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.

learn more… | top users | synonyms

0
votes
0answers
11 views

How to solve this mixed pde/finite-difference equation?

I have the following mixed pde/finite-difference equation for $f(t,x,y)$: $a x^2 f_{xx} + bxf_x + f_t - bxy + ce^{d\delta}-re^{-s\delta} = 0$ subject to $f(T,x,y)=0$, $x>0,\ t\geq 0,\ ...
0
votes
0answers
17 views

What is the discretization matrix of 2D Poisson equation of finite diffence with checkerboard (black and red) pattern?

Given the problem$-\Delta u(x,y)=f(x,y)$ on unit rectangle $\Omega=[0,1]^{2}$ and $u(x,y)=g(x,y)$ on $\partial\Omega$, what is the finite difference matrix associated with step size $h=1/(2N+1)$ where ...
3
votes
1answer
24 views

Explicit Finite Difference Scheme For Approxating a p.d.e

$\frac{du}{dt} = \frac{d}{dx}[\frac{1}{x^2+1}\frac{du}{dx}]$ I am trying to approximate this pde with a finite difference scheme but I am confused with the d/dx. Do I just take the derivative of ...
1
vote
0answers
17 views

Finite difference method and division by zero problem with no flux boundary condition

I am trying to implement an angionesis model described by Anderson and Chaplin in 1998. The model is based on a set of PDEs defined on an unit square with the following no-flux boundary condition ...
1
vote
0answers
35 views

Finite difference method for DEq's of form dy/dx = f(x,y)

I was watching a math tutorial on the finite difference method that stated that the method could be used for ODEs of the form: $dy/dx = f(x) + g(y)$ but not for equations where the function is a ...
2
votes
2answers
63 views

Finite Difference Approximation of Derivative [closed]

I want to build a finite-difference approximation of this derivative: $\frac{\partial^2T }{\partial x^2}$ There are given an error of approximation: $O(\Delta x^{4})$ and nodal values of function:$ ...
0
votes
2answers
47 views

Solving $u'' - 5u = 6$ with finite difference methods.

I have a task: For an equation: $$u'' - 5u = 6, x \in (0, 1)$$ $$u(0) = 0, u'(1) - 3u(1) = 1$$ construct a recurrence relation("scheme" in the original) with second order approximation on a two-point ...
1
vote
0answers
35 views

Using Finite Differences and Integration to prove result

If $f(x)$ is a polynomial in $x$ of third degree and: $$u_{-1}=\int_{-3}^{-1}f(x)dx\ ;\ u_{0}=\int_{-1}^{1}f(x)dx\ ; u_{1}=\int_{1}^{3}f(x)dx$$ then show that $$f(0) = ...
1
vote
0answers
14 views

How to build a matrix in MATLAB with the next characteristics?

Let $\lambda_1=\frac{k D_u}{2h^2}$ a constant value. How to generate a matrix in MATLAB with the next entries: $A= \begin{pmatrix} 1+\lambda_1 & -\lambda_1 & 0 & 0 & \cdots & 0 ...
0
votes
0answers
23 views

Finite difference of radial Laplace operator doesn't give a symmetric (hermition in general) matrix

I'm using the central difference to convert the radial part of Laplace operator into a matrix. $\nabla^2 u = \frac{\partial^2 u}{\partial r^2}+$ $\frac{1}{r}$ $\frac{\partial u}{\partial r}$ which ...
1
vote
1answer
36 views

A finite difference

I read that the $n$-th finite difference of the sequence $1^n, 2^n, 3^n,\dots$ is $n!$, but I'm not able to prove this. Could someone give an idea of why this is true?
0
votes
0answers
11 views

Stability problem for finite difference scheme

u (x,y,a,t) a analytical solution of a PDE $$ u_t+u_a = \nabla_{x,y} \cdot \left(d(x,y,a,t)\nabla_{x,y}u\right) -\mu\left(x,y,a,t\right)u $$ $ u : \left[0,1\right]^2 \times \left[0,A\right] \times ...
0
votes
1answer
22 views

