Initial Value Problem

Please give me detailed, step by step hints to solving the problems. Thanks very much!

Fourier (sine) Series Solution - Wave Equation, Interval, Boundary Conditions, Initial Conditions etc...

Please see attachment for complete questions (for the below "..." indicates equation to be found in attachment). Thanks! (a) Write down the Fourier (sine) series solution u(x,t) of the wave equation ... on the interval ... satisfying the boundary conditions ... and the initial conditions ... (b) Use the identity ... to show that the above series solution u(x,t) can be transformed into the form ... where F(x) is the odd periodic extension of f(x) ... (c) The last result is no surprise. Why not?

Polar Coordinates; Laplace's Equation; Boundary Conditions; Wedge Domain

I need some clues on figuring out these questions. Please see attachment for complete problems (regarding the below: "..." indicates an equation to be found in the attachment. Thanks!) (a) Using polar coordinates, find all the separated solutions of Laplace's equation satisfying the attached boundary conditions in the "wedge domain" ... (b) Use these seperated solutions to find a series solution of Laplace's equation in the given wedge domain subject to the boundary conditions ... (c) Using polar coordinates, find all the separated solutions of Laplace's equation satisfying the following boundary conditions in the "wedge domain" ... (d) Use these seperated solutions to find a se... click for more

Heat Equation; Boundary Conditions; Steady State Condition; Initial Value Problem

(a) Find all the separated solutions of the attached heat equation (satisfying the attached boundary condition) (b) Use these separated solutions to write a series solution for the initial value problem posed by the attached pde and the attached boundary conditions, with the initial condition given by {see attachment} (c) Find the steady state solution for the inital value problem, taking into account the initial condition. (d) Show that the series state solution for the initial value problem approaches the steady state solution of the initial value problem as {see attachment} (e) Give a brief physical interpretation of this limiting behaviour as {see attachment}

Wave Equation on Semi-Infinite Domain (Neumann BC) - Dirichlet and Neumann conditions

Dirichlet and Neumann conditions Solve the following PDE explicitly in terms of...and...in each region...and...: Please see attached for full question.