfies for all then there exists a differentiable function satisfying for any such that for all This result of alsina and Ger has been generalized by Takahasi et al. [12] to the case of the complex Banach space valued differential equation

Furthermore, the results of Hyers-Ulam stability of differential equations of first order were also generalized by Miura et al. [13], Jung [14] and Wang et al. [15].

Li [16] established the stability of linear differential equation of second order in the sense of the Hyers and Ulam Li and Shen [17] proved the stability of nonhomogeneous linear differential equation of second order in the sense of the Hyers and Ulam while Gavruta et al. [18] proved the Hyers-Ulam stability of the equation with boundary and initial conditions. Jung [19] proved the Hyers-Ulam stability of first-order linear partial differential equations. Gordji et al. [20] generalized Jung’s result to first order and second order Nonlinear partial differential equations. Lungu and Craciun [21] established results on the Ulam-Hyers stability and the generalized Ulam-HyersRassias stability of nonlinear hyperbolic partial differential equations.

In this paper we consider the Hyers-Ulam-Rassias stability of the heat equation

(1)

with the initial condition

(2)

where and

We also use a similar argument to establish the HyersUlam-Rassias for the heat equation in higher dimension

(3)

with the initial condition

(4)

where

Moreover we have proved theorems on Hyers-UlamRassias-Gavruta stability for the heat equation in a finite rod.

Definition 1 We will say that the Equation (1) has the Hyers-Ulam-Rassias stability with respect to if there exists K > 0 such that for each and for each solution of the inequality

(5)

with the initial condition (2) then there exists a solution of the Equation (1), such that

,

where is a constant that does not depend on nor on and

Definition 2 We will say that the equation (1) has the Hyers-Ulam-Rassias-Gavruta (HURG) stability with respect to if there exists K > 0 such that for each and for each solution of the inequality

(6)

with the initial condition (2), then there exists a solution of the Equation (1), such that

,

where is a constant that does not depend on nor on and

Definition 3 We will say that the solution of the initial value problem (1), (2) has the Hyers-Ulam-Rassias asymptotic stability with respect to, if it is stable in the sense of Hyers and Ulam with respect to and

Definition 4 Assume the functions and defined on are continuously differentiable and absolutely integrable, then the Fourier transform of is defined as

and the inverse Fourier transform of is

Example 1 Let

We find the Fourier transform of the function.

Since

Then

and by defintion 4 we have

(7)

where

(8)

Differentiating with respect to, we get

Integrating by parts gives

Hence

Putting gives and from (8) one has

Using that, we have

(9)

Therefore, from (7), (9) we obtain

Theorem 1 (See Evans [22]) Assume that and are continuously differentiable and absolutely integrable on. Then 1) for each such that

2) where

is the convolution of and

2. On Hyers-Ulam-Rassias Stability for Heat Equation on an Infinite Rod

Theorem 2 If then the initial value problem (1), (2) is stable in the sense of HyersUlam-Rassias.

Proof. Let and be an approximate solution of the initial value problem (1), (2). We will show that there exists a function satisfying the Equation (1) and the initial condition (2) such that

If we take then from inequality (5), we have

(10)

Applying Fourier Transform to inequality (10), we get

(11)

Or, equivalently

Integrating the inequality from 0 to we obtain

From which it follows

(12)

where and In Example 1, we have established

. Putting n = 1, and, we obtain

Now, Using the convolution theorem, from inequality (12) one has

Applying inverse Fourier transform to the last inequality and using convolution theorem we have

Let us take

(13)

Applying arguments shown above to initial-value problem (1), (2), one can show that (13) is an exact solution of Equation (1).

To show that we put Then so that

Hence, as we find

Therefore the initial value problem (1), (2) is stable in the sense of Hyers-Ulam-Rassias.

More generally, the following Theorem was established for the Hyers-Ulam-Rassias stability of heat equation in

Theorem 3 If then the initial value problem (3), (4) is stable in the sense of Hyers-Ulam-Rassias.