Calculate $(\Delta^2+\Delta-2)^{-1} (n^3+1)$

Is this part correct: $$(\Delta^2+\Delta -2)^{-1} =\left( -2\left(I-\frac{\Delta^2+\Delta}{2}\right)\right)^{-1}=-\frac{1}{2} ...
1
vote
1answer
30 views

Why, for simple nth degree polnomials' finite difference tables, does the nth (constant) difference set, equal the nth derivative

For example with the equation $f(x)=x^4+2x^3+4x^2+2x+1$ the fourth derivative is $f''''(x)=24$ and when you construct a difference table the fourth difference is 24
0
votes
0answers
32 views

On the Identity matrix minus discrete laplacian

I am trying to figure out a way to bound the following expression $\| (I - \alpha^2 D)^{-1} \| $, where: # $\alpha>0$ # $I$ is the identity matrix $(n\times n)$ # $D$ is the matrix obtained by ...
0
votes
1answer
31 views

Richardson extrapolation - deriving methods for forward difference

I am rather stuck on a question from a textbook that I am practicing from which goes as follows: The forward difference formula can be expressed as $$f'(x_{0}) = \frac{1}{h}[f(x_{0} + h) - f(x_{0})] ...
0
votes
2answers
19 views

Show that there is an equilibrium point

Show that if $x_{k+1}=f(x_{k})$, where $f$ is continuous, has two stable equilibrium points, then there's a third equilibrium point between then. I'm trying to approach this problem using the ...
1
vote
1answer
23 views

Neumann boundary conditions at corner of rectangular domain

If we have a rectangular domain $\Omega$ and we are approximating the derivative of $u(x,y)$ by a finite difference $$u_{xy} \approx \frac{(u_{i+1,j+1} + u_{i-1,j-1} - u_{i+1,j-1} - ...
0
votes
3answers
41 views

Most accurate finite difference result for first derivative [closed]

I got a problem in my assignment: obtain the most accurate finite difference results possible for the first derivative of f (x) = exp(cos(x)) at x=1, h = 0.5, 0.25, 0.125,...2^{16}. I have to do this ...
1
vote
0answers
24 views

Coupled linear PDE equations (2nd and 1st order) - Numerical Method

I am trying to solve a coupled reaction-diffusion equations, using Crank Nicolson (implicit Finite Differences Method). I know how to solve them separately, but not simultaneously. Coupled PDE's ...
1
vote
1answer
50 views

Numerically Solving a Poisson Equation with Neumann Boundary Conditions

The Problem Suppose I have an equation of the form $\nabla^2 \phi(x) = f(x)$ on the interval $A \le x \le B$, where $f(x)$ is known and $\phi(x)$ is unknown. I have Neumann-type boundary conditions: ...
1
vote
0answers
12 views

Implicit scheme for a nonlinear PDE system

I have the next PDE system: $u_t=D_u u_{xx}-u+av+u^2v\\ v_t=D_v v_{xx}+b-av-u^2v$ Where $D_u,D_v,a,b$ are constants. I want to build or find a implicit SECOND ORDER scheme for this PDE system. I ...
4
votes
0answers
119 views

Implicite finite differences in two dimensions

Consider the following PDE \begin{align} V(x,y) = F(x,y) + \frac{\partial V}{\partial x} f(x) + \frac{\partial V}{\partial y} g(y) \end{align} where $x$ and $y$ are states, $V$ is the value ...
3
votes
2answers
59 views

Difference operator: Proof by induction that $\Delta^k (X_t)= k!a_k+\Delta^k (Y_t)$

Hello I am having issues with the following exercise. I have to prove that $$\Delta^k (X_t)= k!a_k+\Delta^k (Y_t)$$ where $X_t = m_t +Y_t=\sum_{j=0}^ka_jt^j+Y_t$ for $t \in \mathbb {Z}$. Note: ...
1
vote
0answers
16 views

