how can i solve the following wave equation analytically and plot figure)?

D(u(x,t),t,t)=c^2*D(u(x,t),x,x);

Where u(x,0)=f(x),D[2](u)(x,0)=g(x),u(0,t)=0=u(L,t),L=2

here D[2](u) (x,0) means two times derivative of u with respect to x at t=0

[David Goodmanson]

Hello Pulak,

what is the meaning of D[2](u)(x,0) ?

[PULAK Kumer]

Two times derivative of u at t=0,sir

[David Goodmanson]

with respect to what?

[PULAK Kumer]

with respect to x sir,please solve this math in matlab analytically with plot and animation,please

[David Goodmanson]

Hello Pulak,

Ordinarily the solution would go along these lines. Assume you are to the right of the origin and have a leftward-traveling wave f1(x+c*t). Saying u(0,t) = 0 is equivalent to having a reflection at the origin. So a solution would be

u(x,t) = f1(x+c*t) - f1(-x+c*t)

where the second term is a rightward-moving wave that starts out to the left of the origin. This automatically satisfies u(0,t) = 0, and setting f1 = f also satisfies u(x,0) = f(x), to the right of the origin. The problem is that you could still have rightward-moving wave f2 to the right of the origin and there are not enough boundary conditions to determine that. Since this is a second order equation, ordinarily a second b.c. would be du/dt(x,0) = g(x), <corrected>, not something involving d2u/dx2. After that you would have to take care of the boundary condition at x=L by imposing functions with periodicity L. That is the basic idea, which you would have to work out before invoking Matlab.

[PULAK Kumer]

I can not understand your code sir,please give me matlab code for solving this..please help me

[Walter Roberson]

How does this differ from https://www.mathworks.com/matlabcentral/answers/518609-how-can-i-solve-the-following-wave-equation-analytically-in-matlab-with-plot-and-animation-2d-plot-a ?

[Walter Roberson]

There is no solution for arbitrary functions f(x) .

When you let g(x) = 0, and ignore u(L,t)=0, then the solution is u(x,t) = f(x). For the equations to work, there is requirement that f(0) = 0, which is incompatible with the requirement that f(x) is an arbitrary function. Your boundary condition u(L,t)=0 becomes f(L) = 0, which is incompatible with the requirement that f(x) is an arbitrary function.

If g(x) is not 0 then you get constraints in the form of fourier transforms.

... However, it depends what is meant by D[2](u)(x,0), whether the derivative is with respect to x or with respect to t. My calculation was based on it being with respect to t.

[PULAK Kumer]

sir ,let ,f(x)=(8*x*(L-x)^2)/L^3

then the code will be what?

[David Goodmanson]

Hi Pulak,

It doesn't really work that way. Since this appears to be homework, you need to make an effort to use the course material and come up with some code. If the code doesn't work as it should that's all right, because then there will be something to comment on. No one at this site is predisposed to provide code out of thin air for homework problems.

[Walter Roberson]

I had a typing mistake in my code; my conclusions above are not valid.