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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Numerical computation of the $n^{\mathrm {th}}$ derivative of a multivariate function

From a multivariate function $f$, depending on $n\geq 1$ variables, which can be computed numerically, but which does not admit simple analytic expression, I would like to approximate numerically the quantity: $$ \frac{\partial^n f}{\partial…
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In the numerical solution of the Wave Equation, using finite differences, where do I obtain the spatial values from?

In trying to implement a simplistic numerical solver for wave equations, I have run into a conceptual problem that I haven't been able to solve. Consider a one-dimensional wave equation of a quantity $y$ $$ \frac{\partial^2y(x,t)}{\partial t^2} =…
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Finite differences of function composition

I'm trying to express the following in finite differences: $$\frac{d}{dx}\left[ A(x)\frac{d\, u(x)}{dx} \right].$$ Let $h$ be the step size and $x_{i-1} = x_i - h$ and $x_{i+ 1} = x_i + h$ If I take centered differences evaluated in $x_i$, I…
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Discrete Bessel Functions

I am reading the paper "Discrete Bessel Functions" by R.H. Boyer (Journal of Mathematical Analysis and Applications. Vol 2, Issue 3, June 1961, pg. 509-524) and he begins the paper by discussing Laplace's equation in cylindrical coordinates with…
tomcuchta
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Stability of the BTCS scheme for the heat equation in a disk

Consider the $1$-D heat equation: $$ u_t = a \Delta u = au_{xx} \\ u(0,t) = u(1,t) = 0 \\ u(x,0) = u_0(x) $$ where $a > 0$ is constant and $u_0$ is given. It is a classic result that the implicit finite difference method BTCS is unconditionally…
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Smoothly connecting PDEs with finite differences

A PDE with non-smooth inhomogeneity Let $\mathcal{L}$ be a second-order, linear, elliptic differential operator acting on $\mathcal{C}^2([0,2]^2)$. I'm numerically solving the inhomogeneous…
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Finite difference implicit schema for wave equation 1D not unconditionally stable?

The wave equation 1D with constant density is defined as: \begin{equation} \frac{\partial^2 U}{\partial t ^2} = V^2 \frac{\partial^2 U}{\partial x ^2} \label{eqa} \end{equation} And the implicit difference schema: \begin{equation} U_j^{n+1} = …
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$\Delta^ny = n!$ , difference operator question.

I was looking in a numerical analysis book and found the statement: if $y=x^n$ and the difference is $h=1$ then $\Delta^ny = n!$ and $\Delta^{n+1}y = 0$. Here $\Delta y = y(x+1)-y(x)=(x+1)^n -x^n$, ($\Delta$ is the difference operator). …
Sam Forbes
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Obtaining linear tridiagonal system from PDE in hydraulic fracturing

I'm trying to re-solve the governing equations in hydraulic fracturing modeling $$ \frac{\partial q}{\partial x} + \frac{2hC}{\sqrt{t-\tau(x)}} + \frac{\partial A}{\partial t} = 0 , \qquad 0
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Proving $∆^nf(x_0;h_1,\cdots,h_n)=f^{(n)}(ξ)h_1\cdots h_n$

Let's define finite differences of order $n$ in $x_0$ for a function $f$ as: $$\Delta ^1 f(x_0;h_1)= f(x_0+h_1)-f(x_0)$$ \begin{align*}\Delta ^2 f(x_0;h_1,h_2)&= \Delta f^1 (x_0+h_2;h_1)-\Delta f(x_0;h_1)\\ &=f(x_0+h_2+h_1)-f(x_0+h_2)-\left […
Nameless
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Explicit Euler method for Fokker-Planck equation

I'm trying to obtain an approximation of the solution of the following equation: $$ \left\lbrace \begin{array}{l,l} u_t = \alpha u_{xx} + (\beta u)_x, & u,\alpha ,\beta \in [T_0,T_f]\times [X_0,X_f]\\ u(t=T_0,x) = u_0(x), & \forall x \in [X_0,X_f]…
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The method of Characteristics for Burgers' equation

I'm trying to solve numerically the inviscid Burgers' equation $u_t + u u_x = 0$ with the method of characteristics. Most of all, I want to see how the numerical solution gets "multiple-values" for times $t>1$ as shown here in figure 3.5. The…
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Trying to use Matlab to find Numerical Solution to $u''(x)+e^{u(x)}=0, u(0)=0, u(1)=0$ - Newton's method

I am trying to use Matlab to find Numerical Solution to $u''(x)+e^{u(x)}=0, u(0)=0, u(1)=0$. I can't get my code to work out. I was hoping you could help me. I have added citations to my code to explain what is going on. My confusion is that I am…
MathIsHard
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linear interpolation error estimate for non-smooth function

Suppose I have two points $x_1,x_2$ between which I would like to have a linear interpolation $P_1$. I know the value of the function $f$ at $x_1,x_2$. The error at any point between the two will be bounded by the second derivative in $C$ norm of…
Medan
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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: $\frac{\partial \phi}{\partial x}\big|_{x=A} =…
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