D'Alembert's solution for the one-dimensional wave equation for a semi-infinite string

Hello! Thank you for taking the time to try and solve this problem. Please note that I am unable to use math symbols. Thus, I will use * to symbolize a partial derivative. For example, u*x denotes the partial derivative of u with repsect to x. Here is the problem: (also note that the PDE is the one-dimensional wave equation) Find the solution to: PDE: u*xx-c(to the power of negative k)u*tt=0 , 0=0 BC: u*x(0,t)=0 , t>=0 This BC corresponds to a string with its end point free to move in a vertical direction.

non-linear equation

Hello. Thank you for taking the time to look at my problem. PLease note that I will use * to indicate a partial derivative. Thus, u*x denotes the partial derivative of u with respect to x. In addition, I will abbreviate u*x with p and u*y with q. Thus, u*x=p and u*y=q. Also, the symbol / means division. Here is the problem: The PDE is: xp + yq + p + q -pq = u which gives: p(x+1) + q(y+1) - pq - u = 0 Now, I need to find a complete solution. I have set up my characteristic system to be: dx /x+1-q = dy /y+1-p = du /u-pq = dp /p(pq-u-1) = dq /q(pq-u-1) Help! I cannot solve any of these integrals. I need p = P(x,y,u,a) and q = Q(x,y,u,a) so that... click for more

Riemann's method for solving Cauchy problem

Hello. Thanks for help! I will use * to indicate a partial derivative. For example, u*x denotes the partial derivative of u with respect to x. This is the probelm: Use Riemann's method to solve the Cauchy problem: u*xx + 4u*xy +3u*yy = 1, u=1 and u*n = square root of 5 times x, on the intial curve y=2x. If this problem is too difficult, then perhaps you may solve an easier one, just to give me some direction please. The only example I have is the Riemann function for the telegraph equation. It is very different form this question. Again, if this question is too difficult, then any example, along the same lines, will be fine.

D'Alembert's solution for the one-dimensional wave equation for a semi-infinite string

Hello! Thank you for taking the time to try and solve this problem. Please note that I am unable to use math symbols. Thus, I will use * to symbolize a partial derivative. For example, u*x denotes the partial derivative of u with respect to x. Here is the problem: (also note that the PDE is the one-dimensional wave equation) Find the solution to: PDE: u*xx-c(to the power of negative k)u*tt=0 , 0 ICs: u(x,0)=f(x) and u*t(x,0)=g(x), x>=0 BC: u*x(0,t)=0 , t>=0 This BC corresponds to a string with its end point free to move in a vertical direction. (Please remember to include to boundary condition in your solution. Thanks very much!)

Green's function

Hello. Thank you for taking the time for helping me. The following is the problem which I need to solve (there are actually two parts): I need to construct Green's function for the Dirichlet problem (Laplace's equation) in the upper half plane R={(x,y) : y>0} and I must derive Poisson's integral formula for the half plane. Please note that I am unable to use mathematical symbols (my computer does not allow me.) I will thus write out the following hints. For the first part of the problem we may consider log(R*/R) where R=|x-y|, R*=|x*-y|, with x* the mirror image of x=(x,y) with respect to y=0. I have also know that the Poisson equation may be: u(x,y) = [y divided by pi] times t... click for more