Questions tagged [finite-differences]
A method in numerical analysis which consists of approximating the derivatives of a solution of an ordinary or a partial differential equation. This leads to the solution of a linear system.
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Difference scheme for time-reversed heat conduction equation
I am working on solving the time-reversed heat conduction equation(assuming a two-dimensional space with Dirichlet boundary conditions). I have implemented the finite difference method, using first-...
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Finite Difference For a Fourth Order PDE With Neumann Boundary Conditions.
I'm trying to implement a method I found for image decomposition that boils down to solving a PDE of fourth order. The equation in question is
$$
u_t = -\frac{1}{2\lambda}\Delta\left(\text{div}\left(\...
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Correctness of Python implementation of numerical difference method for Cahn-Hilliard equations
Currently I am working on trying to get the Cahn-Hilliard equation solved using finite differences. I'm unsure if I am going in the right direction with my code. I have been following section 6.1 of ...
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How to justify the way the ghost nodes are applied in Finite Difference Method?
Boundary problem consists of:
PDE which is fulfilled for the points inside some domain D, and
boundary conditions that apply to the points on the boundary.
In other words, the equation describes ...
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finite diffrence scheme in two dimension
for u solution of :
$$
\frac{\partial u}{\partial t}= \frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}+u(1-u), \quad t>0, \quad(x, y) \in \Omega$$
$$
u(x, y, 0)=u^0(x, y), \...
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Finite difference scheme for non-standart PDE (bicylindrical coords)
Mathematical formulation of the problem:
There is the Aifantis equation for the elasticity gradient
$$
\eta =\lambda \left( \operatorname{tr}\ \varepsilon \right)l+2G\varepsilon -c{{\nabla }^{2}}\...
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How to tell which of several possible asymptotic forms a numerical solution to an ODE is converging to
I've previously mentioned the ordinary differential equation
$$12P\left(f\left(x\right)\right)^3f''''\left(x\right)+12\left(3P-1\right)\left(f\left(x\right)\right)^2f'\left(x\right)f'''\left(x\right)+...
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What is the best algorithm to solve a 2D partial differential equation of helmholtz type with finite differences method?
I need to solve numerically the following partial differential equation, in the unknown $u(x,y)$:
$$\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}=C^2u,\ \ \ \ (1)$$
with $x\in[-...
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Symmetry of 2nd order partial difference operators
In multivariable calculus, we know that 2nd order partial derivative operators are symmetric according to the Clairaut's Theorem, which says that if a binary function $f(x, y)$ has continuous 2nd ...
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Application of Crank-Nicholson for PDE with constant term
I have the differential equation
$$
\lambda^2 \frac{\partial^2 V(x,t)}{\partial x^2}-\tau \frac{\partial V(x,t)}{\partial t}=CV(x,t)+D
$$
which I want to solve using the Crank-Nicholson method, ...
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Finite difference method: Difference in calculations
I'm solving the basic heat equation.
$$ \phi_t = c \phi_{xx}$$
which can be written in implicit FDM as follows:
$$
\frac{\phi^{n+1}_i - \phi^n_i}{\Delta t} = c \left(\frac{\phi^{n+1}_{i+1} - 2\phi^{n+...
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Appendix 1 of "A Mathematical Theory of Communication"
In Appendix 1 of "A Mathematical Theory of Communication", Shannon states:
Let $N_i(L)$ be the number of blocks of symbols of length $L$ ending in state $i$. Then we have
$$N_j(L)=\sum_{i,...
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General solution a population growth difference equation
Hi guys I've been struggling with this for two weeks I finally admitted defeat and am looking for help.
I need to find the general solutions to the following difference
$$
\left\{
\begin{array}{l}
U(n+...
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While solving Wave equation using FDM, how to solve and related eigenvalues
$\ t>0, x\in (0,\pi)$
\begin{cases}
u_{tt}=u_{xx}\\
u(t,0) = u(t,\pi)=0\\
u(0,x)= exp(-32(x-(\pi/2))^2\\
u_t(0,x)=0
\end{cases}
We assume that sufficient smoothness in $u$ and using
\begin{cases}
...
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Application of boundary condition finite difference scheme
I am solving a version of the Laplace equation on a square ($a<x<b$, $0<y<h$) grid using finite differences.
I have an analytical solution to my problem so I can easily check the ...
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