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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11 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
13 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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20 views

Solving Hyperbolic PDE using MATLAB (finite difference Method)

I am given the PDE $u_t + u_x = 0, x \in (-2,3), t > 0$ with initial condition $ u(x,0) = 1 - |x|$ when $|x| \leq 1$ and $u(x,0) = 0$ when $|x| \geq 1$. With boundary condition $u(-2,t) = 0$. How ...
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19 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 : ...
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1answer
26 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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25 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 ...
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39 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: $$ ...
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7 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
60 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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15 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
16 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
16 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 ...
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0answers
11 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 ...
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16 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 ...
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0answers
9 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 + ...
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14 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 ...
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1answer
28 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 ...
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10 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} ...
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23 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 ...
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2answers
47 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?
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3answers
86 views

What is the meaning of delta squared?

I know that $\Delta x = x_2 - x_1$. This is well known Therefore, it follows that $\frac{\Delta a}{\Delta b}$ = $\frac{a_2 - a_1}{b_2 - b_1}$ (Unless I missed an elementary math class!) Now, I found ...
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3answers
48 views

Showing there exists a solution to $x \ln{x} = 1$ in $x \in [1.5, 2]$

This was given as a bonus question on my last quiz, and I couldn't figure it out. We just learned about Newton's method and finite differentiation methods but I don't see how any of that applies to ...
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19 views

Explicit finite difference scheme for discontinues media

I am looking for an explicit finite difference scheme for modelling a transverse wave propagation in discontinues 2D media. Let me pose the problem (little bit simplified one) $$ \begin{cases} ...
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1answer
48 views

Can Poisson equation be solved numerically in one shot?

I am trying to solve the 2-D Poisson heat equation, i.e. $\nabla^2T = C$, with boundary conditions (temperatures) for 3 surfaces, and the north surface is insulated. So I created an $N \times N$ ...
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18 views

How to write $ u(x,t) $ in Forward Time Central Space scheme?

If I want to apply the FTCS scheme to the following equation: $$ \frac{\partial u(x, t)}{\partial t} = f(x)u(x, t) +A\frac{\partial u(x, t)}{\partial x}+B\frac{\partial^{2}u(x, t)}{\partial x^{2}} $$ ...
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10 views

Solve transport of intensity equation using finite difference or finite element

I am trying to solve transport of intensity equation. It is in such following format. Transport of Intensity Eq Where I is intensity matrix and phi is phase matrix. I want to use either Finite ...
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1answer
21 views

What are the initial and boundary conditions for this problem?

I'm trying to solve a heat diffusion problem (with finite difference approximation) with conditions stated like this: A 10cm thin plate is initially at $120^{\circ}C$ then suddenly the right ...
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1answer
40 views

How to solve a second order differential equation with numerical approximation?

This is a homework question, but I'm also asking for self-teaching. So I hope it's OK :) In the question, I am given a 2nd order ODE in the form $y'' = 4$ with two initial (or boundary?) conditions ...
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1answer
26 views

Three point numerical differentiation

Is there any generalized way to calculate numerical differentiation using a certain number of points? I have found 2-point and 5-point methods, but could not find information about using any other ...
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1answer
102 views

Find the approximation for the interpolation of $f(x)$ by a polynomial of second degree

Assume that $f(x)$ has a minimum in the interval $x_{n-1}\leq x\leq x_{n+1}$ where $x_k=x_0+kh$, $k$ being an integer. Show that the interpolation of $f(x)$ by a polynomial of second degree yields ...
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21 views

Is it mandatory to use the forward and the backward interpolation formula

Is it mandatory to use the forward and the backward interpolation formula in order to interpolate a function at the respective locations towards the beginning and at the end of the given data set? ...
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6 views

Numerical accuracy vs numerical diffusion

My question: What is the difference between numerical accuracy and numerical diffusion (or dispersion)? My understanding so far: Numerical accuracy is how fast do the derivatives calculated converge ...
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1answer
41 views

Second order one-sided finite difference approximation to a partial derivative

For the computational grid as in following image, given are values of a variable, let's call it "v", on grid points marked by blue dots: I need to calculate the second order approximation of the ...
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27 views

Method of Modified Equations: Wave equation of second order

How do you derive the modified wave equation of second order. I know how to do it for the transport equation or other first order PDE in time but don't know for second order... I get stuck at the ...
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1answer
29 views

Numerical derivative of compoiste function

sorry for the very basic question. I am writing a Fortran program in which I have a quite complicated function in a non-linear system of equations and I need to differentiate it numerically in order ...
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4answers
97 views

Find $f(x)$ if $\Delta f(x)=e^x$

Find $f(x)$ if $\Delta f(x)=e^x$, where $\Delta f(x)$ is the first order forward difference of $f(x)$, step size $=h=1$. Attempt: We have the definition $\Delta f(x)=f(x+h)-f(x)=f(x+1)-f(x)$ ...
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1answer
20 views

Finite difference method - Why does the relation hold?

