A temperature problem for a square: An exact solution

2021 ◽  
pp. 108128652110204
Author(s):  
Mikhail D. Kovalenko ◽  
Irina V. Menshova ◽  
Alexander P. Kerzhaev ◽  
Guangming Yu
Keyword(s):  
Exact Solution ◽  
Boundary Conditions ◽  
Exact Solutions ◽  
Infinite Plane ◽  
Inhomogeneous Problem ◽  
Temperature Problem

We construct examples of exact solutions of the temperature problem for a square: the sides of the square are (i) free and (ii) firmly clamped. Initially, we solve the inhomogeneous problem for an infinite plane. The known exact solutions for a square, with which the boundary conditions on the sides of the square are satisfied, are added to this solution. The solutions are represented as series in Papkovich–Fadle eigenfunctions whose coefficients are determined from simple formulas.

An inhomogeneous problem for an elastic half-strip: An exact solution

2021 ◽  
pp. 108128652199641
Author(s):  
Mikhail D Kovalenko ◽  
Irina V Menshova ◽  
Alexander P Kerzhaev ◽  
Guangming Yu
Keyword(s):  
Exact Solution ◽  
Boundary Conditions ◽  
Exact Solutions ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Theory Of Elasticity ◽  
Orthogonality Relation ◽  
Infinite Strip ◽  
Inhomogeneous Problem ◽  
Half Strip

We construct exact solutions of two inhomogeneous boundary value problems in the theory of elasticity for a half-strip with free long sides in the form of series in Papkovich–Fadle eigenfunctions: (a) the half-strip end is free and (b) the half-strip end is firmly clamped. Initially, we construct a solution of the inhomogeneous problem for an infinite strip. Subsequently, the corresponding solutions for a half-strip are added to this solution, whereby the boundary conditions at the end are satisfied. The Papkovich orthogonality relation is used to solve the inhomogeneous problem in a strip.

scholarly journals An exact solution method for Fredholm integro-differential equations

2019 ◽  
pp. 2-8 ◽  
Author(s):  
Kyriaki Tsilika
Keyword(s):  
Exact Solution ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Exact Solutions ◽  
Solution Method ◽  
Initial Conditions ◽  
Integral Boundary Conditions ◽  
Integral Boundary ◽  
Abstract Operator ◽  
Nonlocal Integral Boundary Conditions

Introduction: Linear boundary value problems for Fredholm ordinary integro-differential equations are seldom consideredwith integral boundary conditions in the literature. In our case, integro-differential equations are subject to multipoint or nonlocalintegral boundary conditions. It should be noted that finding exact solutions even for multipoint problems or problems with nonlocalintegral boundary conditions with a differential equation is a difficult task. Purpose: Finding the uniqueness and existencecriterion of solutions for Fredholm ordinary integro-differential equations with multipoint or nonlocal integral boundary conditionsand obtaining exact solutions in closed form of such problems. Results: Within the class of abstract operator equations, for thespecial case of Fredholm integro-differential equations with multipoint or nonlocal integral boundary conditions, a criterion for theexistence and uniqueness of an exact solution is proved and the analytical representation of the solution is given. A direct methodanalytically solving such problems is proposed, in which all calculations are reproducible in any program of symbolic calculations.If the user sets the input parameters and the initial conditions of the problem, the computer codes check the conditions of existenceand uniqueness and of solution generate the analytical solution. The stages of the solution method are illustrated by twoexamples. The article uses computer algebra system Mathematica to demonstrate the results.

scholarly journals A tale of two nested elastic rings

2017 ◽  
Vol 473 (2204) ◽  
pp. 20170340 ◽  
Author(s):  
G. Napoli ◽  
A. Goriely
Keyword(s):  
Exact Solution ◽  
Boundary Conditions ◽  
Exact Solutions ◽  
Variational Formulation ◽  
Contact Point ◽  
Elliptic Functions ◽  
Elastic Rods ◽  
Elastic Bodies ◽  
Static Solutions

