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A method has been developed for the solution of inverse heat diffusion problems to find the initial condition, boundary condition, and the source and sink function in the heat diffusion equation. The method has been used in the development of a source‐and‐sink method to find the boundary conditions in inverse Stefan problems. Green's functions have been used in the solution, and the problems are solved by using two approaches: a series solution approach, and a time incremental approach. Both can be used to find the boundary conditions without reliance on the flux information to be supplied at both sides of the interface. The methods are efficient in that they require less equations to be solved for the conditions. The numerical results have shown to be accurate, convergent, and stable. Most of all, the results do not degrade with time as in other time marching schemes reported in the literature. Algorithms can also be easily developed for the solution of the conditions.

The purpose of this paper is concerned with a reliable treatment of the classical Stephan problem. The Adomian decomposition method (ADM) is used to carry out the analysis, Moreover, the authors extend the work to examine the Stefan problem with variable latent heat. The study confirms the accuracy and efficiency of the employed method.

The new technique, as presented in this paper in extending the applicability of the ADM, has been shown to be very efficient for solving the Stefan problem.

The Stefan problem with variable latent heat was examined as well. The ADM was effectively used for analytic treatment of the Stefan problem with and without variable latent heat.

The paper presents a new solution algorithm for the Stefan problem.

The purpose of this paper is to provide an insight and to solve numerically the identification of timewise terms and free boundaries coefficient appearing in the heat equation from over-determination conditions.

The formulated coefficient identification problem is inverse and ill-posed, and therefore, to obtain a stable solution, a nonlinear Tikhonov regularization least-squares approach is used. For the numerical discretization, the finite difference method combined with a regularized nonlinear minimization is performed using the MATLAB subroutine lsqnonlin.

The numerical results presented for two examples show the efficiency of the computational method and the accuracy and stability of the numerical solution even in the presence of noise in the input data.

The mathematical formulation is restricted to identify coefficients in unknown components dependent on time, and this may be considered as a research limitation. However, there is no research implication to overcome this, as the known input data is also limited to single temperature in heat equation with Stefan conditions, and the first- and second-order heat moments measurements at a particular time location.

As noisy data are inverted, the study models real situations in which practical measurements are inherently contaminated with noise.

The identification of the timewise terms and free boundaries will be of great interest in the heat transfer community and related fluid flow applications.

The current investigation advances previous studies, which assumed that the coefficient multiplying the lower order temperature term depends on time. The knowledge of this physical property coefficient is very important in heat transfer and fluid flow. The originality lies in the insight gained by performing for the numerical simulations of inversion to find the timewise terms and free boundaries coefficient dependent on time in the heat equation from noisy measurements.

The purpose of the study is to solve numerically the inverse problem of determining the time-dependent convection coefficient and the free boundary, along with the temperature in the two-dimensional convection-diffusion equation with initial and boundary conditions supplemented by non-local integral observations. From the literature, there is already known that this inverse problem has a unique solution. However, the problem is still ill-posed by being unstable to noise in the input data.

For the numerical discretization, this paper applies the alternating direction explicit finite-difference method along with the Tikhonov regularization to find a stable and accurate numerical solution. The resulting nonlinear minimization problem is solved computationally using the MATLAB routine lsqnonlin. Both exact and numerically simulated noisy input data are inverted.

The numerical results demonstrate that accurate and stable solutions are obtained.

The inverse problem presented in this paper was already showed to be locally uniquely solvable, but no numerical solution has been realized so far; hence, the main originality of this work is to attempt this task.

This paper aims to propose a semi-analytical benchmarking framework for enthalpy-based methods used in problems involving phase change with latent heat. The benchmark is based on a class of semi-analytical solutions of spatially symmetric Stefan problems in an arbitrary spatial dimension. Via a public repository this study provides a finite element numerical code based on the FEniCS computational platform, which can be used to test and compare any method of choice with the (semi-)analytical solutions. As a particular demonstration, this paper uses the benchmark to test several standard temperature-based implementations of the enthalpy method and assesses their accuracy and stability with respect to the discretization parameters.

The class of spatially symmetric semi-analytical self-similar solutions to the Stefan problem is found for an arbitrary spatial dimension, connecting some of the known results in a unified manner, while providing the solutions’ existence and uniqueness. For two chosen standard semi-implicit temperature-based enthalpy methods, the numerical error assessment of the implementations is carried out in the finite element formulation of the problem. This paper compares the numerical approximations to the semi-analytical solutions and analyzes the influence of discretization parameters, as well as their interdependence. This study also compares accuracy of these methods with other traditional approach based on time-explicit treatment of the effective heat capacity with and without iterative correction.

