Hyers-Ulam-Rassias Stability for the Heat Equation

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. n 


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][3][4][5][6][7][8][9][10]).
Alsina and Ger [11] were the first mathematicians who investigated the Hyers-Ulam stability of the differential equation .g g   : They proved that if a differentiable function y I  R satisfies y y     for all , t I  then there exists a differentiable function : for any t 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 .

t I 
. y y 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 .y y 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.
    0, In this paper we consider the Hyers-Ulam-Rassias stability of the heat equation with the initial condition where     We also use a similar argument to establish the Hyers-Ulam-Rassias for the heat equation in higher dimension 2 0 , with the initial condition .
Moreover we have proved theorems on Hyers-Ulam-Rassias-Gavruta stability for the heat equation in a finite rod.
Definition 1 We will say that the Equation (1) has the with the initial condition (2) , then there exists a solution of the Equation ( 1), such that where K is a constant that does not depend on  nor on   , Definition 2 We will say that the equation ( 1) has the Hyers-Ulam-Rassias-Gavruta (HURG) stability with respect to 0,

 
if there exists K > 0 such that for each 0 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 K  nor on and We find the Fourier transform of the function.Since and by defintion 4 we have where and from (8) one has Therefore, from ( 7), ( 9) we obtain Theorem 1 (See Evans [22]) Assume that   f x and   g x are continuously differentiable and absolutely integrable on .Then Copyright © 2013 SciRes.AM and  .

On Hyers-Ulam-Rassias Stability for Heat Equation on an Infinite Rod
then the initial value problem (1), ( 2) is stable in the sense of Hyers-Ulam-Rassias.
Proof.Let 0   and   , u x t be an approximate solution of the initial value problem (1), ( 2).We will show that there exists a function   Applying Fourier Transform to inequality (10), we get Or, equivalently Integrating the inequality from 0 to we obtain t From which it follows where , and x n e    .Putting n = 1, and Applying inverse Fourier transform to the last inequality and using convolution theorem we have 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 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 .
Proof.Let 0   and be an approximate solution of the initial value problem (3), (4).We will show that there exists a function , 0 , w x satisfying the Equation (3) and the initial condition (4) such that then from the inequality (5), we have Applying Fourier Transform to inequality ( 14), we get Or, equivalently Integrating the inequality from 0 to we obtain t From which it follows where and 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 One can find that ( 16) is a solution of Equation (3).
To show that Proof.Indeed, if we take   Applying Fourier Transform to inequality (17), we get 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.
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 4 0 ,0 e , Applying the same argument used in the proof of the Theorem 4 to the inequality One can show that the function is a solution of the problem (19), (20).Or, equivalently ,0 e .
x w x   Hence, from ( 21) and ( 23) we get Hence the initial value problem ( 19), ( 20) is stable in the sense of HURG.Moreover, since is asymptotically stable in the sense of HURG.

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 with the initial condition and the boundary conditions where     We introduce the notation We also have where where is a constant that does not explicitly depend on k From the definition of Hyers-Ulam stability we have where for t < c and for t > c, .
  0,    c  By applying the Laplace transform to (26), ( 27) we obtain with the boundary conditions One can easily verify that the function which is given by with boundary condition (32).Now consider the difference Using Gronwall's inequality, we get the estimation , , e x p 2


Hence the initial value problem (3), (4) is stable in the sense of Hyers-Ulam-Rassias.Then the the initial-value problem (1), (2) is stable in the sense of HURG. d Hence the initial-boundary value problem (33)-(35) is stable in the sense of HURG.