28
votes
Nonlinear differential equation: numerical solution
Introduction
I think there are several questions on this site about ODEs of the form
$$(x-a)^2 u''(x) = F(x,u,u')$$
with an initial condition at $x=a$.
There is no general guarantee that solutions ...
- 234k
23
votes
Accepted
Numerically solve the initial value problem for the 1-D wave equation
I've waited for this question for a long time :)
Fully NDSolve-based Numerical Solution
There actually exist 2 issues here:
NDSolve can't handle unsmooth i.c. ...
- 63.8k
20
votes
How to solve ODE with boundary at infinity
Spectral methods
I present two general ways to approach a second-order linear BVP of the form
$$\gamma(x)\, y''(x) + \beta(x)\, y'(x) + \alpha(x)\, y(x) + \varphi(x) = 0,\ y(0) = y_1,\ y(\infty) = ...
- 234k
16
votes
Why does NDSolve blow up when given my ODE but bvp4c in Matlab does not?
First, there is a mistake in the sign k. And secondly, for such tasks we use a special method that expands the possibilities of the shooting method. I will ...
- 42k
15
votes
Numerically solve the initial value problem for the 1-D wave equation
Here is a FEM-based solution:
...
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15
votes
Accepted
${\frac {\partial^{2} u}{\partial {x}^{2}}} +{\frac {\partial ^{2} u}{\partial {y}^{2}}} =0$ with one boundary at infinity
Using the symbolic boundary condition
u[t, y] == 0
instead of the infinite one, and then taking the limit $$ t \to \infty,$$ inside the sum seems to do the trick ...
14
votes
Where is the numerical solving breaking down?
Some experimentation shows that the NDSolve problem is associated exclusively with z with no feedback to ...
- 60.8k
12
votes
Numerical solution of coupled ODEs with boundary conditions
This is the most difficult of the nearly two dozen nonlinear ODE separatrix computations that I have encountered on Mathematica.SE. Nonetheless, it can be can be solved by a systematically refined ...
- 60.8k
12
votes
Accepted
Heat convection differential equations from 1952 - Mathematica "fails to converge"
The problem is with the default starting initial conditions used by the shooting method in NDSolve. The shooting method is where ...
- 234k
12
votes
Accepted
Numerical solutions of active 1D wave equations
This problem can be solved with method of lines and with FDM as well. Using NDSolve we have
...
- 42k
11
votes
Accepted
Frequency domain Maxwell equations with PML boundary conditions
I'm not that familiar with electromagnetism, either, but I think there're at least 4 issues in your solving process:
There's no need to "consider only the magnetic field", because electric ...
- 63.8k
11
votes
Boundary Condition for Schrödinger Equation in Infinite Range
Let's remember Schrodinger's equation:
$i\hbar\frac{\partial}{\partial t} \Psi(\mathbf{r},t) = \left [ \frac{-\hbar^2}{2\mu}\nabla^2 + V(\mathbf{r},t)\right ] \Psi(\mathbf{r},t)$
For the harmonic ...
- 2,591
11
votes
Solving stiff boundary value problem
Relaxation solution
Your ODE is
$$
k \frac{d^2 T}{dx^2} - T^4 = 0, \qquad T(0) = 0.9, \qquad T(1) = 1
$$
where $T$ is a function of $x$. We attempt instead to solve the related PDE
$$
k \frac{\...
- 15k
10
votes
Accepted
Using DSolve with a boundary condition at -Infinity
This is the solution of your equation without the boundary conditions:
...
- 38.7k
10
votes
Accepted
Boundary Condition for Schrödinger Equation in Infinite Range
Since OP has found this interesting post, let me try to implement the exterior complex scaling method mentioned there.
First, make the transform $x=
\left\{\begin{array}{cc}
&
\begin{array}{cc}...
- 63.8k
10
votes
Accepted
Error when solving 't Hooft-Polyakov radial equations using NDSolve
Once again, compared to "Shooting" method that is the default and currently the only available method for solving nonlinear boundary value problem (BVP) ...
- 63.8k
10
votes
An ODE system easily polluted with spurious eigenvalues
NDEigenValues handles the pair of first-order equations in the question much more accurately, when it is converted into a single second-order equation.
...
- 60.8k
10
votes
Accepted
An ODE system easily polluted with spurious eigenvalues
The additional problem added to the end of the question can be solved in a similar manner. Begin with
...
- 60.8k
10
votes
Accepted
Where is the numerical solving breaking down?
For this problem, you must specify a solution method. Since the system of equations is nonlinear and the equation for y does not contain derivatives with respect to ...
- 42k
10
votes
Where is the numerical solving breaking down?
Fully NDSolve-based Solution
Adjustion for spatial step size together with temporal step size helps. I've used parameters mentioned in the comment for testing:
<...
- 63.8k
10
votes
9
votes
Nonlinear differential equation: numerical solution
This question seeks the separatrix of a nonlinear ODE over the range {0, Infinity}. As noted by Michael E2, many such problems have been presented on this site. (...
- 60.8k
9
votes
Accepted
Numerical solution of nonlinear boundary value problem
This system of second-order nonlinear ODEs, like many others in this site, is difficult to solve numerically, because the desired asymptotic solution is a separatrix. As a consequence, infinitesimal ...
- 60.8k
9
votes
Accepted
Instability, Courant Condition and Robustness about solving 2D+1 PDE
OK, since neither of the default difference scheme nor the fix function works properly on this PDE, let's discretize the spatial derivative all by ourselves in a ...
- 63.8k
9
votes
Accepted
Step size is effectively zero
I think that ODE systems with $\| X'\| \sim O(\| X \|^2)$ tend to be unstable, that is, a small rounding error has a chance to cause a solution to blow up. First, boundary-value problems (BVPs) are ...
- 234k
9
votes
Methods of Numerically Finding Function Minimizing Functional
I'd typically go for "discretize and optimize".
Here is a quick and dirty implementation of this strategy that avoids all symbolic computation and that utilizes Sobolev gradient descent with ...
- 104k
8
votes
Accepted
Trouble with shooting method for a 4th-order differential equation
Interesting equation. It seems to be necessary to use the asymptotic solution as the boundary at $x=x_0$ if you want to solve the equation correctly.
Thanks to this answer, we can easily get the ...
- 63.8k
8
votes
Setting Up Boundary Conditions for Magnetostatic PDE
Well, I'm not quite familiar with electromagnetism and it's not immediately clear to me how to compare the numeric solution with the analytic solution, but I've circumvented the ...
- 63.8k
8
votes
Accepted
Handling "ill-conditioned" system of ODE's with NDSolve
Because the homogeneous ODE is singular at its endpoints, begin by determining the asymptotic solution at one of them, here u == 1.
...
- 60.8k
8
votes
Accepted
How to solve this 2nd-order ODE with singularity?
The ODE itself can be solved symbolically. For generality, assume that k and b are undefined and that ...
- 60.8k
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