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 
They
proved 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.