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The purpose of this study is to show the procedure, application and main features of the hybrid numerical-analytical approach known as generalized integral transform technique by using it to study magnetohydrodynamic flow of electrically conductive Newtonian fluids inside flat parallel-plate channels subjected to a uniform and constant external magnetic field.

The mathematical formulation of the analyzed problem is given in terms of a streamfunction, obtained from the Navier–Stokes and energy equations, by considering steady state laminar and incompressible flow and constant physical properties.

Convergence analyses are performed and presented to illustrate the consistency of the integral transformation technique. The results for the velocity and temperature fields are generated and compared with those in the literature as a function of the main governing parameters.

A detailed analysis of the parametric sensibility of the main dimensionless parameters, such as the Reynolds number, Hartmann number, Eckert number, Prandtl number and electrical parameter, for some typical situations is performed.

The purpose of this work is to revisit the integral transform solution of transient natural convection in differentially heated cavities considering a novel vector eigenfunction expansion for handling the Navier-Stokes equations on the primitive variables formulation.

The proposed expansion base automatically satisfies the continuity equation and, upon integral transformation, eliminates the pressure field and reduces the momentum conservation equations to a single set of ordinary differential equations for the transformed time-variable potentials. The resulting eigenvalue problem for the velocity field expansion is readily solved by the integral transform method itself, while a traditional Sturm–Liouville base is chosen for expanding the temperature field. The coupled transformed initial value problem is numerically solved with a well-established solver based on a backward differentiation scheme.

A thorough convergence analysis is undertaken, in terms of truncation orders of the expansions for the vector eigenfunction and for the velocity and temperature fields. Finally, numerical results for selected quantities are critically compared to available benchmarks in both steady and transient states, and the overall physical behavior of the transient solution is examined for further verification.

A novel vector eigenfunction expansion is proposed for the integral transform solution of the Navier–Stokes equations in transient regime. The new physically inspired eigenvalue problem with the associated integmaral transformation fully shares the advantages of the previously obtained integral transform solutions based on the streamfunction-only formulation of the Navier–Stokes equations, while offering a direct and formal extension to three-dimensional flows.

This paper aims to present a meshless technique to find the Green’s functions for solutions of Laplacian boundary value problems on rectangular domains. This paper also investigates a theoretical basis for the Steklov series expansion methods to reduce and estimate the error of numerical approaches for the boundary correction kernel of the Laplace operator.

The main interest is how the Green's functions differ from the fundamental solution of the Laplace operator. Steklov expansion methods for finding the correction term are supported by the analysis that bases of the class of all finite harmonic functions can be formed using harmonic Steklov eigenfunctions. These functions construct a basis of the space of solutions of harmonic boundary value problems and their boundary traces generate an orthogonal basis of the trace space of solutions on the boundary.

The main conclusion is that the boundary correction term for the Green's functions is well-approximated by Steklov expansions with a few Steklov eigenfunctions. The error estimates for the Steklov approximations of the boundary correction term involved in Dirichlet or Robin boundary value problems are found. They appear to provide very good approximations in the interior of the region and become quite oscillatory close to the boundary.

This paper concentrates to document the first attempt to find the Green's function for various harmonic boundary value problems with the explicit Steklov eigenfunctions without concerns regarding discretizations when the region is a rectangle.

The purpose of this paper is to propose the generalized integral transform technique (GITT) to the solution of convection-diffusion problems with nonlinear boundary conditions by employing the corresponding nonlinear eigenvalue problem in the construction of the expansion basis.

The original nonlinear boundary condition coefficients in the problem formulation are all incorporated into the adopted eigenvalue problem, which may be itself integral transformed through a representative linear auxiliary problem, yielding a nonlinear algebraic eigenvalue problem for the associated eigenvalues and eigenvectors, to be solved along with the transformed ordinary differential system. The nonlinear eigenvalues computation may also be accomplished by rewriting the corresponding transcendental equation as an ordinary differential system for the eigenvalues, which is then simultaneously solved with the transformed potentials.

An application on one-dimensional transient diffusion with nonlinear boundary condition coefficients is selected for illustrating some important computational aspects and the convergence behavior of the proposed eigenfunction expansions. For comparison purposes, an alternative solution with a linear eigenvalue problem basis is also presented and implemented.

This novel approach can be further extended to various classes of nonlinear convection-diffusion problems, either already solved by the GITT with a linear coefficients basis, or new challenging applications with more involved nonlinearities.

Different possibilities for the enhancement of convergence rates in eigenfunction expansions are investigated in the realm of integral transform solutions for partial differential equations. A representative parabolic problem is chosen to illustrate two schemes and their combinations; a filtering technique and an integral balance approach. Numerical results are presented to confirm the relative merits in each proposed procedure.

