Wavelet Solutions and Stability Analysis of Parabolic Equations

In this paper wavelet solutions of extended Sideways and non stan-dard parabolic equations have been analyzed along with stabilization and errorsestimation.DOI : http://dx.doi.org/10.22342/jims.16.1.32.69-8

Fourier regularization for a backward heat equation

AbstractIn this paper a simple and convenient new regularization method for solving backward heat equation—Fourier regularization method is given. Meanwhile, some quite sharp error estimates between the approximate solution and exact solution are provided. A numerical example also shows that the method works effectively

A source identification problem in a bi-parabolic equation: convergence rates and some optimal results

This paper is concerned with identification of a spatial source function from final time observation in a bi-parabolic equation, where the full source function is assumed to be a product of time dependent and a space dependent function. Due to the ill-posedness of the problem, recently some authors have employed different regularization method and analysed the convergence rates. But, to the best of our knowledge, the quasi-reversibility method is not explored yet, and thus we study that in this paper. As an important implication, the H{\"o}lder rates for the apriori and aposteriori error estimates obtained in this paper improve upon the rates obtained in earlier works. Also, in some cases we show that the rates obtained are of optimal order. Further, this work seems to be the first one that has broaden the applicability of the problem by allowing the time dependent component of the source function to change sign. To the best of our knowledge, the earlier known work assumed the fixed sign of the time dependent component by assuming some bounded below condition.Comment: Comments are welcome. Typos and some mistakes with sign in the PDE are rectified. Section 4 and 5 are majorly revise

Optimal error bound and Fourier regularization for identifying an unknown source in the heat equation

AbstractIn this paper we investigate a problem of the identification of an unknown source from one supplementary temperature measurement at a given instant of time for the transient heat equation. Under an a priori condition we answer the question concerning the best possible accuracy for the problem. The Fourier regularization method is utilized for solving the problem, and its convergent rate is analyzed. Numerical results are presented to illustrate the accuracy and efficiency of the method

Solution of inverse problem - regularization via thermodynamical criterion

In engineering practice, measuring temperature on both sides of a wall (of, for example, turbine casing or combustion chamber) is not always possible. On the other hand, measurement of both temperature and heat flux on the outer surface of the wall is possible. For transient heat conduction equation, measurements of temperature and heat flux supplemented by the initial condition state the Cauchy problem, which is ill-conditioned In this paper, the stable solution is obtained for the Cauchy problem using the Laplace transformation and the minimisation of continuity in the process of integration of convolution. Test examples confirm proposed algorithm for the inverse problem solution.Papers presented to the 12th International Conference on Heat Transfer, Fluid Mechanics and Thermodynamics, Costa de Sol, Spain on 11-13 July 2016

Determination of the time-dependent reaction coefficient and the heat flux in a nonlinear inverse heat conduction problem

Diffusion processes with reaction generated by a nonlinear source are commonly encountered in practical applications related to ignition, pyrolysis and polymerization. In such processes, determining the intensity of reaction in time is of crucial importance for control and monitoring purposes. Therefore, this paper is devoted to such an identification problem of determining the time-dependent coefficient of a nonlinear heat source together with the unknown heat flux at an inaccessible boundary of a one-dimensional slab from temperature measurements at two sensor locations in the context of nonlinear transient heat conduction. Local existence and uniqueness results for the inverse coefficient problem are proved when the first three derivatives of the nonlinear source term are Lipschitz continuous functions. Furthermore, the conjugate gradient method (CGM) for separately reconstructing the reaction coefficient and the heat flux is developed. The ill-posedness is overcome by using the discrepancy principle to stop the iteration procedure of CGM when the input data is contaminated with noise. Numerical results show that the inverse solutions are accurate and stable