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

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PULAK Kumer 2020-4-16

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评论： Walter Roberson 2020-4-18

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

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David Goodmanson 2020-4-17

编辑：David Goodmanson 2020-4-18

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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.

David Goodmanson 2020-4-18

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.