The purpose of this paper is to provide an analysis of two‐dimensional laminar flow in the entrance region of wavy wall ducts as obtained from the solution of the steady Navier‐Stokes equations for incompressible flow.

The study is undertaken by application of the generalized integral transform technique in the solution of the steady Navier‐Stokes equations for incompressible flow. The streamfunction‐only formulation is adopted, and a general filtering solution that adapts to the irregular contour is proposed to enhance the convergence behavior of the eigenfunction expansion.

A few representative cases are considered more closely in order to report some numerical results illustrating the eigenfunction expansions convergence behavior. The product friction factor‐Reynolds number is also computed and compared against results from discrete methods available in the literature for different Reynolds numbers and amplitudes of the wavy channel.

The proposed methodology is fairly general in the analysis of different channel profiles, though the reported results are limited to the wavy channel configuration. Future work should also extend the analysis to geometries represented in the cylindrical coordinates with longitudinally variable radius.

The error‐controlled converged results provide reliable benchmark results for the validation of numerical results from computational codes that address the solution of the Navier‐Stokes equations in irregular geometries.

Although the hybrid methodology is already known in the literature, the results here presented are original and further challenges application of the integral transform method in the solution of the Navier‐Stokes equations.

This paper seeks to analyze transient convection‐diffusion by employing the generalized integral transform technique (GITT) combined with an arbitrary transient filtering solution, aimed at enhancing the convergence behavior of the associated eigenfunction expansions. The idea is to consider analytical approximations of the original problem as filtering solutions, defined within specific ranges of the time variable, which act diminishing the importance of the source terms in the original formulation and yielding a filtered problem for which the integral transformation procedure results in faster converging eigenfunction expansions. An analytical local instantaneous filtering is then more closely considered to offer a hybrid numerical‐analytical solution scheme for linear or nonlinear convection‐diffusion problems.

The approach is illustrated for a test‐case related to transient laminar convection within a parallel‐plates channel with axial diffusion effects.

The developing thermal problem is solved for the fully developed flow situation and a step change in inlet temperature. An analysis is performed on the variation of Peclet number, so as to investigate the importance of the axial heat or mass diffusion on convergence rates.

This paper succeeds in analyzing transient convection‐diffusion via GITT, combined with an arbitrary transient filtering solution, aimed at enhancing the convergence behaviour of the associated eigenfunction expansions.

Analytical or hybrid numerical‐analytical solutions of multidimensional diffusion problems involve the evaluation of nested multiple infinite summations, which require the computation of eigenvalues and related quantities, from associated auxiliary eigenvalue‐type problems. A substantial reduction of the total computational effort in the construction of the final solution for the original potential can be achieved through the proper reorganization of the multiple summations into a single series representation. Such reordering of terms should be carefully accomplished, in order to account for the most significant contributions to the final numerical result, up to a truncated finite order that meets the user prescribed tolerance for the relative error. Presents an algorithm for an optimized scheme with consequent reduction on the number of eigenquantities to be evaluated. This approach is illustrated through representative two and three‐dimensional transient heat conduction problems.

The integral transform method is employed for the hybrid numerical‐analytical solution of two‐dimensional, steady‐state heat conduction within extended surfaces of variable longitudinal profile and temperature dependent thermal conductivity. Numerical results are then obtainable with automatic accuracy, allowing for the establishment of benchmark results and for the validation of approximate solutions. Convergence rates are illustrated for longitudinal fins with trapezoidal and parabolic profiles, and for different values of the governing parameters, Biot number and aspect ratio. In addition, the classical one‐dimensional approximate solutions are critically examined for these typical non‐straight profiles, and the applicability limits are investigated.

The purpose of this study is to propose the generalised integral transform technique to investigate the natural convection behaviour in a vertical cylinder under different boundary conditions, adiabatic and isothermal walls and various aspect ratios.

GITT was used to investigate the steady-state natural convection behaviour in a vertical cylinder with internal uniformed heat generation. The governing equations of natural convection were transferred to a set of ordinary differential equations by using the GITT methodology. The coefficients of the ODEs were determined by the integration of the eigenfunction of the auxiliary eigenvalue problems in the present natural convection problem. The ordinary differential equations were solved numerically by using the DBVPFD subroutine from the IMSL numerical library. The convergence was achieved reasonably by using low truncation orders.

GITT is a powerful computational tool to explain the convection phenomena in the cylindrical cavity. The convergence analysis shows that the hybrid analytical–numerical technique (GITT) has a good convergence performance in relatively low truncation orders in the stream-function and temperature fields. The effect of the Rayleigh number and aspect ratio on the natural convection behaviour under adiabatic and isothermal boundary conditions has been discussed in detail.

The present hybrid analytical–numerical methodology can be extended to solve various convection problems with more involved nonlinearities. It exhibits potential application to solve the convection problem in the nuclear, oil and gas industries.