scholarly journals New finite integral transform for the Laplace equation in an arbitrary domain

2020 ◽  
Vol 3 ◽  
pp. 115-124
Author(s):  
M.G. Berdnyk ◽  
◽  
Keyword(s):  
Analytical Solution ◽  
Laplace Equation ◽  
Integral Transformation ◽  
Boundary Value ◽  
Inverse Transformation ◽  
Maximum Deviation ◽  
Arbitrary Domain ◽  
First Order ◽  
Finite Integral Transformation ◽  
Finite Integral

Reliability, survivability, as well as the optimal operating mode of operation of the supercomputer will depend on the architecture and efficiency of the cooling system of the hot components of the supercomputer. That is why the number of problems, of great theoretical and practical interest, is the problem of studying the temperature fields arising in elements of arbitrary configuration, cooling a supercomputer. To solve this class of heat conduction problems, the method of finite integral transformations turned out to be the most convenient. This article is the first to construct a new finite integral transformation for the Laplace equation in an arbitrary domain bounded by several closed piecewise-smooth contours. An inverse transformation formula is given. Finding the core of the constructed new finite integral transformation by the finite element method in the Galerkin form for simplex first-order elements reduces to solving a system of algebraic equations. To test the operability of the new integral transformation, calculations were carried out of solutions of the boundary value problem for the Laplace equation obtained using the developed new integral transformation and the well-known analytical solution. The results of comparison the calculations of the solution of the Laplace equation are presented. In the case of a square with a side length equal to one and on one side of the square, the temperature is unity, and on the other, the temperature is zero, with a well-known analytical solution and a solution obtained using the new integral transformation. These results were obtained for 228 simplex first-order elements and 135 nodes. The maximum deviation modulo of these solutions is 0,096, the mathematical expectation of deviations is 0,009, and the variance of the type is 0,001. The developed integral transformation makes it possible to obtain a solution to complex boundary value problems of mathematical physics.

Analytical Solution of Forced Vibration of a Column Subjected to Conservative and Nonconservative Forces

2001 ◽  
Author(s):  
Zhixiang Xu ◽  
Kunisato Seto ◽  
Hideyuki Tamura
Keyword(s):  
Analytical Solution ◽  
Boundary Value Problem ◽  
Integral Transform ◽  
Boundary Value ◽  
Suspension Bridge ◽  
Follower Force ◽  
Analysis Process ◽  
Finite Integral ◽  
Excitation Force ◽  
Integral Transform Technique

Abstract This paper presents analytical results of forced transverse vibration of a column with a mass attached at free-end subjected to a tangential follower force and a transverse distributed excitation force, that is a simplified model of some structures in civil and mechanical engineering, e.g., a column of a suspension bridge, a launched rocket in the atmosphere. Because the tangential follower force is nonconservative, it is very difficult to get the analytical solution of the problem by usually-used analysis methods with which the adjoint boundary value problem can not be directly obtained. However, by applying the finite integral transform technique, we directly obtained the adjoint boundary value problem in the analysis process, and successfully obtained the analytical solution of the column’s vibration excited by the transverse distributed force.

scholarly journals Heat Conduction in Hollow Cylinders with Radiation

1964 ◽  
Vol 14 (2) ◽  
pp. 159-164 ◽  
Author(s):  
E. Marchi ◽  
G. Zgrablich
Keyword(s):  
Heat Conduction ◽  
Boundary Conditions ◽  
Hollow Cylinder ◽  
Integral Transformation ◽  
Finite Integral Transformation ◽  
Finite Integral ◽  
Hollow Cylinders ◽  
Radiation Type

AbstractA new finite integral transformation (an extension of those given by Sneddon (1)), whose kernel is given by cylindrical functions, is used to solve the problem of finding the temperature at any point of a hollow cylinder of any height, with boundary conditions of radiation type on the outside and inside surfaces, with independent radiation constants. It is to be noticed that all possible problems on boundary conditions in hollow cylinders can be solved by particularising the method described here.

scholarly journals NON-STATIONARY AXISYMMETRIC PROBLEM OF INVERSE PIEZOEFFECT FOR CIRCULAR BIMORPHOUS PLATE OF STEPPED VARIABLE THICKNESS AND RIGIDITY

2017 ◽  
Vol 19 (6) ◽  
pp. 133-140
Author(s):  
D.A. Shlyakhin
Keyword(s):  
Variable Thickness ◽  
Arbitrary Function ◽  
Axisymmetric Problem ◽  
Elastic System ◽  
Integral Transformation ◽  
Variable Rigidity ◽  
Closed Solution ◽  
Finite Integral Transformation ◽  
Finite Integral ◽  
Deformed State

Non-stationary axisymmetric problem for a thin circular bimorphous plate under the action on the end surfaces of electric potential, which is an arbitrary function of time is viewed. On the basis of the theory of Tymoshenko by method of finite integral transformation a new closed solution for the viewed electricity and elastic system of stepped variable rigidity and thickness is built. The obtained calculated correlations allow to explore the frequency characteristics and stress-deformed state of bimorphous elements.

