A Modified Method to Solve the One-Dimensional Heat Conduction Problem

The article focuses on the tasks of the mathematical physics – one-dimensional diffusion-convection boundary-value problem (BVP) for solving the heat conduction equation with piece-wise smooth coefficients in the multi-layer media. For this purpose the conservative averaging method (CAM) is using with special created integral splines of exponential type that interpolate the middle integral values of piece-wise smooth function through averaging in zdirection. Thus BVP is reduced to the system of ordinary differential equations (ODE) dependent on time – this enables to find out the averaged solutions of BVP – non-stationary and stationary.


Introduction
The numerical modelling of mathematical physics 1-D problems in layered medium using engineering-technical calculations of sufficient accuracy is important in numerous areas of the applied sciences.
Therefore we are studying the conservative averaging method (CAM) by using special integral exponential type splines with parameters in every layer, which means that the values of these parameters have to be chosen to decrease the error of approximation of the solution.
In the limit case when parameters tend to zero we have the integral parabolic type spline, developed by A. Buikis (Buikis, 1994a;Buikis,1994b).
CAM can be applied both to linear processes (Kalis, 2016) and non-linear processes (the dependency of mathematical model equation coefficients on the process characteristics, such as temperature in the combustion process) (Aboltins, 2017), (Weber, 2012).

Formulation of the problem
The non-stationary diffusion-convection problem is studied in 1-D domain . The domain  consists of N -layered medium.We will consider the non-stationary 1-D problem of the linear diffusion theory for layered piece-wise homogenous materials of one   is the height of layer We can find the distribution of concentrations -concentrations functions in every layer, 0 0 , , , , 0 , -layer domain is considered in (Kalis, 2016).

The conservative averaging method (CAM) in z-direction using integral spline with two fixed exponential type functions
Using CAM with respect to z with fixed parametrical functions For exponential functions we use following parameters: , where  The non-stationary solution of (3.2) can be represented in the following form: ( 2

The CAM in one layer
In one layer we have following problem Using averaged method with respect to z we have The functions are in the following form: , where We integrate the equation of the system (4.1) by variable z between 0 , L , and divide it by layer height L then we insert function (4.2) and use the system (4.1)boundary conditions thus obtaining an initial value problem (4.3) for the ODE: Then, the non-stationary averaged solution is      

Some numerical results
The results of calculations are obtained by MATLAB.We use the discrete values In the following Figs.1-3 there are represented the numerical and analytical (for stationary problem) results obtained by CAM using exponential type splines and "pdepe" for one and two layers.MATLAB routine "pdepe" solves nonlinear PDEs of the following form ( ) , ( t z u u  ) (WEB, a): The error of approximation for stationary solutions with exponential type spline is 10 -7 , with parabolic type spline (Buikis, 1994b) -0.212 (Figure1, b)), for non-stationary solution with exponential type splines -0.016 (Figure2, a)).For one layer the maximal error of approximation for non-stationary solution is 0.018 (Figure2, b)).Figure3.Stationary solution (analytical and generated by exponential spline) and non-stationary generated by exponential spline, t=t f : a) for two layers; b) for one layer

Conclusions
The 1-D non-stationary diffusion-convection problem in a layered domain applying the conservative averaging method (CAM) is reduced to initial value problem (IVP) of ODEs using the created integral exponential type splines with two different functions each of them contain the parameter.
The error of approximation using the splines depends on these parameters.It was established that, to obtain a minimal error of approximation, the parameters of spline function must be equal to characteristic values of the solution of homogenous ODEs for the above mentioned IVP.
The stationary problems are solved analytically but the solutions of corresponding averaged non-stationary initial-boundary-value problems are obtained numerically also applying MATLAB routine "pdepe".

F
concentration on the boundary for the boundary, f t -the final time, 0 i u -the given initial condition.It must be added, that in present paper a specific diffusion-convection process is investigated, for which the constancy of the source-function i is inherent.For 1  N the conditions on the contact line are deleted.Similarly 3-D initial-boundary problem in 1  N

Figure1.Figure2.
Figure1.Surface of solution generated by "pdepe" (a), stationary solution (analytical, generated by exponential and parabolic splines) (b) The parameters are also used by designing the stationary analytic solution of the above mentioned BVP. i).