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This meshless collocation method is applicable not only to the Helmholtz equation with Dirichlet boundary condition but also mixed boundary conditions. It can calculate not only the high wavenumber problems, but also the variable wave number problems.

In this paper, the authors developed a meshless collocation method by using barycentric Lagrange interpolation basis function based on the Chebyshev nodes to deduce the scheme for solving the three-dimensional Helmholtz equation. First, the spatial variables and their partial derivatives are treated by interpolation basis functions, and the collocation method is established for solving second order differential equations. Then the differential matrix is employed to simplify the differential equations which is on a given test node. Finally, numerical experiments show the accuracy and effectiveness of the proposed method.

The numerical experiments show the advantages of the present method, such as less number of collocation nodes needed, shorter calculation time, higher precision, smaller error and higher efficiency. What is more, the numerical solutions agree well with the exact solutions.

Compared with finite element method, finite difference method and other traditional numerical methods based on grid solution, meshless method can reduce or eliminate the dependence on grid and make the numerical implementation more flexible.

The Helmholtz equation has a wide application background in many fields, such as physics, mechanics, engineering and so on.

This meshless method is first time applied for solving the 3D Helmholtz equation. What is more the present work not only gives the relationship of interpolation nodes but also the test nodes.

This purpose of this paper is to find the approximate analytical solutions of a multidimensional partial differential equation such as Helmholtz equation with space fractional derivatives α,β,γ (1<α,β,γ≤2). The fractional derivatives are described in the Caputo sense.

By using initial values, the explicit solutions of the equation are solved with powerful mathematical tools such as He's homotopy perturbation method (HPM).

This result reveals that the HPM demonstrates the effectiveness, validity, potentiality and reliability of the method in reality and gives the exact solution.

The most important part of this method is to introduce a homotopy parameter (p), which takes values from [0,1]. When p=0, the equation usually reduces to a sufficiently initial form, which normally admits a rather simple solution. When p→1, the system goes through a sequence of deformations, the solution for each of which is close to that at the previous stage of deformation. Here, we also discuss the approximate analytical solution of multidimensional fractional Helmholtz equation.

The purpose of this paper is to demonstrate how Monte Carlo methods can be applied to the solution of field theory problems.

This objective is achieved by building insight from Laplacian field problems. The point solution of a Laplacian field problem can be viewed as the solid angle average of the Dirichlet potentials from that point. Alternatively it can be viewed as the average of the termination potential of a number of random walks. Poisson and Helmholtz equations add the complexity of collecting a number of packets along this walk, and noting the termination of a random walk at a Dirichlet boundary.

When approached as a Monte Carlo problem, Poisson type problems can be interpreted as collecting and summing source packets representative of current or charge. Helmholtz problems involve the multiplication of packets of information modified by a multiplier reflecting the conductivity of the medium.

This method naturally lends itself to parallel processing computers.

This is the first paper to explore random walk solutions for all classes of eddy current problems, including those involving velocity. In problems involving velocity, the random walk direction enters depending on the walk direction with respect to the local velocity.

The purpose of this paper is to describe the development of an electroosmotic dynamic model to simulate the transport phenomena in association with the electric therapy in modern medicine.

The present study builds a new model by employing SUPG finite element method to solve the electroosmotic transport equation in microchannels of human body.

The present electroosmotic finite element analysis demonstrated that the electric treatment has a better curative effect.

The governing electric field equations for tissue fluids in microchannel include the Laplace equation for the effective electrical potential and the Helmholtz equation for the electrical potential established in the electric double layer (EDL). The transport equations governing the hydrodynamic field variables include the mass conservation equation for the electrolyte and the equations of motion for the incompressible charged fluids subject to an electroosmotic body force.

The phenomena of microchannels are dominated by elliptic equations, Laplace, Helmholtz and diffusion equations (Navier Stokes equations at Re=0.0259). These governing equations explain why the reaction of electric treatment is very fast, even immediate.

