Research on Initial and Boundary Conditions for Differential and Partial Differential Equations

1989 ◽  
pp. 80-88
Author(s):  
George Adomian
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Partial Differential ◽  
Initial And Boundary Conditions

scholarly journals Effects of radiation and magnetohydrodynamic on unsteady Casson fluid over accelerated plate

2021 ◽  
Vol 85 (1) ◽  
pp. 93-100
Author(s):  
Nur Fatihah Mod Omar ◽  
Husna Izzati Osman ◽  
Ahmad Qushairi Mohamad ◽  
Rahimah Jusoh ◽  
Zulkhibri Ismail
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Casson Fluid ◽  
Dimensionless Form ◽  
Laplace Transform Method ◽  
Partial Differential ◽  
Transform Method ◽  
Initial And Boundary Conditions ◽  
Effects Of Radiation

The effects of radiation and magnetohydrodynamic on unsteady Casson fluid through an accelerated plate is analysed. The problem is formulated in the form of Partial Differential Equations (PDE) with imposed initial and boundary conditions. The Partial Differential Equations are transformed into dimensionless form by introducing suitable non-dimensional variables. Laplace transform method is used to derive the exact solutions for temperature and velocity profiles, fulfilling all initial and boundary conditions. The effects of parameters are depicted and illustrated graphically for radiation, Casson fluid and time, as well as Magnetohydrodynamics (MHD). It is found that the thermal radiation rises due to an increase in temperature. Besides, the increasing of Casson fluid and MHD parameter has decreasing effect on velocity. Finally, the influence of time will increase the velocity of the fluid.

scholarly journals Solving partial differential equations in deformed grids by estimating local average gradients with planes

2021 ◽  
Vol 2090 (1) ◽  
pp. 012069
Author(s):  
Aarne Pohjonen
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Analytical Solutions ◽  
Physical Science ◽  
Partial Differential ◽  
Average Gradient ◽  
Initial And Boundary Conditions ◽  
Local Average

Abstract For constructing physical science based models in irregular numerical grids, an easy-to-implement method for solving partial differential equations has been developed and its accuracy has been evaluated by comparison to analytical solutions that are available for simple initial and boundary conditions. The method is based on approximating the local average gradients of a field by fitting equation of plane to the field quantities at neighbouring grid positions and then calculating an estimate for the local average gradient from the plane equations. The results, comparison to analytical solutions, and accuracy are presented for 2-dimensional cases.

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.

scholarly journals On the theory and practice of thin-walled structures

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Tamaz Vashakmadze
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Complex Analysis ◽  
Theory And Practice ◽  
Linear Case ◽  
Wide Sense ◽  
Partial Differential ◽  
Thin Walled Structures ◽  
Thin Walled

Abstract The basic problem of satisfaction of boundary conditions is considered when the generalized stress vector is given on the surfaces of elastic plates and shells. This problem has so far remained open for both refined theories in a wide sense and hierarchic type models. In the linear case, it was formulated by I. N. Vekua for hierarchic models. In the nonlinear case, bending and compression-expansion processes do not split and in this context the exact structure is presented for the system of differential equations of von Kármán–Mindlin–Reisner (KMR) type, constructed without using a variety of ad hoc assumptions since one of the two relations of this system in the classical form is the compatibility condition, but not the equilibrium equation. In this paper, a unity mathematical theory is elaborated in both linear and nonlinear cases for anisotropic inhomogeneous elastic thin-walled structures. The theory approximately satisfies the corresponding system of partial differential equations and the boundary conditions on the surfaces of such structures. The problem is investigated and solved for hierarchic models too. The obtained results broaden the sphere of applications of complex analysis methods. The classical theory of finding a general solution of partial differential equations of complex analysis, which in the linear case was thoroughly developed in the works of Goursat, Weyl, Walsh, Bergman, Kolosov, Muskhelishvili, Bers, Vekua and others, is extended to the solution of basic nonlinear differential equations containing the nonlinear summand, which is a composition of Laplace and Monge–Ampére operators.

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.

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