scholarly journals A Simple Example for Linear Partial Differential Equations and Its Solution Using the Method of Separation of Variables

2019 ◽  
Vol 27 (1) ◽  
pp. 25-34
Author(s):  
Sora Otsuki ◽  
Pauline N. Kawamoto ◽  
Hiroshi Yamazaki
Keyword(s):  
Wave Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Partial Derivative ◽  
Separation Of Variables ◽  
Partial Differential ◽  
One Dimensional ◽  
Linear Partial Differential Equations ◽  
Dimensional Wave

Summary In this article, we formalized in Mizar [4], [1] simple partial differential equations. In the first section, we formalized partial differentiability and partial derivative. The next section contains the method of separation of variables for one-dimensional wave equation. In the last section, we formalized the superposition principle.We referred to [6], [3], [5] and [9] in this formalization.

Beam Vibrations With Time-Dependent Boundary Conditions

1950 ◽  
Vol 17 (4) ◽  
pp. 377-380
Author(s):  
R. D. Mindlin ◽  
L. E. Goodman
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Separation Of Variables ◽  
Boundary Value ◽  
Time Dependent ◽  
Partial Differential ◽  
Linear Partial Differential Equations ◽  
Vibration Problems

Abstract A procedure is described for extending the method of separation of variables to the solution of beam-vibration problems with time-dependent boundary conditions. The procedure is applicable to a wide variety of time-dependent boundary-value problems in systems governed by linear partial differential equations.

Pressure Transients Across Constrictions

1999 ◽  
Vol 122 (1) ◽  
pp. 34-41 ◽  
Author(s):  
Michael A. Adewumi ◽  
E. S. Eltohami ◽  
W. H. Ahmed
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Single Phase ◽  
Tvd Scheme ◽  
Eulerian Model ◽  
Partial Differential ◽  
One Dimensional ◽  
First Order ◽  
Linear Partial Differential Equations ◽  
Phase Representation

The modeling of pressure transient across constrictions is achieved by using a one-dimensional, isothermal, noncompositional, single-phase representation of the Eulerian model. A TVD scheme was used to solve the ensuing nonhomogeneous hyperbolic set of first-order quasi-linear partial differential equations. Three types of constrictions were modeled and in each case the behavior of the transient was analyzed. This analysis was used to interpret the pressure response at the inlet resulting from the reflection of the transient at the constrictions. The comparisons between the predicted and inputted data are very good, suggesting that the technique has much promise. [S0195-0738(00)00301-0]

scholarly journals Existence of weak solutions to SPDEs with fractional Laplacian and non-Lipschitz coefficients

2021 ◽  
Author(s):  
Shohei Nakajima
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Weak Solutions ◽  
Stochastic Partial Differential Equations ◽  
Fractional Laplacian ◽  
Existence Of Solutions ◽  
Partial Differential ◽  
Continuity Theorem ◽  
Existence Of Weak Solutions ◽  
One Dimensional

AbstractWe prove existence of solutions and its properties for a one-dimensional stochastic partial differential equations with fractional Laplacian and non-Lipschitz coefficients. The method of proof is eatablished by Kolmogorov’s continuity theorem and tightness arguments.

scholarly journals Approximation of linear one dimensional partial differential equations including fractional derivative with non-singular kernel

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Raheel Kamal ◽  
Kamran ◽  
Gul Rahmat ◽  
Ali Ahmadian ◽  
Noreen Izza Arshad ◽  
...  
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Laplace Transform ◽  
Fractional Derivative ◽  
Meshless Method ◽  
Inverse Laplace Transform ◽  
Contour Integral ◽  
Partial Differential ◽  
One Dimensional ◽  
The Laplace Transform

AbstractIn this article we propose a hybrid method based on a local meshless method and the Laplace transform for approximating the solution of linear one dimensional partial differential equations in the sense of the Caputo–Fabrizio fractional derivative. In our numerical scheme the Laplace transform is used to avoid the time stepping procedure, and the local meshless method is used to produce sparse differentiation matrices and avoid the ill conditioning issues resulting in global meshless methods. Our numerical method comprises three steps. In the first step we transform the given equation to an equivalent time independent equation. Secondly the reduced equation is solved via a local meshless method. Finally, the solution of the original equation is obtained via the inverse Laplace transform by representing it as a contour integral in the complex left half plane. The contour integral is then approximated using the trapezoidal rule. The stability and convergence of the method are discussed. The efficiency, efficacy, and accuracy of the proposed method are assessed using four different problems. Numerical approximations of these problems are obtained and validated against exact solutions. The obtained results show that the proposed method can solve such types of problems efficiently.

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.

A method for constructing solutions of homogeneous partial differential equations: localized waves

1992 ◽  
Vol 437 (1901) ◽  
pp. 673-692 ◽  
Keyword(s):  
Wave Equation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Gordon Equation ◽  
Partial Differential ◽  
Wave Solutions ◽  
Propagation Axis ◽  
Potential Applications ◽  
Klein Gordon ◽  
Localized Waves

We introduce a method for constructing solutions of homogeneous partial differential equations. This method can be used to construct the usual, well-known, separable solutions of the wave equation, but it also easily gives the non-separable localized wave solutions. These solutions exhibit a degree of focusing about the propagation axis that is dependent on a free parameter, and have many important potential applications. The method is based on constructing the space-time Fourier transform of a function so that it satisfies the transformed partial differential equation. We also apply the method to construct localized wave solutions of the wave equation in a lossy infinite medium, and of the Klein-Gordon equation. The localized wave solutions of these three equations differ somewhat, and we discuss these differences. A discussion of the properties of the localized waves, and of experiments to launch them, is included in the Appendix.

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