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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Numerically Solving a 3d PDE with Stochastic Terms

I'm getting a bit confused if the procedure I'm doing is correct so any feedback would be great! It's just a standard deterministic PDE for the price of a theoretic option, even if it's quite a ...
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0answers
21 views

Question on the solution to obtaining the truncation error for the Crank-Nicholson finite-difference scheme

I'm working on an exercise in a textbook that asks to derive the local truncation error for the Crank-Nicholson finite-difference scheme at the point $(ih, jk)$ for the partial differential equation $\...
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0answers
11 views

What are the different ways in which I can find Lipschitz constant for:

What are the different ways in which I can find Lipschitz constant for $$|| \bigtriangledown(X)||_F^2$$ where $$|| \bigtriangledown(X)||_F^2 = \sum_{i,j}|| \bigtriangledown(X)_{i,j})||_F^2$$ and $\...
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1answer
20 views

Analytical Solution of 3D Heat Equation - FDM

I'm writing a simple FDM algorithm for solving the well known 3D heat equation $$ \frac{\partial u}{\partial t} = \alpha \nabla^2 u + \frac{q}{c_p \rho} $$ where $q(x,y,z,t)$ represents the ...
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0answers
15 views

finite difference time domain grid question

The finite difference time domain method is a finite difference method for solving maxwell equations numerically. There are several pieces to it, but this is the root of my question $H_{i +1/2 , j+1/...
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0answers
13 views

Numerical scheme and boundary condition for 2D Fokker Planck equation

$\newcommand{\P}{\mathbb{P}}$ I have a 2D stationary Fokker-Planck equation $$\frac{\partial^2 \P(A,B)}{\partial A^2}+\frac{\partial^2 \P(A,B)}{\partial B^2}=\frac{\partial f_1(A,B) \P(A,B)}{\partial ...
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27 views

Heat Equation 1D in Cylindrical Region

I have heat equation 1D in cylindrical coordinates: $$u_{\rho\rho}+\frac{1}{\rho}u_{\rho}=u_{t},\;0<\rho<1,t>0 $$ with boundary conditions $u(1,t)=0,\;t>0$ and initial condition $u(\rho,0)=...
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20 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 ...
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1answer
27 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/(x^...
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0answers
26 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 ...
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38 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 ...
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2answers
74 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:$ ...
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2answers
52 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 ...
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0answers
36 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) = \frac{1}{2}\Bigg(u_0-\frac{\...
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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 &...
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26 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 ...
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1answer
38 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?
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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 \...
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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} \left(I+\frac{\Delta^2+\Delta}{2}+\left(\frac{\Delta^2+\Delta}{2}\right)^...
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1answer
33 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
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36 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 ...
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1answer
33 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})] ...
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2answers
25 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 ...
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1answer
34 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} - u_{i-1,j+1})}{...
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3answers
43 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 ...
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0answers
27 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 ...
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1answer
73 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: $...
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14 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 ...
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1answer
127 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 function,...
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2answers
61 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: $\...
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0answers
18 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 u_{xx}-u+av+u^...
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0answers
32 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 $...
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1answer
32 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 ...
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24 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. ...
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0answers
105 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 ...
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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 ...
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2answers
134 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 ...
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0answers
23 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: ...
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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 $\...
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0answers
23 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 : $u(0,...
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1answer
73 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 ...
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33 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 u}{\...
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0answers
49 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: $$ \frac{\...
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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 $...
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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 ...
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0answers
27 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 $\...
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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 {\...
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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
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0answers
16 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
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0answers
32 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 ...