Monotone Finite Difference Schemes for Quasilinear Parabolic Problems with Mixed Boundary Conditions

2016 ◽  
Vol 16 (2) ◽  
pp. 231-243
Author(s):  
Francisco José Gaspar ◽  
Francisco Javier Lisbona ◽  
Piotr P. Matus ◽  
Vo Thi Kim Tuyen
Keyword(s):  
Finite Difference ◽  
Boundary Conditions ◽  
Mixed Boundary ◽  
Finite Difference Methods ◽  
Neumann Boundary ◽  
Numerical Approach ◽  
Parabolic Problems ◽  
Finite Difference Schemes ◽  
Quasilinear Parabolic Problems ◽  
Quasilinear Parabolic

AbstractIn this paper, we consider finite difference methods for two-dimensional quasilinear parabolic problems with mixed Dirichlet–Neumann boundary conditions. Some strong two-side estimates for the difference solution are provided and convergence results in the discrete norm are proved. Numerical examples illustrate the good performance of the proposed numerical approach.

scholarly journals Blow-Up Solutions and Global Existence for Quasilinear Parabolic Problems with Robin Boundary Conditions

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Juntang Ding
Keyword(s):  
Boundary Conditions ◽  
Global Solution ◽  
Blow Up ◽  
Sufficient Conditions ◽  
Parabolic Problems ◽  
Robin Boundary Conditions ◽  
Quasilinear Parabolic Problems ◽  
Robin Boundary ◽  
Upper Estimate ◽  
Quasilinear Parabolic

We study the blow-up and global solutions for a class of quasilinear parabolic problems with Robin boundary conditions. By constructing auxiliary functions and using maximum principles, the sufficient conditions for the existence of blow-up solution, an upper bound for the “blow-up time,” an upper estimate of the “blow-up rate,” the sufficient conditions for the existence of global solution, and an upper estimate of the global solution are specified.

Biharmonic navigation using radial basis functions

Robotica ◽  
2021 ◽  
pp. 1-12
Author(s):  
Xu-Qian Fan ◽  
Wenyong Gong
Keyword(s):  
Finite Difference ◽  
Boundary Conditions ◽  
Radial Basis Functions ◽  
Real World ◽  
Biharmonic Equation ◽  
Finite Difference Methods ◽  
Basis Functions ◽  
Large Size ◽  
Radial Basis ◽  
Difference Methods

Abstract Path planning has been widely investigated by many researchers and engineers for its extensive applications in the real world. In this paper, a biharmonic radial basis potential function (BRBPF) representation is proposed to construct navigation fields in 2D maps with obstacles, and it therefore can guide and design a path joining given start and goal positions with obstacle avoidance. We construct BRBPF by solving a biharmonic equation associated with distance-related boundary conditions using radial basis functions (RBFs). In this way, invalid gradients calculated by finite difference methods in large size grids can be preventable. Furthermore, paths constructed by BRBPF are smoother than paths constructed by harmonic potential functions and other methods, and plenty of experimental results demonstrate that the proposed method is valid and effective.

scholarly journals Reliable Efficient Difference Methods for Random Heterogeneous Diffusion Reaction Models with a Finite Degree of Randomness

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 206
Author(s):  
María Consuelo Casabán ◽  
Rafael Company ◽  
Lucas Jódar
Keyword(s):  
Stochastic Process ◽  
Finite Difference ◽  
Coming Out ◽  
Finite Difference Methods ◽  
Diffusion Reaction ◽  
Finite Difference Schemes ◽  
Finite Degree ◽  
Mean Square Stability ◽  
Reaction Models ◽  
Difference Methods

This paper deals with the search for reliable efficient finite difference methods for the numerical solution of random heterogeneous diffusion reaction models with a finite degree of randomness. Efficiency appeals to the computational challenge in the random framework that requires not only the approximating stochastic process solution but also its expectation and variance. After studying positivity and conditional random mean square stability, the computation of the expectation and variance of the approximating stochastic process is not performed directly but through using a set of sampling finite difference schemes coming out by taking realizations of the random scheme and using Monte Carlo technique. Thus, the storage accumulation of symbolic expressions collapsing the approach is avoided keeping reliability. Results are simulated and a procedure for the numerical computation is given.

scholarly journals A Laplace transform finite difference scheme for the Fisher-KPP equation

2021 ◽  
Vol 15 ◽  
pp. 174830262199958
Author(s):  
Colin L Defreitas ◽  
Steve J Kane
Keyword(s):  
Finite Difference ◽  
Laplace Transform ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Finite Difference Methods ◽  
Reaction Diffusion ◽  
Numerical Approach ◽  
Travelling Wave ◽  
Reaction Diffusion Equation ◽  
Kpp Equation

This paper proposes a numerical approach to the solution of the Fisher-KPP reaction-diffusion equation in which the space variable is developed using a purely finite difference scheme and the time development is obtained using a hybrid Laplace Transform Finite Difference Method (LTFDM). The travelling wave solutions usually associated with the Fisher-KPP equation are, in general, not deemed suitable for treatment using Fourier or Laplace transform numerical methods. However, we were able to obtain accurate results when some degree of time discretisation is inbuilt into the process. While this means that the advantage of using the Laplace transform to obtain solutions for any time t is not fully exploited, the method does allow for considerably larger time steps than is otherwise possible for finite-difference methods.

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