Finite difference scheme in time and space for the Sel'kov model

I have the next nonlinear PDE system, and I want to apply a SECOND ORDER CENTRAL finite difference scheme in TIME and a SECOND ORDER CENTRAL finite difference scheme in SPACE: $u_t=D_u ...
1
vote
0answers
31 views

Finite difference second order central in time

Is it valid to approximate $u_t$ for a second order central finite difference?. I mean $u_t=\frac{u_m^{n-1}-2u_m^{n}+u_m^{n+1}}{k^2}$, does it approximate the first partial differential derivate of ...
2
votes
1answer
30 views

Is there any explcit methods to solve ''stiff'' coupled differential equations?

I am trying to solve coupled ordinary,partial differential equations, reaction diffusion equations with finite difference mehtod. But I found that these equations are 'stiff'. I searched that when I ...
0
votes
0answers
22 views

Stability for the FTCS method (PDE inside)

For the following PDE - I worked out the derivation of the FTCS method but I do not know how to construct the stability. I would really appreciate some help. Let me know if I need to add more info. ...
4
votes
0answers
93 views

Finite differences of power functions

I'm interested in finite differences, to be precise, finite differences $\Delta^n f(x)$, where $n \in \mathbb{N}$ and $f$ is a real function given by $f(x) = x^a$ for some $a \in \mathbb{R}$. I use ...
0
votes
4answers
46 views

Identify the misprint in a sequence

Given the sequence, 1,3,11,31,69,113,223,351,521,739. Identify the misprint. I tried reasoning out but couldn't. Do I have to do some numerical analysis like forward differences? Have no clue. Kindly ...
9
votes
2answers
128 views

Why isn't finite calculus more popular?

I'm reading through Concrete Math, and learning about finite calculus. I've never heard of it anywhere else, and a Google search found very few relevant sources. It seems to me an incredibly powerful ...
0
votes
0answers
20 views

Wide stencil for Second Derivative in finite difference - stability in maximum norm

I am given the problem $-u'' + a*u = f$. I already derived a 5-point wide stencil for finite difference with fourth order convergence, and then the matrix $A$ for the problem has a stencil like this: ...
0
votes
1answer
17 views

How to use the finite difference method to solve a transport equation with a source term?

I am going to use the finite difference method to solve a transport equation with a source term. In order to solve $u_t+u_x=0$, we can use $u_j^{n+1}=j_j^n-\lambda*(u_{j+1}^n-u_{j}^n)$, where ...
0
votes
0answers
22 views

Finite difference scheme for piecewise domain heat equation

We have piecewise heat equation $u_{t}(x,t)=\left\{\begin{matrix} D_{1}u_{xx}(x,t) &x\in(0,1) \\ D_{2}u_{xx}(x,t) & x\in (1,2)\end{matrix}\right.$, with IC: $u(x,0)=2x+1$ and BC : ...
2
votes
1answer
66 views

Probability that a biased asymmetric random walk reaches the origin

I am working on the following problem for my probability class and I am a little stuck: A particle moves at each step two units to the right or one unit to the left, with corresponding ...
1
vote
0answers
31 views

LTE for the Cahn Hilliard Equation

I'm trying to Calculate the LTE of a first order finite difference (central difference) scheme for the following 1D equation: $u_t = \frac{\partial^2}{\partial x^2}(u^3 - u - \frac{\partial^2 ...
1
vote
0answers
48 views

Finite difference discretisation of the heat equation

Here is the equation to be discretised: $$ k\left(\frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2}\right) = \dot{q} $$ Using the following discretisation scheme: $$ ...
0
votes
0answers
9 views

Reference for finite difference scheme for elliptic PDEs

Is there somewhere a gentle presentation for the numerical analysis of the finite difference method for elliptic PDEs. For instance in $\mathbb{R}^2$, $\Omega = (-L_x,L_x) \times (-L_y,L_y)$ for some ...
1
vote
1answer
66 views

a nonlinear difference equation limit

let $$a_1=1, a_{n+1} = a_n + \frac{1}{a_n^2}$$, and it seems $$\lim_{n\to\infty} (a_n^3-3n-\log n) = \text{C}(constant) $$ I use computer program to verify that $$C = 1.13525585...$$,and expecting ...
0
votes
0answers
24 views

Why is the forward difference operator $\Delta $ also called descending difference operator?

