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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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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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Python code for Second Order ODE Initial Value problem using Finite Difference methods

I am trying to solve the following ODE $$\begin{aligned} & y^{\prime \prime}+2 y^{\prime}+y=0 \\ & y(0)=2 \quad y^{\prime}(0)=-1 \\ & 0 \leqslant x \leqslant 1 \end{aligned}$$ Using ...
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finite difference method for reaction diffusion equation

I am implementing a finite difference scheme for the following PDE: $$ u_t=u_{xx}+f(u) $$ with the nonflux boundary condition and a given initial condition. The scheme is backward in time and central ...
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why can the result of the four point finite difference formula of a function, be greater than the exact derivative of the function?

if i have a function like $f(x) = e^{\sqrt{x+\frac{2}{x}}}$. why is the result of using the formula below, where $x$ is $2$ and $h$ is $0.1$, greater than the exact derivative of $f(x)$?
conrad szyman's user avatar
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Second order central difference formula with difference step sizes

The well-known equation $$u''(x)= \frac{u(x+h) - 2 u(x) + u(x-h)}{h^2}+\mathcal{O}(h^2)$$ is derived by adding $$u(x+h)= u(x)+u'(x)h+\frac{1}{2}u''(x)h^2+\mathcal{O}(h^2)$$ $$u(x-h)= u(x)-u'(x)h+\frac{...
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Artificial Smoothing, Diffusion and other effects of numerical Schemes

I am looking for an explanation for the terminology "artificial smoothing" and "artificial diffusion" in the context of numerical schemes. For instance, consider the upwind scheme ...
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Unexpected result in finite difference solution beam deflection problem

I try to numerically solve (by finite difference method) the beam deflection equation $$EJ\cdot\frac{\partial^4 v}{\partial x^4} = -g\cdot F\cdot\rho$$ with following boundary conditions $$v(0)=0;\...
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Constructing the matrices in finite difference method

I am trying to solve the following PDE using finite difference: \begin{aligned} \frac{\partial}{\partial t} u(x, t)-\frac{\partial^2}{\partial x^2} u(x, t) & =x, \quad x \in(0,1), \quad t \in(0,1),...
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Kronecker product and finite difference discretization for poisson equation

In this notebook from MIT's Intro to Linear PDEs course, it is unclear to me why the Kronecker product is used to formulate the coefficients matrix $A$ for solving the linear system of equations $A u =...
Jared Frazier's user avatar
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How do we find the weights for finite differences?

My professor gave us this question to solve, but I don't know have much familiarity with the topic. Question The forward difference is first order accurate and is defined to be $$ D_{+} = \frac{f(x+h) ...
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Convergence of "partial" PDE finite difference scheme

Suppose I have a PDE $$ \frac\partial{\partial t}f(x,t) = F\left(f(x,t), \frac\partial{\partial x} f(x,t)\right). $$ I would like to discretize $f(x,t)$ as $$f^\epsilon(x,t) = \sum_{n=0}^{N_\epsilon} ...
Thorstein's user avatar
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Final value of a recursion

Problem Given $p_1, \sigma > 0$, consider the following recursion \begin{equation*} p_{i}=(1-L_i)p_{i-1} \qquad i=2,\dots,k \end{equation*} where \begin{equation*} L_i \triangleq \frac{p_{i-1}}{p_{...
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Numerical Analysis of Differential Equation with Boundary Conditions

Note I have decided to edit this post so it is more specific and also because I was incorrect the first time. I didn't wanna make another post about the same subject, but if this is not allowed then ...
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Numerical method for a free boundary problem

I am consider the following free boundary problem: $u_t + \mathcal{L}u = 0, 0<x<s(t)$, with the boundary conditions: $u(T,x) = f_T(x)$, $u(t,s(t)) = g(t)$ and $u_x(t,s(t))=C$. Here, $\mathcal{L}...
Kenneth Ng's user avatar
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Convergence of Finite Difference Scheme for Helmholtz equation

Consider the 2-dimensional Helmholtz equation $$-\Delta u-k^2 u= f.$$ I'm trying to proof convergence of the standard finite difference scheme (using the 5-point-stencil for the Laplacian). I have ...
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Spectrum of discretized Laplace operator with homogeneous Neumann B.C.s

Consider the Laplace operator over a 1D domain with homogeneous Neumann B.C.s. The cell-centered Finite Volumes discretization of this operator on a uniform grid looks as follows, which is a well-...
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How to solve this coupled PDE eigenvalue problem numericallly?

I'm solving a system of two coupled partial integro-differential equations for two functions $ {\phi _0^\alpha (R,r')} $ and $ {\phi _0^\beta (R,r)} $: $$ \frac{{{\partial ^2}\phi _0^\alpha }}{{\...
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Do higher order differences within explicit schemes negatively effect stability?

I was reading the paragraph in section E.4 of this paper, where the authors discuss adding longer histories in neural/numerical PDE solvers and claim that "using higher-order differences [in time]...
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finite difference estimator of the derivative of the inverse distribution function

let $x_1,...,x_n$, $n\geq 3$ be a sample form the distribution $F$. We want to estimate $\frac{dF^{-1}(p)}{dp}$, replacing the distribution function $F$ by empirical distribution function $F_n$, and ...
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Expasion of Second Order Difference

I'm currently exploring a problem related to a two-parameter estimator, and I've simplified it for ease of discussion, eliminating the need for an extensive probability background. The problem is as ...
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How does local and global error relate for finite difference methods?

