A fast algorithm for solving the Cauchy problem for ODE using the Sobolev orthogonal polynomials generated by Chebyshev polynomials of the first kind

Purpose The purpose of this paper is to present the method for solving the inverse Cauchy-type problem for the Laplace’s equation. Calculations were made for the rectangular domain with the target temperature on three boundaries and, additionally, on one of the boundaries, the heat flux distribution was selected. The purpose of consideration was to determine the distribution of temperature on a section of the boundary of the investigated domain (boundary Γ1) and find proper method for the problem regularization. Design/methodology/approach The solution of the direct and the inverse (Cauchy-type) problems for the Laplace’s equation is presented in the paper. The form of the solution is noted as the linear combination of the Chebyshev polynomials. The collocation method in which collocation points had been determined based on the Chebyshev nodes was applied. To solve the Cauchy problem, the minimum of functional describing differences between the target and the calculated values of temperature and the heat flux on a section of the domain’s boundary was sought. Various types of the inverse problem regularization, based on Tikhonov and Tikhonov–Philips regularizations, were analysed. Regularization parameter α was chosen with the use of the Morozov discrepancy principle. Findings Calculations were performed for random disturbances to the heat flux density of up to 0.01, 0.02 and 0.05 of the target value. The quality of obtained results was next estimated by means of the norm. Effect of the disturbance to the heat flux density and the type of regularization on the sought temperature distribution on the boundary Γ1 was investigated. Presented methods of regularization are considerably less sensitive to disturbances to measurement data than to Tikhonov regularization. Practical implications Discussed in this paper is an example of solution of the Cauchy problem for the Laplace’s equation in the rectangular domain that can be applied for determination of the temperature distribution on the boundary of the heated element where it is impossible to measure temperature or the measurement is subject to a great error, for instance on the inner wall of the boiler. Authors investigated numerical examples for functions with singularities outside the domain, for which values of gradients change significantly within the calculation domain what corresponds to significant changes in temperature on the wall of the boiler during the fuel combustion. Originality/value In this paper, a new method for solving the Cauchy problem for the Laplace’s equation is described. To solve this problem, the Chebyshev polynomials and nodes were used. Various types of regularization of this problem were considered.

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Cauchy Problem for Equations with Fractional Differentiation Bessel Operator in the Space of Temperate Distributions

Nonlinear Analysis Modelling and Control ◽

10.15388/na.2003.8.1.15178 ◽

2003 ◽

Vol 8 (1) ◽

pp. 61-75

Author(s):

V. Litovchenko

Keyword(s):

Functional Analysis ◽

Cauchy Problem ◽

Fundamental Solution ◽

Initial Function ◽

Well Posedness ◽

Fractional Differentiation ◽

Basic Tool ◽

Conjugate Space ◽

Analysis Technique ◽

The Cauchy Problem

The well-posedness of the Cauchy problem, mentioned in title, is studied. The main result means that the solution of this problem is usual C∞ - function on the space argument, if the initial function is a real functional on the conjugate space to the space, containing the fundamental solution of the corresponding problem. The basic tool for the proof is the functional analysis technique.

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The constructive solution of the cauchy problem for linear systems of partial difference or differential equations with constant coefficients

2001 European Control Conference (ECC) ◽

10.23919/ecc.2001.7076158 ◽

2001 ◽

Author(s):

Ulrich Oberst ◽

Franz Pauer

Keyword(s):

Cauchy Problem ◽

Differential Equations ◽

Linear Systems ◽

Partial Difference ◽

Constructive Solution ◽

The Cauchy Problem ◽

Constant Coefficients

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