scholarly journals Analytical Solution for Electrical Problem Forced by a Finite-Length Needle Electrode: Implications in Electrostimulation

2019 ◽  
Vol 2019 ◽  
pp. 1-10
Author(s):  
Ricardo Romero-Méndez ◽  
Francisco G. Pérez-Gutiérrez ◽  
Francisco Oviedo-Tolentino ◽  
Enrique Berjano
Keyword(s):  
Analytical Solution ◽  
Electric Field ◽  
Boundary Conditions ◽  
Separation Of Variables ◽  
Needle Electrode ◽  
Analytical Models ◽  
Infinite Length ◽  
Principle Of Superposition ◽  
Clinical Procedures ◽  
Sturm Liouville

Needle electrodes, widely used in clinical procedures, are responsible for creating an electric field in the treated biological tissue. This is achieved by setting a constant voltage along the length of their metallic section. In accordance with Laplace’s equation, the electric field is spatially non-uniform around the electrode surface. Mathematical modelling can provide useful information on the spatial distribution of electrical fields. Indeed, exact solutions for the electrical problem are indispensable for validating numerical codes. All the analytical models developed to date to solve the needle electrode electrical problem have been one-dimensional models, which assumed an electrode of infinite length. We here propose the first analytical solution based on a two-dimensional model that considers the real length of the electrode in which the Laplace equation is solved through the method of separation of variables, dealing with the nonhomogeneous source term and boundary conditions by Green’s functions. On assuming a needle electrode of given length, the problem combines boundary conditions on the electrode boundary (of the first and second kind). Since this rules out using the Sturm-Liouville Theorem, the problem is decomposed into two different problems and the principle of superposition is used. The solution obtained can reproduce a reasonable electric field around the electrode, especially the edge effect characterized by an extremely high gradient around the electrode tip.

scholarly journals Analytical Solution for Free Vibrations of a Moderately Thick Rectangular Plate

2013 ◽  
Vol 2013 ◽  
pp. 1-13 ◽  
Author(s):  
Ivo Senjanović ◽  
Marko Tomić ◽  
Nikola Vladimir ◽  
Dae Seung Cho
Keyword(s):  
Analytical Solution ◽  
Boundary Conditions ◽  
Thick Plate ◽  
Mixed Boundary ◽  
Separation Of Variables ◽  
Equations Of Motion ◽  
Mixed Boundary Conditions ◽  
Free Vibrations ◽  
Force Displacement ◽  
Good Agreement

In the present thick plate vibration theory, governing equations of force-displacement relations and equilibrium of forces are reduced to the system of three partial differential equations of motion with total deflection, which consists of bending and shear contribution, and angles of rotation as the basic unknown functions. The system is starting one for the application of any analytical or numerical method. Most of the analytical methods deal with those three equations, some of them with two (total and bending deflection), and recently a solution based on one equation related to total deflection has been proposed. In this paper, a system of three equations is reduced to one equation with bending deflection acting as a potential function. Method of separation of variables is applied and analytical solution of differential equation is obtained in closed form. Any combination of boundary conditions can be considered. However, the exact solution of boundary value problem is achieved for a plate with two opposite simply supported edges, while for mixed boundary conditions, an approximate solution is derived. Numerical results of illustrative examples are compared with those known in the literature, and very good agreement is achieved.

Analytical solutions of some internal boundary value problems of elasticity for domains with hyperbolic boundaries

2018 ◽  
Vol 24 (6) ◽  
pp. 1726-1748 ◽  
Author(s):  
Natela Zirakashvili
Keyword(s):  
Analytical Solution ◽  
Boundary Conditions ◽  
Analytical Solutions ◽  
Strain State ◽  
Separation Of Variables ◽  
Internal Boundary ◽  
Test Problems ◽  
Two Dimensional ◽  
Elliptic Coordinates ◽  
Hyperbolic Boundary

An analytical solution of two-dimensional problems of elasticity in the region bounded by a hyperbola in elliptic coordinates is constructed using the method of separation of variables. The stress–strain state of a homogenous isotropic hyperbolic body and that with a hyperbolic cut is studied when there are non-homogenous (non-zero) boundary conditions given on the hyperbolic boundary. The graphs for the numerical results of some test problems are presented.

scholarly journals Exact semi-separation of variables in waveguides with non-planar boundaries

2017 ◽  
Vol 473 (2201) ◽  
pp. 20170017 ◽  
Author(s):  
G. A. Athanassoulis ◽  
Ch. E. Papoutsellis
Keyword(s):  
Boundary Conditions ◽  
Series Expansion ◽  
Water Waves ◽  
Separation Of Variables ◽  
Convergent Series ◽  
Nonlinear Water Waves ◽  
Coupled Mode ◽  
Sturm Liouville ◽  
Dirichlet To Neumann Operator ◽  
Physical Boundary

