Partial Differential Equations

Q1: Which of the following is the correct classification of the PDE: u_xx + 2u_xy + u_yy = 0?

Parabolic Hyperbolic Elliptic Mixed type

Q2: What is the general solution of the wave equation u_tt = c²u_xx?

u(x,t) = f(x-ct) + g(x+ct) u(x,t) = f(x-ct) - g(x+ct) u(x,t) = f(x) + g(t) u(x,t) = f(x)cos(ct) + g(x)sin(ct)

Q3: Which of the following is the Laplace equation?

u_xx + u_yy = 0 u_xx - u_yy = 0 u_xx + u_yy = u_t u_xx + u_yy = f(x,y)

Q4: For the heat equation u_t = α²u_xx on a finite interval with zero boundary conditions, which of the following is TRUE?

The solution decays exponentially in time The solution grows exponentially in time The solution oscillates in time The solution is constant in time

Q5: What type of PDE is u_xx - u_yy + u_x = 0?

Hyperbolic Parabolic Elliptic Mixed type

Q6: In separation of variables for the 1D wave equation, what form is assumed for the solution?

u(x,t) = X(x)T(t) u(x,t) = X(x) + T(t) u(x,t) = X(x)/T(t) u(x,t) = X(x) - T(t)

Q7: What is the fundamental solution of the Laplace equation in 2D?

(1/2π)ln|r| 1/(4πr) 1/(2πr) (1/4π)ln|r|

Q8: Which of the following is TRUE about the maximum principle for Laplace's equation?

The maximum and minimum occur on the boundary The maximum occurs in the interior The minimum occurs in the interior There is no maximum principle

Q9: What is the method of characteristics used for?

Solving first-order PDEs Solving second-order elliptic PDEs Solving integral equations Solving non-linear algebraic equations

Q10: For the PDE u_xx + u_yy + u_zz = 0, what is the equation type?

Elliptic Hyperbolic Parabolic Mixed type

Q11: Which of the following boundary conditions is NOT homogeneous?

u(0,t) = 0 u_x(0,t) = 0 u(0,t) = f(t) u(0,t) + u_x(0,t) = 0

Q12: In the heat equation, what is the physical meaning of the term α²?

Thermal diffusivity Thermal conductivity Specific heat Density

Q13: What is the Green's function G(x,ξ) for the 1D Poisson equation -u_xx = f(x) on [0,L] with Dirichlet boundary conditions?

G(x,ξ) = { (x(L-ξ))/L for x < ξ, (ξ(L-x))/L for x > ξ } G(x,ξ) = { (x(ξ-L))/L for x < ξ, (ξ(x-L))/L for x > ξ } G(x,ξ) = { (x(L+ξ))/L for x < ξ, (ξ(L+x))/L for x > ξ } G(x,ξ) = { (x(ξ+L))/L for x < ξ, (ξ(x+L))/L for x > ξ }

Q14: Which of the following PDEs is NOT linear?

u_t + uu_x = 0 u_t + u_x = 0 u_t = α²u_xx u_xx + u_yy = 0

Q15: What is the solution of the wave equation u_tt = c²u_xx with initial conditions u(x,0) = f(x) and u_t(x,0) = g(x)?

u(x,t) = 1/2[f(x-ct) + f(x+ct)] + 1/(2c)∫_{x-ct}^{x+ct} g(s)ds u(x,t) = f(x-ct) + f(x+ct) + ∫ g(s)ds u(x,t) = 1/2[f(x-ct) + f(x+ct)] + ∫ g(s)ds u(x,t) = f(x-ct) + f(x+ct) + 1/(2c)∫ g(s)ds