We consider the finite difference method for the approximation $\left\{\begin{matrix} -u''(x)+q(x)u(x)=f(x)\\ u'(a)=u'(b)=0 \end{matrix}\right.$ and let $K$ be the $(N+2) \times (N+2)$ matrix of ...
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19 views

Finite difference method to solve first-order, multivariable system of equations

I'm trying to replicate the model presented in this paper, which is basically to model heat and mass transfer along a one-dimensional duct. There are four characteristic equations for this problem : ...
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29 views

About the Mueller's (summation) formula?

Let $f$ be a real function with $\lim_{n\rightarrow\infty}f(x)=0$. (1) What is the necessary conditions for the Mueller's formula: $$ \sum_{x}f(x)=\sum_{n=1}^\infty(f(n)-f(n+x))+C ? $$ (2) What is ...
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32 views

Upper bound on a sequence after a finite number of steps

i need to find an upper bound (as tightest as possible) of the following recurrence for $\mu^{(j)}$ $$ \left\{ \begin{array}{l} t^{(j-1)} = 2 \mu^{(j-1)} - \nu & j \geq 1\\ \mu^{(j)} = ...
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0answers
19 views

Finite difference for nonlinear PDE

I'm trying to solve a complicated PDE for a function $h(x, y,t)$ with mixed derivatives and also a term involving $(h_y)^2$. Just to try and get my basics right first, suppose we consider ( for ...
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27 views

Discrete maximum principle for finite difference to the heat equation

Given a continuous-time, discrete space approximation of $u_t = u_{xx}$ with homogeneous Dirichlet boundary condition. The discrete ODE version for the nodal values is: $u'_j = \frac{u_{j-1}+u_{j+1} - ...
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64 views

Periodic Boundary Conditions 5 Point Laplacian Finite Differences

Short Question As I write this, I realize this is related to: Discretization of Inhomogeneous Dirichlet Boundary Conditions for 2D Poisson's Equations My question can then be summarized as: Do I ...
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0answers
40 views

Square root digit recurrence algorithm

I'm reading a book which expose how to compute a square root using the digit recurrence algorithm. Following the book basically it expose a simple case with each step necessary to compute such square ...
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0answers
11 views

Z transform treating difference equation with boundary condition fixed.

Is there a way to treat a difference equation with boundary condition using the Z transform? In a handbook of mathematics i've found a huge formula, but there's no derivation and i don't understand ...
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1answer
40 views

An error bound for a finite difference approximation to the second derivative

I want to show that $$|\delta_{h,r}f(x)- f''(x)| \leq \frac{11}{12} h^2 \|f^{(4)}\|_{\infty}$$ where $$\delta_{h,r}f(x)=\frac{1}{h^2} (2f(x)-5f(x+h)+4f(x+2h)-f(x+3h))$$ I applied the Taylor expanson ...
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23 views

Ordinary Differential Equation using finite difference

How do I solve the following question? Solve the following ordinary differential equation (ODE) using finite difference with $\Delta ...
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33 views

Represent Dirac Delta function in Finite Difference method

I recently solving $-\Delta u=\delta$ where $\delta$ is dirac delta function using FDM on 2 dimensional space. Since dirac delta function is undefined at origin, and 0 elsewhere, I will use ...
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19 views

Boundary difference equation, monotonicity of the solution properties.

let's say we have a function $f \in C^{\infty}[a,b]$ such that $f,f',f'' > 0 \forall x \in [a,b]$ What i would like to prove is that the solution of this boundary problem $$\left\{ ...
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13 views

obtaining difference equation of ODE

I have an ODE with the following: y''(t) + y'(t) + y(t) = u'(t) + u(t) [1] I am trying to convert this to a difference equation. I am using ...