HYERS-ULAM STABILITY OF HEAT-CONDUCTION EQUATION

We prove the Hyers-Ulam stability of a partial differential equation. That is, if u is an approximate solution of the heat-conduction equation ∂u ∂t = a2 ∂ 2u ∂x2 + f(x, t) and ∂u ∂t = a2( 2u ∂x2 + ∂ 2u ∂y2 ) + f(x, y, t), then there exists an exact solution of the differential equation near to u. AMS Subject Classification: 34K20, 26D10

The aim of this paper is to study the Hyers-Ulam stability of the following partial differential equations : where a ∈ R is a fixed number.

Fourier Transform and Inverse Transform
Firsly, we should have the following prerequisite knowledge.
Fourier transform: inverse transform: we make the notation that: Hence, we have and if both f ′ (x) and f (x) is suitable to fourier transform, we have:

Main Results
To prove the stability of heat-conduction equation, we need to prove the following Lemmas firstly.As we all know, fourier transform can be use to solve some partial differential equation: Lemma 3.1.Suppose that u(x, t) : R × [0, ∞) → R, and ϕ(x) : R → R is continuous and bounded.Then is the solution of the equations Proof.We make the notation that: And then apply Fourier transform to and by using Eq.(2.2), we have It is an ordinary differential equation, the solution is: By using the inverse Fourier transform: By the properties of the Fourier transform Eq.(2.1), we get We can verify that it is exactly the solution.
Lemma 3.2.Suppose that u(x, t) : R × [0, ∞) → R, and f (x) : R → R is continuous and bounded.Then is the solution of the equations Proof.By using homogeneitisation principle, we know that: where w(x, t) is the solution of: Then, by using Lemma 3.1, we obtain the solution of u(x, t).
By Lemma 3.1 and 3.2, it is easy to see the following lemma holds.
Lemma 3.3.Suppose that u(x, t) : R × [0, ∞) → R, and ϕ(x), f (x) : R → R is continuous and bounded.Then is the solution of Eq. ( 1) The main result of this paper is given in the next theorem.
Proof.Let u be a solution of inequality in theorem and put Using Lemma 3.3, we can know that the solution of the above equation is On the other hand, the solution of Eq.( 1) is Moreover, by noticing we have Hence, Remark 3.5.We can know from the above theorem that: for every fixed t ∈ R + , Eq.( 1) has Hyers-Ulam stability.
Corollary 3.6.Let ε be a nonnegative number.If u(x, t) : R × [0, +∞) → R, and ϕ(x) and f (x, t) are continuous and bounded , and u satisfies the following inequality for every (x, t) ∈ R × R + , where g(x, t) satisfies for all t ∈ R, then there exists a solution v of By using the similar method, we can prove the situation of two-dimensional space.