Dynamic Thermoelastic Analysis of a Slab Using Finite Integral Transformation Method

AIAA Journal ◽  
2010 ◽  
Vol 48 (8) ◽  
pp. 1833-1839 ◽  
Author(s):  
Jianli Ruan ◽  
Xue Feng ◽  
Guobing Zhang ◽  
Yong Wang ◽  
Daining Fang
Keyword(s):  
Integral Transformation ◽  
Transformation Method ◽  
Thermoelastic Analysis ◽  
Finite Integral Transformation ◽  
Finite Integral ◽  
Integral Transformation Method

Analytical solution for unsteady flow behind ionizing shock wave in a rotational axisymmetric non-ideal gas with azimuthal or axial magnetic field

2021 ◽  
Vol 76 (3) ◽  
pp. 265-283
Author(s):  
G. Nath
Keyword(s):  
Magnetic Field ◽  
Shock Wave ◽  
Analytical Solution ◽  
Axial Magnetic Field ◽  
Order Approximation ◽  
Ideal Gas ◽  
Field Region ◽  
First Order ◽  
First Order Approximation ◽  
Flow Variables

Abstract The approximate analytical solution for the propagation of gas ionizing cylindrical blast (shock) wave in a rotational axisymmetric non-ideal gas with azimuthal or axial magnetic field is investigated. The axial and azimuthal components of fluid velocity are taken into consideration and these flow variables, magnetic field in the ambient medium are assumed to be varying according to the power laws with distance from the axis of symmetry. The shock is supposed to be strong one for the ratio C 0 V s 2 ${\left(\frac{{C}_{0}}{{V}_{s}}\right)}^{2}$ to be a negligible small quantity, where C 0 is the sound velocity in undisturbed fluid and V S is the shock velocity. In the undisturbed medium the density is assumed to be constant to obtain the similarity solution. The flow variables in power series of C 0 V s 2 ${\left(\frac{{C}_{0}}{{V}_{s}}\right)}^{2}$ are expanded to obtain the approximate analytical solutions. The first order and second order approximations to the solutions are discussed with the help of power series expansion. For the first order approximation the analytical solutions are derived. In the flow-field region behind the blast wave the distribution of the flow variables in the case of first order approximation is shown in graphs. It is observed that in the flow field region the quantity J 0 increases with an increase in the value of gas non-idealness parameter or Alfven-Mach number or rotational parameter. Hence, the non-idealness of the gas and the presence of rotation or magnetic field have decaying effect on shock wave.

On the Boundary Value Problem in A Dihedral Angle for Normally Hyperbolic Systems of First Order

1998 ◽  
Vol 5 (2) ◽  
pp. 121-138
Author(s):  
O. Jokhadze
Keyword(s):  
Partial Differential Equations ◽  
Boundary Value Problem ◽  
Principal Part ◽  
Hyperbolic Systems ◽  
Boundary Value ◽  
Three Dimensional ◽  
Order System ◽  
First Order ◽  
Dimensional Boundary ◽  
Class Of Functions

Abstract Some structural properties as well as a general three-dimensional boundary value problem for normally hyperbolic systems of partial differential equations of first order are studied. A condition is given which enables one to reduce the system under consideration to a first-order system with the spliced principal part. It is shown that the initial problem is correct in a certain class of functions if some conditions are fulfilled.

scholarly journals Regularity of Weakly Well-Posed Characteristic Boundary Value Problems

2010 ◽  
Vol 2010 ◽  
pp. 1-39 ◽  
Author(s):  
Alessandro Morando ◽  
Paolo Secchi
Keyword(s):  
Energy Estimate ◽  
Boundary Value ◽  
A Priori ◽  
Regularity Of Solutions ◽  
First Order ◽  
Characteristic Boundary ◽  
Partial Differential System ◽  
Constant Multiplicity ◽  
Well Posed ◽  
Smooth Data

We study the boundary value problem for a linear first-order partial differential system with characteristic boundary of constant multiplicity. We assume the problem to be “weakly” well posed, in the sense that a uniqueL2-solution exists, for sufficiently smooth data, and obeys an a priori energy estimate with a finite loss of tangential/conormal regularity. This is the case of problems that do not satisfy the uniform Kreiss-Lopatinskiĭ condition in the hyperbolic region of the frequency domain. Provided that the data are sufficiently smooth, we obtain the regularity of solutions, in the natural framework of weighted conormal Sobolev spaces.

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