Recent content by sigmund

Look into the help browser of Mathematica (press F1) under NDSolve. There should be a list of examples (see 'Further examples'). From this you will also see how to plot the solution provided by NDSolve. See also...

HallsofIvy, thank you for the reply. I guess it suffices to substitute the assumed solutions into the system, and show that they satisfy the differential equations. This is of course the straightforward way to do it. As Einstein said: "Everything should be done as simple as possible, but no...

I have a system of two PDEs: y_t+(h_0v)_x=0 \quad (1a) v_t+y_x=0 \quad (1b), where h_0 is a constant. Then I want to show that (1) has traveling wave solutions of the form y(x,t)=f(x-ut) \quad (2a) v(x,t)=g(x-ut) \quad (2b), where u is the propagation velocity...

Well, after all, everything is correct. It just occurred to me that I did not plot the correct arrays against each other. I've corrected that, and now I have got the correct solutions.

Well, maybe this is not a mathematics question after all, but however, I ask it here. I have to implement a semi-difference scheme in Matlab of the perturbed sine-Gordon equation u_{1,t}=u_2 u_{2,t}=u_{1,xx}-\sin(u_1)-\alpha u_2+\gamma. Here u_1 and u_2 are functions of x and t, and...

By "I want a numerical method to solve the problem exactly", I mean a method for which the local error, in theory, vanishes. However, that is not my problem, because I know that with a fourth order numerical method this is achieved. On the contrary, I want some help with the the second problem...

We have the first order ODE y'=4t \sqrt y,~y(0)=1, for which i have found the exact solution, namely a fourth order polynomial. I want a numerical method to solve the problem exactly. This method has to be a fourth order method, since this implies that the local error vanishes. Now...

Well, I think I have got it now. The series for 1 does only consist of a constant term, hence we set n=0. Then k=0 also, and we determine the coefficient a_{-4}, using the Cauchy product. All the other coefficients in the series for 1 are zero. Thus we get the recursion formula (stated in your...

Actually, I cannot follow you here. The sides of which equation are you talking of? If you set n=0 and k=0 in the last equation (the result of the Cauchy multiplication), you should get \frac{-2a_{-4}}{4!}=1\Leftrightarrow a_{-4}=-12. And then, using the recursion formula (the very last...

I have a function 2-z^2-2\cos z, which has a zero at z=0. I have determined the Maclaurin series for f: \sum_{j=2}^\infty(-1)^{j-1}\frac{2z^{2j}}{(2j)!}, and now I have to determine the coefficients a_{-j},~\forall j>0, in the Laurent series for a function h, which is defined as...

The function f has a simple pole at z=z_0 (z_0 has been stated several times in this thread, so I omit it here). Hence the residue at that point is \text{Res}(f;z_0)=\lim_{z\to z_0}(z-z_0)f(z). If we let f(z)=p(z)/q(z), where p and q are both analytic at z_0, and q has a simple zero at z_0...

I am not sure how to do this. The denominator of f has a zero of order 1 at z=i\pi/4, whence it can be written as (z-i\pi/4)g(z), where g(z) is analytic at i\pi/4 and g(i\pi/4)\neq0. I do not know how to do this. Couldn't you give me a hint to how to solve this?

The function f has singularities in z=\frac{i}{4}\left(\pi+k2\pi\right),~k=0,1,2,\dots, and the contour is a rectangle with vertices at z=-r_1, z=r_2, z=r_2+i\pi, and z=-r_1+i\pi, where r_1,r_2>0. It is also oriented counter-clockwise. Moreover, in the first question of the exercise, I am asked...

I have the function f(z)=\frac{e^z}{1+e^{4z}}. and the loop \gamma_{r_1,r_2}=I_{r_1,r_2}+II_{r_2}+III_{r_1,r_2}+IV_{r_1},\quad r_1,r_2>0, which bounds the domain A_{r_1,r_2}=\{z\in\mathbb{C}\mid -r_1<\Re(z)< r_2\wedge0<\Im(z)<\pi\}. Now I have to show that...

CarlB, Thank you for answering. I did not figure out the problem myself, but my teacher has provided a solution for this problem. See page 3 of the following, where the problem has been solved: http://www2.mat.dtu.dk/education/01141/S/7homework05.pdf . Actually, I did not really figure out...