Stable method for solving the Cauchy problem with the use of Chebyshev polynomials

Abstract

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.

Blood flow of MHD non-Newtonian nanofluid with heat transfer and slip effects

“…The domain in the η -direction is [0, η ∞ ], where η ∞ is the end of the boundary layer is transformed into the computational domain [–1, 1], then the system of non-linear equations (13) to (20) can be solved by applying the CPS method usefully used by Elgazery (2008), Elshehawey et al (2004), Elbarbary and Elgazery (2004a, 2004b) and Joachimiak et al (2019). In view of CPS equations and the boundary conditions, we have the following system of equations: …”

Choice of the regularization parameter for the Cauchy problem for the Laplace equation

“…Solving the direct problem, where the temperature on boundaries Γ 1 , Γ 2 , Γ 3 and Γ 4 was known, was reduced to solving the matrix equation what was described in detail in the paper (Joachimiak et al , 2019a). Based on the solution to the direct problem, constants a k [Equation (12)] are of the following equation (14): where Hh,k=∑j=2m−1trueA˜k,jnWh−1true(yjtrue), while A ̃ k,jn ( k = 1, 2, …, mn ; j = 2, 3, …, m − 1) are elements of the matrix A −1 .…”

“…There are many methods used to regularize inverse problems. Among them, there is the Tikhonov regularization (Beck and Woodbury, 2016; Chen et al , 2019; Djerrar et al , 2017; Frąckowiak et al , 2019a; Laneev, 2018; Marin, 2010, 2016; Niu et al , 2014; Sun, 2016; Tikhonov and Arsenin, 1977; Yaparova, 2016), the Tikhonov–Philips regularization (Joachimiak et al , 2019a), the discrete Fourier transform (Frąckowiak and Ciałkowski, 2018; Wróblewska et al , 2015) and SVD algorithm (Hasanov and Mukanova, 2015). In her article, Cheruvu (2017) applied the wavelet regularization of Laplace’s equation in the arbitrarily shaped domain.…”

“…In many cases, the regularization of the inverse problem concerns the problem of choosing the regularization parameter. The regularization parameter can be chosen based on the Morozov principle (Chen et al , 2019; Han et al , 2011; Joachimiak et al , 2019a; Marin, 2016; Morozov, 1984; Sun, 2016) or using the L-curve method (Jin and Zheng, 2006; Marin and Munteanu, 2010; Marin, 2005). In the study of Marin (2011), the optimal regularization parameter was sought based on the generalized cross-validation criterion.…”

“…In this article, the solution to the Cauchy problem for the Laplace’s equation was investigated with the use of the Chebyshev polynomials. To regularize the solution to the inverse problem, the modification of the Tikhonov and of the Tikhonov–Philips regularizations, described in the article (Joachimiak et al , 2019a) was applied. The choice of the regularization parameter was made based on the Morozov principle, the minimum energy integral criterion and the L-curve method.…”