### NUMERICAL SIMULATION OF THE SOME ILL-POSED PROBLEMS FOR THE HEAT TRANSFER EQUATION

#### Abstract

In this paper the regularization methods for solving some one-dimensional ill-posed problems for heat transfer equation are considered. The solution is obtained by solving the retrospective problem for linear and nonlinear initial-boundary value problem of singularly perturbed heat transfer equations.

For regularization of some inverse problems or unstable retrospective problems the well-known Lattes-Lions technique -- introduction in the linear equations the differential operators of higher order with small coefficients -- is used. In this case finite difference approximations in the space are obtainedusing method of lines with two small parameters $\epsilon$ in three ways:

1) the finite difference scheme (FDS method) with the second and the fourth orders of approximation in the uniform grid,

2) the difference scheme with exact spectrum (FDSES method) in the uniform grid,

3) the global approximation (GAN method) in nonuniform grid with grid points on the roots of the Chebyshev polynomials.

The nonlinear heat transfer equations with nonmonotonous heat conductivity without Lattes-Lions technique numerical solution is regularized with two methods: By introducing the differential operator of higher order with mixed derivatives in the special form $\epsilon \frac{\partial^5 u}{\partial ^4 x \partial t}$ andconstructing monotonous continuous functions for approximation the heat conductivity. \pagebreak

To investigate stability of bounded solutions for the continuous and discrete problems, solvability in the Sobolev space and to determine the parameters $\epsilon$ some theoretical estimations are obtained using the methodof logarithmic convexity and integral identity.

For regularization of some inverse problems or unstable retrospective problems the well-known Lattes-Lions technique -- introduction in the linear equations the differential operators of higher order with small coefficients -- is used. In this case finite difference approximations in the space are obtainedusing method of lines with two small parameters $\epsilon$ in three ways:

1) the finite difference scheme (FDS method) with the second and the fourth orders of approximation in the uniform grid,

2) the difference scheme with exact spectrum (FDSES method) in the uniform grid,

3) the global approximation (GAN method) in nonuniform grid with grid points on the roots of the Chebyshev polynomials.

The nonlinear heat transfer equations with nonmonotonous heat conductivity without Lattes-Lions technique numerical solution is regularized with two methods: By introducing the differential operator of higher order with mixed derivatives in the special form $\epsilon \frac{\partial^5 u}{\partial ^4 x \partial t}$ andconstructing monotonous continuous functions for approximation the heat conductivity. \pagebreak

To investigate stability of bounded solutions for the continuous and discrete problems, solvability in the Sobolev space and to determine the parameters $\epsilon$ some theoretical estimations are obtained using the methodof logarithmic convexity and integral identity.

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