Malaysian Journal of Mathematical Sciences, August 2012, Vol. 6(S)
Special Issue: International Workshop on Mathematical Analysis (IWOMA)
Finite Difference Method for the Estimation of a Heat Source Dependent on Time Variable
Allaberen Ashyralyev and Abdullah Said Erdogan
Corresponding Email: [email protected]
Received date: -
Accepted date: -
Abstract:
Well-posedness of difference scheme for the inverse problem of reconstructing the right side of a parabolic equation $$\left\{\begin{matrix}
\frac{\partial u(t,x)}{\partial t}=a(x)\frac{\partial^{2}u(t)}{\partial x^{2}}-\sigma u(t,x)+p(t)q(x)+f(t,x),\\
0 < x < 1, 0 < t \leq T,\\
u_{x}(t,0)=0, u_{x}(t,l)=\psi(t), 0 \leq t \leq T,\\
u(0,x)=\varphi (x), 0 \leq x \leq l,\\
u(t, x^{\star})=\rho(t), 0 \leq x^{\star} \leq l, 0 \leq t \leq T,
\end{matrix}\right.$$ where \(u(t, x)\) and \(p(t)\) are unknown functions, \(f(t, x)\), \(q(x)\), \(\varphi(x)\), \(\psi(t)\) and \(\rho(t)\) are given functions, \(a(x) \geq \delta > 0\) and \(\sigma > 0\) is a sufficiently large number. Numerical methods for estimation of constant terms of coercive stability estimates are described.
Keywords: Parabolic inverse problem, well-posedness, difference scheme