Proof. Let and be an approximate solution of the initial value problem (3), (4). We will show that there exists a function satisfying the Equation (3) and the initial condition (4) such that

Taking then from the inequality (5), we have

(14)

Applying Fourier Transform to inequality (14), we get

Or, equivalently

Integrating the inequality from 0 to we obtain

From which it follows

(15)

where and

Using Example 1, we find that

and applying the convolution theorem, from inequality (15) one has

By applying the inverse Fourier transform to the last inequality, and then using convolution theorem we get

Now, let us take

(16)

One can find that (16) is a solution of Equation (3).

To show that we put Then so that

Hence as we obtain

since

Hence the initial value problem (3), (4) is stable in the sense of Hyers-Ulam-Rassias.

Theorem 4 Suppose that satisfies the inequality (5) with the initial condition Then the the initial-value problem (1), (2) is stable in the sense of HURG.

Proof. Indeed, if we take then from the inequality (5), we have

(17)

Applying Fourier Transform to inequality (17), we get

Now, by applying the same argument used above, we obtain

(18)

One takes

as a solution of initial-value problem (1), (2).

Therefore the initial value problem (1), (2) is stable in the sense of HURG.

Corollary 1 Suppose that satisfies the inequality (5) with the initial condition (2). Then the the initial-value problem (1), (2) is asymptotically stable in the sense of Hyers-Ulam-Rassias.

Proof. It follows from Theorem 4, and letting in (18), we infer that

Remark Using similar arguments it can be shown that the initial-value problem (3), (4) is asymptotically stable in the sense of HURG.

Example 2 We find the solution of the Cauchy problem

(19)

(20)

Applying the same argument used in the proof of the Theorem 4 to the inequality

we get

(21)

One can show that the function

(22)

is a solution of the problem (19), (20).

Or, equivalently

Now, using the change of variables

in the integral

we obtain the integral

Therefore we have

(23)

It is clear that

Hence, from (21) and (23) we get

Hence the initial value problem (19), (20) is stable in the sense of HURG. Moreover, since

then problem (19), (20) is asymptotically stable in the sense of HURG.

3. A Modified Hyers-Ulam-Rassias Stability for Problem of Heat Propagation in a Finite Rod

In this section we show how Laplace transform method can be used to esatblish the Hyers-Ulam-Rassias-Gavruta (HURG) stability of solution for heat equation

(24)

with the initial condition

(25)

and the boundary conditions

(26)

where and

We introduce the notation

where

Theorem 5 If then the initial-boundary value problem (24-26) is stable in the sense of Hyers-Ulam-Rassias.

Proof. Given Suppose is an approximate solution of the initial value problem (24)-(26). We show that there exists an exact solution satisfying the Equation (24) such that

where is a constant that does not explicitly depend on nor on

From the definition of Hyers-Ulam stability we have

(27)

where for t < c and for t > c,.

By applying the Laplace transform to (26), (27) we obtain

(28)

and

Assuming the operation of differentiation with respect to is interchangeable with integration with respect to in Laplace transform, we will get

(29)

We also have

(30)

From the inequality (28), and using (29), (30) it follows that

(31)

Integrating twice inequality (31) from 0 to x, we have

with the boundary conditions

(32)

One can easily verify that the function which is given by

has to satisfy the the equation

with boundary condition (32).

Now consider the difference

Using Gronwall’s inequality, we get the estimation

Or, equivalently

Consequently, we have

Hence the initial-boundary value problem (24)-(26) is stable in the sense of HURG.

Example 3 Consider the problem

(33)

with the initial condition

(34)

with the boundary conditions

(35)

By the definition of HURG stability we have

(36)

By applying the Laplace transform to ( 36) we obtain

(37)

Integrating twice inequality (37) from 0 to x, we have

with the boundary conditions

It is easily to verify that the function

satisfies the boundary value problem

Now consider the difference

Hence, we get the estimation

Or, equivalently

Consequently, we have

Hence the initial-boundary value problem (33)-(35) is stable in the sense of HURG.

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