Series expansions of unknown fields Φ = ∑ φ n Z n in elongated waveguides are commonly used in acoustics, optics, geophysics, water waves and other applications, in the context of coupled-mode theories (CMTs). The transverse functions Z n are determined by solving local Sturm–Liouville problems (reference waveguides). In most cases, the boundary conditions assigned to Z n cannot be compatible with the physical boundary conditions of Φ , leading to slowly convergent series, and rendering CMTs mild-slope approximations. In the present paper, the heuristic approach introduced in Athanassoulis & Belibassakis (Athanassoulis & Belibassakis 1999 J. Fluid Mech . 389 , 275–301) is generalized and justified. It is proved that an appropriately enhanced series expansion becomes an exact, rapidly convergent representation of the field Φ , valid for any smooth, non-planar boundaries and any smooth enough Φ . This series expansion can be differentiated termwise everywhere in the domain, including the boundaries, implementing an exact semi-separation of variables for non-separable domains. The efficiency of the method is illustrated by solving a boundary value problem for the Laplace equation, and computing the corresponding Dirichlet-to-Neumann operator, involved in Hamiltonian equations for nonlinear water waves. The present method provides accurate results with only a few modes for quite general domains. Extensions to general waveguides are also discussed.

Sturm-Liouville Problem for Stationary Differential Operator with Nonlocal Two-Point Boundary Conditions

2006 ◽  
Vol 11 (1) ◽  
pp. 47-78 ◽  
Author(s):  
S. Pečiulytė ◽  
A. Štikonas
Keyword(s):  
Boundary Condition ◽  
Differential Operator ◽  
Boundary Conditions ◽  
Nonlocal Boundary Condition ◽  
Nonlocal Boundary ◽  
Liouville Problem ◽  
Complex Case ◽  
Qualitative Behavior ◽  
Point Boundary ◽  
Sturm Liouville

The Sturm-Liouville problem with various types of two-point boundary conditions is considered in this paper. In the first part of the paper, we investigate the Sturm-Liouville problem in three cases of nonlocal two-point boundary conditions. We prove general properties of the eigenfunctions and eigenvalues for such a problem in the complex case. In the second part, we investigate the case of real eigenvalues. It is analyzed how the spectrum of these problems depends on the boundary condition parameters. Qualitative behavior of all eigenvalues subject to the nonlocal boundary condition parameters is described.

scholarly journals Regular approximation of singular Sturm–Liouville problems with eigenparameter dependent boundary conditions

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Maozhu Zhang ◽  
Kun Li ◽  
Hongxiang Song
Keyword(s):  
Boundary Conditions ◽  
Operator Theory ◽  
Resolvent Convergence ◽  
Spectral Inclusion ◽  
Regular Approximation ◽  
Special Cases ◽  
Abstract Operator ◽  
Norm Resolvent Convergence ◽  
Sturm Liouville ◽  
Spectral Exactness

AbstractIn this paper we consider singular Sturm–Liouville problems with eigenparameter dependent boundary conditions and two singular endpoints. The spectrum of such problems can be approximated by those of the inherited restriction operators constructed. Via the abstract operator theory, the strongly resolvent convergence and norm resolvent convergence of a sequence of operators are obtained and it follows that the spectral inclusion of spectrum holds. Moreover, spectral exactness of spectrum holds for two special cases.

An analytical solution of a two-dimensional temperature field in a hollow cylinder under a time periodic boundary condition using Fourier series

2009 ◽  
Vol 223 (8) ◽  
pp. 1889-1901 ◽  
Author(s):  
G Atefi ◽  
M A Abdous ◽  
A Ganjehkaviri ◽  
N Moalemi
Keyword(s):  
Boundary Condition ◽  
Temperature Field ◽  
Fourier Series ◽  
Temperature Distribution ◽  
Analytical Solution ◽  
Hollow Cylinder ◽  
Periodic Boundary Condition ◽  
Periodic Boundary ◽  
Separation Of Variables ◽  
Two Dimensional

The objective of this article is to derive an analytical solution for a two-dimensional temperature field in a hollow cylinder, which is subjected to a periodic boundary condition at the outer surface, while the inner surface is insulated. The material is assumed to be homogeneous and isotropic with time-independent thermal properties. Because of the time-dependent term in the boundary condition, Duhamel's theorem is used to solve the problem for a periodic boundary condition. The periodic boundary condition is decomposed using the Fourier series. This condition is simulated with harmonic oscillation; however, there are some differences with the real situation. To solve this problem, first of all the boundary condition is assumed to be steady. By applying the method of separation of variables, the temperature distribution in a hollow cylinder can be obtained. Then, the boundary condition is assumed to be transient. In both these cases, the solutions are separately calculated. By using Duhamel's theorem, the temperature distribution field in a hollow cylinder is obtained. The final result is plotted with respect to the Biot and Fourier numbers. There is good agreement between the results of the proposed method and those reported by others for this geometry under a simple harmonic boundary condition.

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