The analysis of the coupled hydrodynamic and electrical fields, the externally applied electric potential has been shown to be an aid to accelerate the tissue fluid due to the formation of an EDL. Interaction of plasma and tissue fluids in human body is also revealed.

To develop a numerical technique for solving the inverse source problem associated with the constant coefficients convection‐diffusion equation.

The proposed numerical technique is based on the boundary element method (BEM) combined with an iterative sequential quadratic programming (SQP) procedure. The governing convection‐diffusion equation is transformed into a Helmholtz equation and the ill‐conditioned system of equations that arises after the application of the BEM is solved using an iterative technique.

The iterative BEM presented in this paper is well‐suited for solving inverse source problems for convection‐diffusion equations with constant coefficients. Accurate and stable numerical solutions were obtained for cases when the number of sources is correctly estimated, overestimated, or underestimated, and with both exact and noisy input data.

The proposed numerical method is limited to cases when the Péclet number is smaller than 100. Future approaches should include the application of the BEM directly to the convection‐diffusion equation.

Applications of the results presented in this paper can be of value in practical applications in both heat and fluid flow as they show that locations and strengths for an unknown number of point sources can be accurately found by using boundary measurements only.

The BEM has not as yet been employed for solving inverse source problems related with the convection‐diffusion equation. This study is intended to approach this problem by combining the BEM formulation with an iterative technique based on the SQP method. In this way, the many advantages of the BEM can be applied to inverse source convection‐diffusion problems.

This paper aims to describe a numerical procedure for approximating the potential distribution for a harmonic current point source, which is either buried in horizontally stratified multilayer earth, or positioned in the air. The procedure is very efficient and general. The total number of layers and the source position in relation to the medium model layers are completely arbitrary.

The efficiency of the computation procedure is based on the successful application of the numerical approximation of two kernel functions of the integral expression for the potential distribution within an arbitrarily chosen layer of the medium model. Each kernel function of the observed layer is approximated using a linear combination of 15 real exponential functions. Using these approximations and the analytical integration based on the Weber integral, a simple expression for numerical approximation of potential distribution within boundaries of the observed medium layer is given. Potential retardation is taken into account approximately.

The numerical procedure developed for the approximation of potential distribution for a harmonic current point source, which is positioned arbitrarily in air or in horizontally stratified multilayer earth, is efficient, numerically stable and generally applicable.

Numerical model developed for the harmonic current point source is the basis of a wider numerical models for computation of the harmonic and transient fields of earthing system, which consists of earthing grids buried in horizontally stratified multilayer earth and metallic structures in the air.

This is efficient and numerically stable frequency dependent harmonic current point source model. Potential retardation, which has been neglected at the first step of the approximation, is subsequently added to the potential expression in such a way that the Helmholtz differential equation has been approximately solved without introducing the Sommerfeld integrals.

This paper aims to present a high-order algorithm implemented with the modal spectral element method and simulations of three-dimensional thermal convective flows by using the full viscous dissipation function in the energy equation. Three benchmark problems were solved to validate the algorithm with exact or theoretical solutions. The heated rotating sphere at different temperatures inside a cold planar Poiseuille flow was simulated parametrically at varied angular velocities with positive and negative rotations.

The fourth-order stiffly stable schemes were implemented and tested for time integration. To provide the hp-refinement and spatial resolution enhancement, a modal spectral element method using hierarchical basis functions was used to solve governing equations in a three-dimensional space.

It was found that the direction of rotation of the heated sphere has totally different effects on drag, lateral force and torque evaluated on surfaces of the sphere and walls. It was further concluded that the angular velocity of the heated sphere has more influence on the wall normal velocity gradient than on the wall normal temperature gradients and therefore, more influence on the viscous dissipation than on the thermal dissipation.

This paper concerns incompressible fluid flow at constant properties with up to medium temperature variations in the absence of thermal radiation and ignoring the pressure work.

