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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.

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.

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.

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.

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.

To reduce the heat load of a gas turbine blade, its surface is covered with an outer layer of ceramics with high thermal resistance. The purpose of this paper is the selection of ceramics with such a low heat conduction coefficient and thickness, so that the permissible metal temperature is not exceeded on the metal-ceramics interface due to the loss ofmechanical properties.

Therefore, for given temperature changes over time on the metal-ceramics interface, temperature changes over time on the inner side of the blade and the assumed initial temperature, the temperature change over time on the outer surface of the ceramics should be determined. The problem presented in this way is a Cauchy type problem. When analyzing the problem, it is taken into account that thermophysical properties of metal and ceramics may depend on temperature. Due to the thin layer of ceramics in relation to the wall thickness, the problem is considered in the area in the flat layer. Thus, a one-dimensional non-stationary heat flow is considered.

The range of stability of the Cauchy problem as a function of time step, thickness of ceramics and thermophysical properties of metal and ceramics are examined. The numerical computations also involved the influence of disturbances in the temperature on metal-ceramics interface on the solution to the inverse problem.

The computational model can be used to analyze the heat flow in gas turbine blades with thermal barrier.

A number of inverse problems of the type considered in the paper are presented in the literature. Inverse problems, especially those Cauchy-type, are ill-conditioned numerically, which means that a small change in the inputs may result in significant errors of the solution. In such a case, regularization of the inverse problem is needed. However, the Cauchy problem presented in the paper does not require regularization.

In this paper, various regularization methods are numerically implemented using the boundary element method (BEM) in order to solve the Cauchy steady‐state heat conduction problem in an anisotropic medium. The convergence and the stability of the numerical methods are investigated and compared. The numerical results obtained confirm that stable numerical results can be obtained by various regularization methods, but if high accuracy is required for the temperature, or if the heat flux is also required, then care must be taken when choosing the regularization method since the numerical results are substantially improved by choosing the appropriate method.

In this paper, the Cauchy-type problem for the Laplace equation was solved in the rectangular domain with the use of the Chebyshev polynomials. The purpose of this paper is to present an optimal choice of the regularization parameter for the inverse problem, which allows determining the stable distribution of temperature on one of the boundaries of the rectangle domain with the required accuracy.

The Cauchy-type problem is ill-posed numerically, therefore, it has been regularized with the use of the modified Tikhonov and Tikhonov–Philips regularization. The influence of the regularization parameter choice on the solution was investigated. To choose the regularization parameter, the Morozov principle, the minimum of energy integral criterion and the L-curve method were applied.

Numerical examples for the function with singularities outside the domain were solved in this paper. The values of results change significantly within the calculation domain. Next, results of the sought temperature distributions, obtained with the use of different methods of choosing the regularization parameter, were compared. Methods of choosing the regularization parameter were evaluated by the norm N_max.

Calculation model described in this paper can be applied to determine temperature distribution on the boundary of the heated wall of, for instance, a boiler or a body of the turbine, that is, everywhere the temperature measurement is impossible to be performed on a part of the boundary.

The paper presents a new method for solving the inverse Cauchy problem with the use of the Chebyshev polynomials. The choice of the regularization parameter was analyzed to obtain a solution with the lowest possible sensitivity to input data disturbances.

The purpose of this paper is to develop a mesh-free algorithm based on the least square support vector machines method for numerical simulation of the modified Helmholtz equations.

The proposed method deals with a Cauchy problem for the modified Helmholtz equations. The algorithm converts the problem into a quadratic programming. It can be divided into three steps. First, some training points are allocated. Then, an approximate function is constructed. Finally, the shape parameters are estimated.

The proposed method's stability is discussed. Numerical experiments are conducted to check the efficiency of the algorithm. The proposed method is found to feasible for the ill-posed problems of the modified Helmholtz equations.

The originality lies in that the proposed method is applied to solve the modified Helmholtz equations for the first time, and the expected results are obtained.

The purpose of this paper is to investigate the L^p‐maximal regularity for the abstract incomplete second order problem.

First, the paper gives the definition of the L^p‐maximal regularity for incomplete second‐order Cauchy problems and lists their basic properties based on Chill and Srivastava's recent work for completing second order problem. Second, the paper establishes its characterization by means of Fourier multiplier and the operator‐sum theorem. Finally, it considers an application to quasilinear systems by the regularity and linearization techniques.

Two criteria of L^p‐maximal regularity are obtained, and the existence of the local solution for the second order quasilinear problem is given. In addition, the connection on maximal regularity between second order problems with initial values and that with periodic problems is investigated. A perturbation result is given.

The maximal regularity is an important tool in the theory of non‐linear differential equations. The results obtained in this paper are universal because the operator is not necessarily the generator of a cosine operator function. Using this unifying approach it is possible to clarify the L^p‐maximal regularity and the existence of the solution for some systems described by partial differential equations, such as wave equations.

The purpose of this paper is to introduce a new homotopy perturbation method (NHPM) to solve Cauchy problem of unidimensional non‐linear diffusion equation.

In this paper a modified version of HPM, which the authors call NHPM, has been presented; this technique performs much better than the HPM. HPM and NHPM start by considering a homotopy, and the solution of the problem under study is assumed to be as the summation of a power series in p, the difference between two methods starts from the form of initial approximation of the solution.

In this article, the authors have applied the NHPM for solving nonlinear Cauchy diffusion equation. In comparison with the homotopy perturbation method (HPM), in the present method, the authors achieve exact solutions while HPM does not lead to exact solutions. The authors believe that the new method is a promising technique in finding the exact solutions for a wide variety of mathematical problems.

The basic idea described in this paper is expected to be further employed to solve other functional equations.

The purpose of this paper is to introduce a reverted way to design electrical machines. The authors present a work flow that systematically yields electrical machine geometries from given air gap fields.

The solution process exploits the inverse Cauchy problem. The desired air gap field is inserted to this as the Cauchy data, and the solution process is stabilized with the aid of linear algebra.

The results are verified by solving backwards the air gap fields in the standard way. They match well with the air gap fields inserted as an input to the system.

The paper reverts the standard design work flow of electrical motor by solving directly for a geometry that yields the desired air gap field. In addition, a stabilization strategy for the underlying Cauchy problem is introduced.

To propose and investigate a stable numerical procedure for the reconstruction of the velocity of a viscous incompressible fluid flow in linear hydrodynamics from knowledge of the velocity and fluid stress force given on a part of the boundary of a bounded domain.

Earlier works have involved the similar problem but for stationary case (time‐independent fluid flow). Extending these ideas a procedure is proposed and investigated also for the time‐dependent case.

The paper finds a novel variation method for the Cauchy problem. It proves convergence and also proposes a new boundary element method.

The fluid flow domain is limited to annular domains; this restriction can be removed undertaking analyses in appropriate weighted spaces to incorporate singularities that can occur on general bounded domains. Future work involves numerical investigations and also to consider Oseen type flow. A challenging problem is to consider non‐linear Navier‐Stokes equation.

Fluid flow problems where data are known only on a part of the boundary occur in a range of engineering situations such as colloidal suspension and swimming of microorganisms. For example, the solution domain can be the region between to spheres where only the outer sphere is accessible for measurements.

A novel variational method for the Cauchy problem is proposed which preserves the unsteady Stokes operator, convergence is proved and using recent for the fundamental solution for unsteady Stokes system, a new boundary element method for this system is also proposed.

In this paper, we use the decomposition method for solving the Cauchy problem without using the canonical form of Adomian. We also give proof of convergence by using a new formulation of the Adomain polynomials and we compare our technique with the Picard method.