scholarly journals Solving Fundamental Solution of Non-Homogeneous Heat Equation with Dirichlet Boundary Conditions

2020 ◽  
Vol 22 ◽  
pp. 1-9
Author(s):  
Kahsay Godifey Wubneh
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Boundary Conditions ◽  
Heat Equation ◽  
Constant Coefficient ◽  
Dirichlet Boundary ◽  
Dirichlet Boundary Conditions ◽  
Partial Differential ◽  
Nonhomogeneous Heat ◽  
Interesting Solution

In this study, we developed a solution of nonhomogeneous heat equation with Dirichlet boundary conditions. moreover, the non-homogeneous heat equation with constant coefficient. since heat equation has a simple form, we would like to start from the heat equation to find the exact solution of the partial differential equation with constant coefficient. to emphasize our main results, we also consider some important way of solving of partial differential equation specially solving heat equation with Dirichlet boundary conditions. the main results of our paper are quite general in nature and yield some interesting solution of non-homogeneous heat equation with Dirichlet boundary conditions and it is used for problems of mathematical modeling and mathematical physics.

scholarly journals Explicit Solution for Cylindrical Heat Conduction

2016 ◽  
Vol 13 (2) ◽  
Author(s):  
Kaitlyn Parsons ◽  
Tyler Reichanadter ◽  
Andi Vicksman ◽  
Harvey Segur
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Heat Conduction ◽  
Boundary Conditions ◽  
Heat Equation ◽  
Analytical Method ◽  
Explicit Solution ◽  
Separation Of Variables ◽  
Partial Differential ◽  
Forcing Function

The heat equation is a partial differential equation that elegantly describes heat conduction or other diffusive processes. Primary methods for solving this equation require time-independent boundary conditions. In reality this assumption rarely has any validity. Therefore it is necessary to construct an analytical method by which to handle the heat equation with time-variant boundary conditions. This paper analyzes a physical system in which a solid brass cylinder experiences heat flow from the central axis to a heat sink along its outer rim. In particular, the partial differential equation is transformed such that its boundary conditions are zero which creates a forcing function in the transform PDE. This transformation constructs a Green’s function, which admits the use of variation of parameters to find the explicit solution. Experimental results verify the success of this analytical method. KEYWORDS: Heat Equation; Bessel-Fourier Decomposition; Cylindrical; Time-dependent Boundary Conditions; Orthogonality; Partial Differential Equation; Separation of Variables; Green’s Functions

Lyapunov-type inequalities for partial differential equations with 𝑝-Laplacian

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Robert Stegliński
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Nonlinear Partial Differential Equations ◽  
Dirichlet Boundary Conditions ◽  
Partial Differential ◽  
Linear Partial Differential Equations ◽  
Lyapunov Inequalities ◽  
Funct Anal

Abstract The aim of this paper is to extend results from [A. Cañada, J. A. Montero and S. Villegas, Lyapunov inequalities for partial differential equations, J. Funct. Anal. 237 (2006), 1, 176–193] about Lyapunov-type inequalities for linear partial differential equations to nonlinear partial differential equations with 𝑝-Laplacian with zero Neumann or Dirichlet boundary conditions.

Large Time Behavior of Solutions to One Nonlinear Integro-Differential Equation

2008 ◽  
Vol 15 (3) ◽  
pp. 531-539
Author(s):  
Temur Jangveladze ◽  
Zurab Kiguradze
Keyword(s):  
Magnetic Field ◽  
Differential Equation ◽  
Boundary Conditions ◽  
Large Time ◽  
Dirichlet Boundary ◽  
Large Time Behavior ◽  
Dirichlet Boundary Conditions ◽  
Time Behavior ◽  
Integro Differential Equation ◽  
Homogeneous Data

Abstract Large time behavior of solutions to the nonlinear integro-differential equation associated with the penetration of a magnetic field into a substance is studied. The rate of convergence is given, too. Dirichlet boundary conditions with homogeneous data are considered.

Nonlinear Vibrations of Non-Uniform Beams Including Inertia Effects

2005 ◽  
Author(s):  
Dumitru I. Caruntu ◽  
Ion Stroe
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Boundary Conditions ◽  
Nonlinear Vibrations ◽  
Multiple Scales ◽  
Nonlinear Partial Differential Equation ◽  
Nonlinear Effects ◽  
Order Partial Differential Equation ◽  
Partial Differential ◽  
Inertia Effects

This papers deals with nonlinear vibrations of non-uniform beams with geometrical nonlinearities such as moderately large curvatures, and inertia nonlinearities such as longitudinal and rotary inertia forces. The nonlinear fourth-order partial-differential equation describing the above nonlinear effects is presented. Using the method of multiple scales, each effect is found by reducing the nonlinear partial-differential equation of motion to two simpler linear partial-differential equations, homogeneous and nonhomogeneous. These equations along with given boundary conditions are analytically solved obtaining so-called zero-and first-order approximations of the beam’s nonlinear frequencies. Since the effect of mid-plane stretching is ignored, any boundary conditions could be considered as long as the supports are not fixed a constant distance apart. Analytical expressions showing the influence of these three nonlinearities on beam’s frequencies are presented up to some constant coefficients. These coefficients depend on the geometry of the beam. This paper can be used to study these influences on frequencies of different classes of beams. However, numerical results are presented for uniform beams. These results show that as beam slenderness increases the effect of these nonlinearities decreases. Also, they show that the most important nonlinear effect is due to moderately large curvature for slender beams.

Calculation of Elastic Induced Stress of Surrounding Rocks

2012 ◽  
Vol 170-173 ◽  
pp. 37-40
Author(s):  
Bo Qian
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Boundary Conditions ◽  
Analytical Solutions ◽  
Superposition Principle ◽  
Compatibility Conditions ◽  
Partial Differential ◽  
Induced Stress ◽  
Stress Functions ◽  
Round Section

In accordance with equilibrium differential equations and compatibility conditions of deformation, the partial differential equation of induced stress is achieved for elastic surrounding rocks of tunnels and chambers of round section. By method of the superposition principle, elastic analytical solutions of induced stress of surrounding rocks is derived from the partial differential equation, which is based on stress functions and boundary conditions.

Sign in / Sign up

Export Citation Format

Share Document