When solving a (one-dimensional linear) boundary value problem by finite differences, how does the local and global error relate? I would have expected that if the local error was $O(h^k)$ in the step ...
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Understanding a formula for the solution to an inhomogeneous linear difference equation

My question comes from section 8.4 of Analysis of Numerical Methods by Isaacson and Keller. I am trying to get a better understanding of the formula stated in Theorem 3, p.409. First, some definitions ...
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Ensuring Symmetry in Mixed Derivatives Using RBF-FD Method

Hello Mathematics Stack Exchange Community, I'm working on a numerical problem where I have the first-order partial derivatives $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ of a ...
Rule's user avatar
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How best to generalize finite difference Laplacian matrix from 1D to 2D (and beyond)

To numerically solve the Poisson equation in one dimension we might transform the problem into a linear system $A u = b$, where $A$ is a finite difference approximation of the Laplace operator, say $$ ...
ummg's user avatar
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Did I get Godunov's scheme right?

I want to implement Godunov's scheme in order to simulate the nonlinear LWR-type equation $$ \partial_t u + \partial_x (u(1-u)) = 0, \quad u(0, \cdot) = u_0. $$ The update step is ($n$ denotes time ...
Hyperbolic PDE friend's user avatar
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Boundary Value Problems Solving with Central Difference

I have differential equation $u''(x)+9u(x) = \cos(2x), \; x \in [0, π/2]$ the boundary conditions are $u(0)=1$, $u(\pi / 2) = -1$, with $h = 0.2$ Now I replaced $u''(x)$ with centered difference ...
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Curve Fitting and Parameter Estimation Study

I have a problem with an experimental data analysis to obtain unknown parameters. The experimental data can be described by the equation: $y_k-y_{k-1}=\eta_k (x_k^{1-\alpha_k}-x_{k-1}^{1-\alpha_k})$ ...
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Numerical Partial Differential Equation

The solution of the difference diagram for some partial differential equations from the Fourier transform and Fourier analysis can be written in ${U_{n}}^{k}=q^{k}e^{in\xi}$ form. The condition for ...
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Relationship Between Shift Operator and Backward Difference Operator (Finite Differences)

Consider $z_1,...,z_n$ and let $E(z_k) = z_{k+1}$ be a forward shift operator and let $\Delta_-(z_k) = z_k - z_{k-1}$ be a backward difference operator. I'm trying to show that $E = \sum_{j=0}^{\infty}...
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Numerical Treatment of Dirac Delta in Finite Difference Scheme

I'm considering a particle in a spatial domain $\Omega \subset \mathbb{R}^2$ which secretes some chemical substance with production rate $q_0$, and is subject to a force proportional to the gradient ...
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Eigenvalue problem with boundary condition

I am playing with the one dimensional massless Dirac equation $$-i\partial_x \sigma_x \Psi(x) = \epsilon \Psi(x) \, .$$ Solving it analytically with some boundary conditions is an easy task. I am more ...
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How to derive this finite difference scheme?

My professor gave me the following discretization scheme to use to discretize the mass conservation equation $\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0$. I haven't seen this ...
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Finite Difference method, ADI Scheme of Douglas and Rachford

I am trying to implement the ADI scheme of Douglas and Rachford. For $p(X,Z,t)$, there is: $$ \begin{gathered} A=p^{n-1}+\Delta t_n\left[F_0\left(p^{n-1}, t_{n-1}\right)+F_1\left(p^{n-1}, t_{n-1}\...
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Finding second-order finite difference stencil

If you have the anisotropic diffusion equation to find $u(x,y)$ $$\left(\mu_{x}\frac{\partial^{2}u}{\partial x^{2}}+\mu_{y}\frac{\partial^{2}u}{\partial y^{2}}\right)=f(x,y) \tag 1$$ and you ...
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Can't get proper numerical convergence for complicated Advection-Diffusion-Reaction PDE

I have trying for quite some time to write a finite difference solver for the following advection-diffusion-reaction differential equation: $$ \frac{\partial C}{\partial x} = \frac{1}{u(z) + u_e(x,z)}\...
David G.'s user avatar
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Restructuring a Discrete Equation System for Compact Representation

I have the following system of equations for $i,j=1,...,n$: $u_{i-1,j}+u_{i+1,j}+u_{i,j-1}+u_{i,j+1}-4u_{i,j} = f_{i,j}$ The right hand side $f$ is known, as well as $u_{0,j},u_{n+1,j},u_{i,0},u_{i,n+...
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The real CFL condition for cylindrical laplacian

I've been exploring the CFL (Courant-Friedrichs-Lewy) condition in polar coordinates and have observed that previous inquiries haven't yielded a satisfactory answer. I've come across this paper which ...
Manuel Borra's user avatar
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Transform Lagrangian with a square root

I have an action given by, \begin{equation} S = \int^{\tau_f}_{\tau_0} d\tau \left(\frac{1}{z^3}\sqrt{-f(u,z) \dot{u}^2 + 2 \dot{u} \dot{z} + \dot{x}^2} + \frac{2 \dot{z}}{z^3} \right), \end{equation} ...
mathemania's user avatar
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Coupled non-linear PDE analytical solution to verify numerical solution.

as part of a computational module we were given the following coupled PDEs and solved them numerically using finite difference methods: I got the following graphs representing the numerical solution ...
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Regarding Crank-Nicolson Scheme functions on the LHS

Assuming I have a differential equation $$ \frac{\partial u}{\partial t} = \frac{1}{f(x,t)} F(u,x,t) $$ The Crank-Nicolson scheme would have the equation discretized as such $$ \Big[ \frac{\partial u}{...
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