Differential Equation : Space Factor, Time Factor, Eigenvalues and Sturm-Liouville

7. Consider the differential equation ut =1/2 uxx + ux for 0 0 with boundary conditions u(0,t) = u(pi,t) = 0. (a) Separate variables and write the ordinary differential equations that the space factor X(x) and the time factor T(t) must satisfy. (b) Show that 0 is not an eigenvalue of the Sturm-Liouville proble ...continues

Initial-Value Problem : First Order PDE - Method of Characteristics

Solve the IVP ut + xux = -u u(x,0) = sin x Please see the attached file for the fully formatted problem.

Initial-Value Problem : First Order PDE - Method of Characteristics

Solve the initial-value problem: (1-t)ut + xux = 0 u(x,0) = e^(-x^2) Please see the attached file for the fully formatted problem.

Diffusion Equation : Energy Decreasing as a Function of Time

3. Suppose that u(x. t) satisfies the diffusion equation ut = kuxx for 0 < x < L and t > 0, and the Robin boundary conditions ux(0, t) — aou(0, t) = 0 and ux(L, t) + aLu(L, t) = 0 where k, L, a0 and aL are all positive constants. Show that ... is a decreasing function of t. Please see the attached file for the fully for ...continues

Use Separation of Variables to Find Solution to Heat Conduction Problem with Mixed Boundary Conditions

3. Use seperation of variables to find the solution, in the form of an infinite series, of the homogenous heat conduction problem with mixed boundary conditions... Partial Differential Equation, Boundary Conditions Initial Conditions.... Please see the attached file for the fully formatted problems.

Diffusion Equations : Classify the PDE's as Elliptic, Parabolic or Hyperbolic

Classify the following PDE's as elliptic, parabolic or hyperbolic. If mixed, identify the regions and classify within each region. (b) xuxx - uxy + yuxy +3uy = 1 Please see the attached file for the fully formatted problem.

Separation of Variables and Non-Cartesian Coordinates : Isotherms

Please see the attached file for the fully formatted problems.

Separation of Variables and Non-Cartesian Coordinates : Isotherms

Please see the attached file for the fully formatted problems.

Differential Equations : Solution to Heat Equation

Consider the heat equation . Show that if where and is a constant, then satisfies the ordinary differential equation , (where ). Show that = is independent of only if . Further, show that if then where C is an arbitrary constant. From this last ordinary differential eq ...continues

Wave Equation : Triangular Pulse

Let u(x; y) be the solution on 0 < x < 2 and 0 < y < 2 of uxx + uyy = 0 with u(0, y) = u(2, y) = u(x, 2) = 0 and u(x, 0) = f(x) the triangular pulse with f(0) = f(2) = 0 and f(1) = 2. f(x) = {2x 0 ...continues