This paper contributes a viable high-order algorithm in time and space for modeling convective heat transfer involving an internal heated rotating sphere with the effect of viscous heating.

Results of this paper could provide reference for related topics such as enhanced heat transfer forced convection involving rotating spheres and viscous thermal effect.

The merits include resolving viscous dissipation and thermal diffusion in stationary and rotating boundary layers with both h- and p-type refinements, visualizing the viscous heating effect with the full viscous dissipation function in the energy equation and modeling the forced advection around a rotating sphere with varied positive and negative angular velocities subject to a shear flow.

The purpose of this paper is to provide an analytical solution to the plane eddy current problem inside conductive bars with rectangular cross‐section.

Eddy currents inside conductive materials have been investigated for a very long time, using measurements and mathematical modelling. This paper provides an analytical solution to the plane eddy current problem inside conductive bars with rectangular cross‐section.

The paper's approach solves the given plane eddy current problem with the boundary integral equation method. The Helmholtz' equation for vector potential inside the rectangle is solved by separation. The solution is inserted into the remaining boundary integral equation for the exterior vector potential in the domain surrounding the conductor yielding a system of linear equations. Results match existing solutions.

The method discussed provides a new way to solve the EC problem and is slightly faster than the available commercial codes; yet, it is limited to rectangular bars of cross‐section.

This study aims to understand the difference between irreversibility in heat and work transfer processes. It also aims to explain that Helmholtz or Gibbs energy does not represent “free” energy but is a measure of loss of Carnot (reversible) work opportunity.

The entropy of mass is described as the net temperature-standardised heat transfer to mass under ideal conditions measured from a datum value. An expression for the “irreversibility” is derived in terms of work loss (W_loss) in a work transfer process, unaccounted heat dissipation (Q_loss) in a heat transfer process and loss of net Carnot work (CW_net) opportunity resulting from spontaneous heat transfer across a finite temperature difference during the process. The thermal irreversibility is attributed to not exploiting the capability for extracting work by interposing a combination of Carnot engine(s) and/or Carnot heat pump(s) that exchanges heat with the surrounding and operates across the finite temperature difference.

It is shown, with an example, how the contribution of thermal irreversibility, in estimating reversible input work, amounts to a loss of an opportunity to generate the net work output. The opportunity is created by exchanging heat with surroundings whilst transferring the same amount of heat across finite temperature difference. An entropy change is determined with a numerical simulation, including calculation of local entropy generation values, and results are compared with estimates based on an analytical expression.

A new interpretation of entropy combined with an enhanced mental image of a combination of Carnot engine(s) and/or Carnot heat pump(s) is used to quantify thermal irreversibility.

So far the proposed analytical methods for calculation of copper losses are rather simplified and do not include the time component in the basic partial differential equations, which describe current density distribution. Moreover, when the physical parameters of the transformer (wire dimensions) are out of the certain range, the current density distribution approaches infinity. The purpose of this paper is to offer a generally applicable analytical method. The main goal of the proposed modification of the solution to the current density is improvement of the accuracy and stability of the analytical results.

This paper deals with the calculation of copper losses with various methods, which are based on a time‐dependent electromagnetic field. Analytical method is based on Maxwell equations and Helmholtz equation. Numerical calculation is performed with finite element method (FEM).

Analytical method is a very accurate and it gives results, which are very similar to the actual behaviour of the current density in the winding. However, the FEM analysis is easier to comprehend, but yet very dependent on input parameters.

The numerical analysis may not be accurate enough, because of the problems with the oscillation of the output welding current amplitude. To calculate copper losses correctly, the output welding current must be equal in all test cases, especially during the measurements.

When the physical properties exceed a certain range, the copper losses of the analyzed welding transformer cannot be calculated with existing analytical methods. The new analytical approach gives a far more realistic solution to the current density distribution and improves the accuracy and stability of the results.