This study shows that the quantitative comparison between the semi-analytical and numerical solutions of the symmetric Stefan problems can serve as a robust tool for identifying the optimal values of discretization parameters, both in terms of accuracy and stability. Moreover, this study concludes that, from the performance point of view, both of the semi-implicit implementations studied are equivalent, for optimal choice of discretization parameters, they outperform the effective heat capacity method with iterative correction in terms of accuracy, but, by contrast, they lose stability for subcritical thickness of the mushy region.

The proposed benchmark provides a versatile, accessible test bed for computational methods approximating multidimensional phase change problems. The supplemented numerical code can be directly used to test any method of choice against the semi-analytical solutions.

While the solutions of the symmetric Stefan problems for individual spatial dimensions can be found scattered across the literature, the unifying perspective on their derivation presented here has, to the best of the authors’ knowledge, been missing. The unified formulation in a general dimension can be used for the systematic construction of well-posed, reliable and genuinely multidimensional benchmark experiments.

A method has been developed for conjugate heat transfer analysis of fluid flow inside parallel channels formed by a phase change material (PCM) separated from the fluid by a wall. The phase change in the PCM is two dimensional and a hybrid analysis consisting of an analytical solution in one direction and a finite‐difference method in another direction is used to solve for the temperature in the PCM. The heat transfer fluid (HTF) inlet temperature is given and the heat transfer between the HTF and the PCM is treated as a conjugate problem that requires no iterations to obtain a solution. The numerical results are found to be stable, convergent, and accurate. Application of the method to the solution of heat extraction from a phase‐change energy storage unit is given in detail and the numerical results are shown to be accurate, based on an energy conservation analysis, to within 3%.

This paper discusses an algorithm for phase change front identification in continuous casting. The problem is formulated as an inverse geometry problem, and the solution procedure utilizes temperature measurements inside the solid phase and sensitivity coefficients. The proposed algorithms make use of the boundary element method, with cubic boundary elements and Bezier splines employed for modelling the interface between the solid and liquid phases. A case study of continuous casting of copper is solved to demonstrate the main features of the proposed algorithms.

The purpose of this paper is to introduce a new modified version of the homotopy perturbation method (NMHPM) and Adomian decomposition method (ADM) for solving the nonlinear ordinary differential equation arising in MHD non-Newtonian fluid flow over a linear stretching sheet.

The governing equation is solved analytically by applying a newly developed optimal homotopy perturbation approach and ADM. This optimal approach contains convergence-control parameter and is computationally rather efficient. The results of numerical example are presented and only a few terms are required to obtain accurate solutions.

A new modified optimal and ADM methods accelerate the rapid convergence of the series solution. These methods dramatically reduce the size of work. The obtained series solution is combined with the diagonal Padé approximants to handle the boundary condition at infinity. Results derived from these methods are shown graphically and in tabulated forms to study the efficiency and accuracy.

Non-Newtonian flow processes play a key role in many types of polymer engineering operations. The formulation of mathematical model for these processes can be based on the equations of non-Newtonian fluid mechanics. The flow of an electrically conducting fluid in the presence of a magnetic field is of importance in various areas of technology and engineering such as MHD power generation, MHD flow meters, MHD pumps, etc. It is generally admitted that a number of astronomical bodies (e.g. the sun, Earth, Jupiter, Magnetic stars, Pulsars) posses fluid interiors and (or least surface) magnetic fields.

The present results are original and new for the MHD non-Newtonian fluid flow over a linear stretching sheet. The results attained in this paper confirm the idea that NMHPM and ADM are powerful mathematical tools and that can be applied to a large class of linear and nonlinear problems arising in different fields of science and engineering.

This paper aims to suggest a novel modified Laplace decomposition method (MLDM) for MHD flow over a non-linear stretching sheet with slip condition by suitable choice of an initial solution.

The governing partial differential equations are converted into dimensionless non-linear ordinary differential equation by similarity transformation, which is solved by MLDM. The method is based on the application of Laplace transform to boundary layers in fluid mechanics. The non-linear term can be easily handled by the use of He's polynomials.

The series solution of the MHD flow of an incompressible viscous fluid over a non-linear stretching sheet subject to slip condition is obtained. An excellent agreement between the MLDM and HPM is achieved. Convergence of the obtained series solution is properly checked by using the ratio test.

Stretching surface is an important type of flow occurring in a number of engineering processes such as heat-treated materials travelling between a feed roll and a wind up roll, aerodynamic extrusion of plastic sheets, glass fiber and paper production, cooling of an infinite metallic plate in a cooling path, manufacturing of polymeric sheets are few examples of flow due to stretching surfaces. This work provides a very useful source of information for researchers on this subject.

Such flow analysis is even not available yet for the hydrodynamic fluid. The series solution for MHD boundary layer problem with slip condition by means of MLDM is yet not available in the literature.