The forward difference operator $\Delta$ defined as: $$\Delta\ f(x) = f(x+h) - f(x)$$ is also referred to as the descending difference operator. Similarly, the backward difference operator ...
1
vote
1answer
18 views

Example of forward difference approaching derivative only up to $O(h)$

According to the article on Wikipedia about finite differences, the forward and backward difference by $h$ of a function $f(x)$ divided by $h$ approach the derivative to order $O(h)$, i.e. $$\frac ...
0
votes
1answer
28 views

Taylor Series from General Finite Difference Scheme

"For a 3-point stencil $[x_{i-1},x_{i+1}]$, we can write a generic expression as $\frac{\partial u}{\partial x}|_{x_i}=au_{i-1}+bu_i+cu_{i+1} + O(h^m)\qquad (1)$ where a,b, and c are unknowns to be ...
1
vote
0answers
15 views

How to adjust finite differencing method for mapping from $\mathbb{R}^{n} \rightarrow \mathbb{R}^{m}$ where $n = m^{2}$?

So I'm supposing $F:\mathbb{R}^{n} \rightarrow \mathbb{R}^{m}$ is differentiable, and I have MatLab code that evaluates $F$ at an arbitrary $x$ in $q$ flops. I know that given $F(\bar{x})$ where ...
0
votes
0answers
31 views

Stability of Lax-Friedrichs scheme by energy norm analysis

Considere the linear advection equation $$ u_t + a\,u_x = 0. $$ I want to proof the stability of Lax-Friedrichs method for this equation by the energy norm analysis. The Lax-Friedrichs scheme is given ...
0
votes
0answers
10 views

Determine the system of difference equation for $(k_t, p_t)$

Now I have $ f : \mathbf{R}_{+} \rightarrow \mathbf{R} $ be defined as $f(k_{t}) = \frac{k_{t}}{\alpha + (1-\alpha)k_{t}}$ $ p_{t+1} = \frac{r}{n}p_{t} + \frac{\beta}{n}k_t + ...
0
votes
0answers
16 views

Finite difference along a line of $xyz$ coordinates

I have a line of $x,y,z$ coordinates. I'm looking at using finite differences in order to find local maxima (of $z$)..i.e. 'hills'. Am I right in thinking that I can use the second derivative? I have ...
1
vote
1answer
32 views

accurate vibration analysis (finding eigenvalues) of a large, sparse, non-symmetric matrix

I have a large, sparse, non-symmetric matrix $M$, and I need to get accurate eigenvalues $\lambda_i$ and eigenvectors $\vec{s}_i$ for all $i$. (An example system is below; it's based on a ...
0
votes
0answers
20 views

Obtaining the weak form of Allen-Cahn equation

\begin{equation} \frac{\partial\phi(\mathbf{x},t)}{\partial t}=g(\mathbf{x})(\varepsilon^{2}\Delta\phi-F^{'}(\phi)),\ \ \ \mathbf{x}\in \Omega,t>0\ \ (*) \end{equation} \begin{equation} ...
1
vote
0answers
53 views

Forward-time, centered space evalaution of the heat equation: numerical stability and unique solution

I have a script of code which models a planetesimal that is accreted into a planetary atmosphere. In the code, I include the physics of frictional ablation and thermal ablation. Frictional ablation is ...
0
votes
2answers
56 views

Where did the Finite Difference Coefficients come from?

Finite Difference Coefficients allows one to estimate various derivatives. My question is, where did they come from? How do you derive the finite difference coefficients?