Solution of the Hamilton – Jacobi Equation in a Central Potential Using the Separation of Variables Method with Staeckel Boundary Conditions

This manuscript aims at solving Hamilton-Jacobi equation in a central potential using the separation of variables technique with Staeckel boundary conditions. Our results show that the Hamilton – Jacobi variables can be completely separated, which agrees with other results employing different methods. Keywords: Lagrangian mechanics, Hamilton-Jacobi, Staeckel boundary conditions, Staeckel matrix, Staeckel vector, Hamilton's characteristic function, Hamilton's principal function.

Solution of the Hamilton – Jacobi Equations in an Electromagnetic Field Using Separation of Variables Method – Staeckel Boundary Conditions

This manuscript aims to resolve the Hamilton-Jacobi equations in an electromagnetic field by two methods. The first uses the separation of variables technique with Staeckel boundary conditions, whereas the second uses the Newtonian formalism to solve the same example. Our results demonstrate that the Hamilton-Jacobi variables can be completely detached by using separation of variables technique with Staeckel boundary conditions that correspond to other results using Newtonian formalism.

Conformable Derivatives in Laplace Equation and Fractional Fourier Series Solution

In this paper the solution of conformable Laplace equation, \frac{\partial^{\alpha}u(x,y)}{\partial x^{\alpha}}+ \frac{\partial^{\alpha}u(x,y)}{\partial y^{\alpha}}=0, where 1 < α ≤ 2 has been deduced by using fractional fourier series and separation of variables method. For special cases α =2 (Laplace's equation), α=1.9, and α=1.8 conformable fractional fourier coefficients have been calculated. To calculate coefficients, integrals are of type "conformable fractional integral".

A Cauchy problem for the asymmetric Parabolic equation in polar coordinates with the perturbed diffusivity

The inverse problem for the heat equation plays an important area of study and application. Up to now, the backward heat problem (BHP) in Cartesian coordinates has been arisen in many articles, but the BHP in different domains such as polar coordinates, cylindrical one or spherical one is rarely considered. This paper’s purpose is to investigate the BHP on a disk, especially, the problem is associated with the perturbed diffusivity and the space-dependent heat source. In order to solve the problem, the authors apply the separation of variables method, associated with the Bessel’s equation and Bessel’s expansion. Based on the exact solution, the regularized solution is constructed by using the modified quasi-boundary value method. As a result, a Holder type of convergence rate has been obtained. In addition, a numerical experiment is given to illustrate the flexibility and effectiveness of the used method.

О сфероидальной модели рассеяния света несферическими частицами

We have constructed a spheroidal model to solve the problem of light scattering by nonspherical particles. The semiaxes of the model spheroid are determined based on the requirement that the volumes of initial and model particles are equal, as well as the ratios of their longitudinal and transverse dimensions. This ensures the closeness of the optical properties of initial and model particles. This approach has been applied to prolate and oblate parallelepipeds, cylinders, and cones with the ratios between their larger and smaller dimensions equal to 2 or 10. The direction of propagation of the incident TE or TM plane wave was either parallel or perpendicular to the symmetry axis of particles and model spheroid. The particle size has been determined by dimensionless parameter $${{x}_{{v}}}$$ = $$2\pi {{r}_{{v}}}$$ /λ, which depends on the particle volume, since $${{r}_{{v}}}$$ is the radius of the equivolume sphere. In calculations, this parameter has been varied from small values to fairly large ones, $${{x}_{{v}}}$$ = 10. The applicability range of the model has been determined by comparing the results of numerical calculations performed by the rigorous separation of variables method for spheroids and the method of discrete dipoles for other nonspherical particles. It has been shown that the applicability range of the model for parallelepipeds, cylinders, and cones is wide enough for different parameters of the problem, in particular, if the parameter $${{x}_{{v}}}$$ ≤ 6, then the relative error of the model does not exceed 10–15%. To a large extent, this is related to the fact that the first maximum of the dependence of scattering factor Q _sca on $${{x}_{{v}}}$$ is similar for particles of different shapes approximated by one and the same model spheroid.

Permanent magnet linear induction heating device: new topology enhancing performances

Purpose This work aims to study a new design of linear permanent magnet transverse flux induction heating devices of nonmagnetic parallelepipedic workpiece. In these topologies, the permanent magnet inductor produces a static magnetic field, and the workpiece to be heated is subjected to a linear movement. To study the magnetothermal process, a new analytical coupling method between the magnetic and thermal phenomena is developed. This analytical model described in this study takes into account the variation of the physical properties of the heated workpiece. The analytical results are compared with good agreement to those issued from finite elements simulations, as well as those issued from measurements on an actual prototype. Design/methodology/approach The research methodology is based on analytical development of coupled problem, including the electromagnetic and thermal boundary problems. A strongly coupled magneto-thermal analytical model is developed; the time dependent magnetic problem is first solved by using the separation of variables method to evaluate the induced currents in the nonmagnetic plate and the resulting power density loss distribution. The plate temperature profile is then obtained, thanks to strong involvement of this magnetic model in a new analytical thermal model based on a synergy of separation of variables method and Green’s function transient regime analysis method. Findings The results show that an efficient transient magneto-thermal analytical model was developed allowing fast analysis of permanent magnet induction heater for deep heating of parallelepipedic workpieces. Developed model allows also fast and precise simulations of nonlinear and transient magneto-thermal phenomena for different types of permanent magnet induction heating devices. Practical implications The developed magneto-thermal analytical model can be used for fast designing of permanent magnet linear induction heating devices for moving parallelepipedic nonmagnetic workpiece. Originality/value A new analytical coupled model, including the electromagnetic and transient thermal boundary problem with additional algebraic equations and taking into account the nonlinearity, has been developed. The developed model accuracy was validated with a permanent magnet linear induction heating device. Developed coupled analytical model allows fast analysis and designing of such permanent magnet linear induction heating devices.