scholarly journals On the Multidomain Bivariate Spectral Local Linearisation Method for Solving Systems of Nonsimilar Boundary Layer Partial Differential Equations

2019 ◽  
Vol 2019 ◽  
pp. 1-18 ◽  
Author(s):  
Vusi Mpendulo Magagula
Keyword(s):  
Boundary Layer ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Large Time ◽  
Nonlinear Partial Differential Equations ◽  
Computation Time ◽  
General System ◽  
Partial Differential ◽  
Boundary Layer Equations ◽  
Grid Points

In this work, a novel approach for solving systems of nonsimilar boundary layer equations over a large time domain is presented. The method is a multidomain bivariate spectral local linearisation method (MD-BSLLM), Legendre-Gauss-Lobatto grid points, a local linearisation technique, and the spectral collocation method to approximate functions defined as bivariate Lagrange interpolation. The method is developed for a general system of n nonlinear partial differential equations. The use of the MD-BSLLM is demonstrated by solving a system of nonlinear partial differential equations that describe a class of nonsimilar boundary layer equations. Numerical experiments are conducted to show applicability and accuracy of the method. Grid independence tests establish the accuracy, convergence, and validity of the method. The solution for the limiting case is used to validate the accuracy of the MD-BSLLM. The proposed numerical method performs better than some existing numerical methods for solving a class of nonsimilar boundary layer equations over large time domains since it converges faster and uses few grid points to achieve accurate results. The proposed method uses minimal computation time and its accuracy does not deteriorate with an increase in time.

APPROXIMATE SOLUTION OF MODIFIED CAMASSA–HOLM AND DEGASPERIS–PROCESI EQUATIONS USING WAVELET OPTIMIZED FINITE DIFFERENCE METHOD

2013 ◽  
Vol 11 (02) ◽  
pp. 1350019
Author(s):  
RATIKANTA BEHERA ◽  
MANI MEHRA
Keyword(s):  
Approximate Solution ◽  
Partial Differential Equations ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Differential Equations ◽  
Difference Method ◽  
Nonlinear Partial Differential Equations ◽  
Partial Differential ◽  
Grid Points ◽  
Scientific Phenomena

In this paper, we apply wavelet optimized finite difference method to solve modified Camassa–Holm and modified Degasperis–Procesi equations. The method is based on Daubechies wavelet with finite difference method on an arbitrary grid. The wavelet is used at regular intervals to adaptively select the grid points according to the local behaviour of the solution. The purpose of wavelet-based numerical methods for solving linear or nonlinear partial differential equations is to develop adaptive schemes, in order to achieve accuracy and computational efficiency. Since most of physical and scientific phenomena are modeled by nonlinear partial differential equations, but it is difficult to handle nonlinear partial differential equations analytically. So we need approximate solution to solve these type of partial differential equation. Numerical results are presented for approximating modified Camassa–Holm and modified Degasperis–Procesi equations, which demonstrate the advantages of this method.

The boundary layer on a fixed sphere on the axis of an unbounded rotating fluid

1982 ◽  
Vol 123 ◽  
pp. 219-236 ◽  
Author(s):  
S. C. R. Dennis ◽  
D. B. Ingham
Keyword(s):  
Boundary Layer ◽  
Steady State ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Numerical Procedure ◽  
Steady State Solution ◽  
State Solution ◽  
Partial Differential ◽  
Boundary Layer Equations ◽  
Finite Set

The problem of determining both the steady and unsteady axially symmetrical motion of a viscous incompressible fluid outside a fixed sphere when the fluid at large distances rotates as a solid body is considered. It is assumed that the Reynolds number for the motion is so large that the boundary-layer equations may be assumed to hold. The steady-state boundary-layer equations are solved using backward- forward differencing and the terminal solutions at the equator and the pole of the sphere are generatedas part ofthe numerical procedure. To check that this steady-state solution can be approached from an unsteady situation, the case of a sphere that is initially rotating with the same constant angular velocity as the fluid and is then impulsively brought to rest is investigated. I n this case the motion is governed by a coupled set of three nonlinear time-dependent partial differential equations, which are solved by employing the semi-analytical method of series truncation to reduce the number of independent variables by one and then solving by numerical methods a finite set of partial differential equations in one space variable and time. The physical properties of the flow are calculated as functions of the time and compared with the known solution at small times and the steady-state solution.

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 Analytical mathematical schemes: Circular rod grounded via transverse Poisson’s effect and extensive wave propagation on the surface of water

Open Physics ◽  
2020 ◽  
Vol 18 (1) ◽  
pp. 545-554
Author(s):  
Asghar Ali ◽  
Aly R. Seadawy ◽  
Dumitru Baleanu
Keyword(s):  
Wave Propagation ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Longitudinal Wave ◽  
Symbolic Computation ◽  
Boussinesq Equation ◽  
Nonlinear Partial Differential Equations ◽  
Wave Model ◽  
Simple Equation ◽  
Partial Differential

AbstractThis article scrutinizes the efficacy of analytical mathematical schemes, improved simple equation and exp(-\text{Ψ}(\xi ))-expansion techniques for solving the well-known nonlinear partial differential equations. A longitudinal wave model is used for the description of the dispersion in the circular rod grounded via transverse Poisson’s effect; similarly, the Boussinesq equation is used for extensive wave propagation on the surface of water. Many other such types of equations are also solved with these techniques. Hence, our methods appear easier and faster via symbolic computation.

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