Elastic rods in contact provide a rich paradigm for understanding shape and deformation in interacting elastic bodies. Here, we consider the problem of determining the static solutions of two nested elastic rings in the plane. If the inner ring is longer than the outer ring, it will buckle creating a space between the two rings. This deformation can be further influenced by either adhesion between the rings or if pressure is applied externally or internally. We obtain an exact solution of this problem when both rings are assumed inextensible and unshearable. Through a variational formulation of the problem, we identify the boundary conditions at the contact point and use the Kirchhoff analogy to give exact solutions of the problems in terms of elliptic functions. The role of both adhesion and pressure is explored.

scholarly journals Analytical exact solution of telegraph equation using HPM

BIBECHANA ◽  
2016 ◽  
Vol 14 ◽  
pp. 30-36
Author(s):  
Jamshad Ahmad ◽  
Ghulam Mohiuddin
Keyword(s):  
Exact Solution ◽  
Partial Differential Equations ◽  
Boundary Conditions ◽  
Perturbation Method ◽  
Exact Solutions ◽  
Homotopy Perturbation Method ◽  
Iterative Scheme ◽  
Nonlinear Partial Differential Equations ◽  
Telegraph Equation ◽  
Homotopy Perturbation

In this paper, exact solutions of different variants of second order hyperbolic telegraph equation are investigated with Homotopy Perturbation Method (HPM). The results determined by the proposed method are quite satisfactory and shows that HPM technique is very effective and useful for solving the nonlinear partial differential equations (PDEs) with given initial or boundary conditions. The proposed iterative scheme finds the solution without any discretization, linearization or restrictive assumptions.BIBECHANA 14 (2017) 30-36

The Exact Solution to an Ablation Problem With Arbitrary Initial and Boundary Conditions

1984 ◽  
Vol 51 (4) ◽  
pp. 821-826 ◽  
Author(s):  
L. N. Tao
Keyword(s):  
Exact Solution ◽  
Boundary Conditions ◽  
Surface Temperature ◽  
Exact Solutions ◽  
Melting Temperature ◽  
Physical Interpretation ◽  
Existence And Uniqueness ◽  
Frictional Heating ◽  
Convective Motions ◽  
Initial And Boundary Conditions

The problem of ablation by frictional heating in a semi-infinite solid with arbitrarily prescribed initial and boundary conditions is investigated. The study includes all convective motions caused by the density differences of various phases of the materials. It is found that there are two cases: (i) ablation appears immediately and (ii) there is a waiting period of redistribution prior to ablation. The exact solutions of velocities and temperatures of both cases are derived. The solutions of the interfacial positions are also established. Existence and uniqueness of the solutions are examined and proved. The conditions for the occurrence of these two cases are expressed by an inequality. Physical interpretation of the inequality is explored. Its implication coincides with one’s expectation. Ablation appears only when the surface temperature is at or above the melting temperature.

scholarly journals Exact solutions for the total variation denoising problem of piecewise constant images in dimension one

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Riccardo Cristoferi
Keyword(s):  
Exact Solution ◽  
Exact Solutions ◽  
Total Variation ◽  
Piecewise Constant ◽  
Geometrical Properties ◽  
Total Variation Denoising ◽  
Fidelity Term ◽  
Dimension One

AbstractA method for obtaining the exact solution for the total variation denoising problem of piecewise constant images in dimension one is presented. The validity of the algorithm relies on some results concerning the behavior of the solution when the parameter λ in front of the fidelity term varies. Albeit some of them are well-known in the community, here they are proved with simple techniques based on qualitative geometrical properties of the solutions.

Inhomogeneous Problem of the Theory of Elasticity in a Half-Strip. Exact Solution

2020 ◽  
Vol 55 (6) ◽  
pp. 784-790
Author(s):  
M. D. Kovalenko ◽  
I. V. Menshova ◽  
A. P. Kerzhaev ◽  
G. Yu
Keyword(s):  
Exact Solution ◽  
Theory Of Elasticity ◽  
Inhomogeneous Problem ◽  
Half Strip
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