PDEs 1

Question 1

when is a PDE linear

Answer 1

• if u and its partial derivatives only occur linearly
• and possibly with coefficients that are functions of independent variables

Question 2

when is a PDE homogeneous

Answer 2

• when the function f(x1,…,xn) = 0
• otherwise it is nonhomogeneous

Question 3

what is the type, order and characteristic of the PDE du/dt + c*du/dx = 0

Answer 3

• advection equation
• first-order
• linear
• homogeneous

Question 4

what is the type, order and characteristic of the PDE d^2u/dt^2 = c^2*d^2u/dx^2

Answer 4

• wave equation
• second-order
• linear
• homogenous

Question 5

what is the type, order and characteristic of the PDE du/dt = k*d^2u/dx^2 + f(x,y)

Answer 5

• heat equation
• second-order
• linear
• nonhomogeneous

Question 6

when is a PDE semilinear

Answer 6

• when all derivatives of u occur linearly
• but u itself occurs nonlinearly

Question 7

when is a PDE with order k quasilinear

Answer 7

• if its partial derivatives of order k appear linearly
• possibly with coefficients that are functions of u and derivatives of u of order less than k

Question 8

what is the type, order and characteristic of the PDE du/dt + f(u)*du/dx = 0

Answer 8

• burgers equation
• first-order
• quasilinear
• homogeneous

Question 9

when solving an ODE and you have the roots p and q from the characteristic λ equation, what is the solution if you have 2 real roots, 1 repeated root or complex roots in the form y(x)

Answer 9

• 2 real roots: y = Ae^px + Be^qx
• 1 repeated root: y = Ae^px + Bxe^px
• complex root p +/- qi: y = Ae^pxcos(qx) + Be^pxsin(qx)

Question 10

how do you find the solution to a second order linear ODE with non-constant coefficient r

Answer 10

• find the complementary solution of form R = r^α where the ODE = 0
• find the particular integral solution depending on the RHS
• CF + PI = solution in form R = Ar^α1 + Br^α2 + kr

Question 11

when finding a variable n which is a solution of the PDE in the general form a(x,y)dn/dx + b(x,y)dn/dy = 0, what is the expression for the characteristic equation dy/dx

Answer 11

• dy/dx = b(x,y)/a(x,y)

Question 12

what are the steps for finding the variable n

Answer 12

• find dy/dx =.. knowing the coefficients of dn/dx and dn/dy already
• integrate to get y =… + constant
• set the constant to n and rearrange for n
• check the solution works by working out dn/dx and dn/dy and putting it into the PDE

Question 13

when you have a PDE in the form a(x,y)du/dx + b(x,y)du/dy = f(x,y,u) subject to boundary conditions u(x,0) = … for x >0 and u(0,y) = … for y > 0 how would you start solving it (first major steps)

Answer 13

• find characteristic dy/dx = b/a
• integrate to get it in x, y and constant C form
• find compatibility condition du/dx = f/a
• integrate to get as a function of u(x,y) with the CONSTANT k(n) BEING A FUNCTION OF n

Question 14

after you have found dy/dx and du/dx, now using the boundary condition u(x,0) for x > 0 as an example, what is the first step solving for the first expression of u(x,t)

Answer 14

• work with the characteristic first
• replace x with n and input x = n and y = 0 into the characteristic to solve for C
• write the full characteristic for this boundary condition

Question 15

what is the next step in solving for the first expression u(x,t)

Answer 15

• work with the compatibility equation
• set x = n and y = 0 for u(n,0) = …
• use the characteristic eqn that has n in it to solve for k(n) in terms of x and y
• then write out u(x,y) = … with k(n) found, no n’s in the expression

Question 16

what should the final answer for u(x,t) look like

Answer 16

• there should be a number of answers equal to the number of boundary conditions youre given
• each solution of u(x,t) needs to be paired with its respective conditions