# Tagged Questions

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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### 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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### 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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### 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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u (x,y,a,t) a analytical solution of a PDE $$u_t+u_a = \nabla_{x,y} \cdot \left(d(x,y,a,t)\nabla_{x,y}u\right) -\mu\left(x,y,a,t\right)u$$ $u : \left[0,1\right]^2 \times \left[0,A\right] \times \... 1answer 22 views ### Calculate$(\Delta^2+\Delta-2)^{-1} (n^3+1)$Is this part correct: $$(\Delta^2+\Delta -2)^{-1} =\left( -2\left(I-\frac{\Delta^2+\Delta}{2}\right)\right)^{-1}=-\frac{1}{2} \left(I+\frac{\Delta^2+\Delta}{2}+\left(\frac{\Delta^2+\Delta}{2}\right)^... 1answer 33 views ### Why, for simple nth degree polnomials' finite difference tables, does the nth (constant) difference set, equal the nth derivative For example with the equation f(x)=x^4+2x^3+4x^2+2x+1 the fourth derivative is f''''(x)=24 and when you construct a difference table the fourth difference is 24 0answers 36 views ### On the Identity matrix minus discrete laplacian I am trying to figure out a way to bound the following expression \| (I - \alpha^2 D)^{-1} \| , where: # \alpha>0 # I is the identity matrix (n\times n) # D is the matrix obtained by ... 1answer 33 views ### Richardson extrapolation - deriving methods for forward difference I am rather stuck on a question from a textbook that I am practicing from which goes as follows: The forward difference formula can be expressed as$$f'(x_{0}) = \frac{1}{h}[f(x_{0} + h) - f(x_{0})] ... 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 ... 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})}{... 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 ... 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 ... 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: ... 0answers 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 ... 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,... 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: \... 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^... 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 ... 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 ... 0answers 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. ... 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 ... 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 ... 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 ... 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: ... 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 \... 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,... 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 ... 0answers 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}{\... 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{\... 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)$for some$...
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 ...