Topic 3 & 4 - Second Order PDEs

Question 1

What makes a PDE linear?

Answer 1

A PDE is linear when it contains no product of dependent-variable terms.

e.g. y(dy/dx) is a non-linear term, as the derivative tells us that ‘y’ is a dependent variable.

Question 2

How can non-linear PDEs be solved?

Answer 2

Non-linear PDEs cannot be solved analytically, however, they can be solved with numerical methods.

Question 3

What does a PDE need to be homogeneous?

Answer 3

If every term in the equation for a PDE contains the dependent variable in some form, it is homogenous.

Question 4

Homogeneous or inhomogeneous?

Answer 4

Inhomogeneous as not all terms contain the dependent variable.

The constant 1 is what makes the equation inhomogeneous.

Question 5

Why is a curly d used for PDEs over a regular dx?

Answer 5

A curly d is used to communicate that a multivariable function is involved.

Question 6

What is the significance of this function?

Answer 6

Instead of having one variable changing the other, and this being graphed, for this equation every point on the plane is an input.

Question 7

Does separation of variables work on non-linear PDEs?

Answer 7

No, it doesn’t.

Question 8

What would this non-linear PDE look like if you plugged in u(x,y) = f(x)*g(y)

Answer 8

The middle term is not the same as d²u/dx².

This middle term has two lots of u.

Which is significant as you have to insert u twice. So you end up with one more f(x) and g(y) compared to d²u/dx².

Question 9

If the discriminant is positive, what sort of PDE is it?

Answer 9

Hyperbolic

Question 10

If the discriminant is 0, what sort of PDE is it?

Answer 10

parabolic

Question 11

If the discriminant is less than 0, what sort of PDE is it?

Answer 11

Elliptic

Question 12

What sort of PDE is laplace’s equation?

Answer 12

Elliptic

Question 13

What sort of PDE is the heat equation?

Answer 13

Parabolic

Question 14

What sort of PDE is the wave equation?

Answer 14

Hyperbolic

Question 15

What is the general form of a 2nd order Linear PDE?

What is the discriminant function of this equation?

Answer 15

Question 16

How do you perform partial fraction decomposition?

Answer 16

Question 17

What are the equations for the three different cases to solve linear ODEs?

Answer 17

Question 18

What do you need to remember about the values found for coefficents (i.e. for A and B) in cases and different BC for linear ODEs?

Answer 18

Between cases coefficient values determined are not the kept.

However, within a case for a BC you can use a coeffcient value obtained from a previous/another BC.

Question 19

Could you use partial fraction decomposition directly for this fraction?

Answer 19

No you cannot. Partial fraction decomposition only works for ‘proper’ fractions, i.e. the numerators degree is not larger than the denominators degree.

Instead you need to use long division to obtain a new fraction which you can perform partial fraction decomposition on.

Question 20

How do you solve a partial fraction that has a factor in the denominator that occurs more than once?

Answer 20

Start solving as you normally would. Plug in values of x to solve coefficients.

For the remaining coefficients that you can’t solve, plug the values you do know in to get an equation of the remaining unknown values.

Depending on the number of unknowns, plug any other x value into the equation to get a system of equations to solve for the unkown coefficients.

Question 21

When you’ve separated your variables solving a linear PDE, how do you decide which equation to solve for first?

Answer 21

Go for the one with the simpler boundary conditions.

Boundary conditions equal to zero are easier to solve than ones equal to constants, so start with those.

Question 22

What do you need to remember about admissable values when determining the solution to a separation of variable ODE after already solving the other one?

Answer 22

The values of K determined for the general solution of the ODE that has been solved limits the other ODE to those values.

E.g. if k_n = (nπ/b)² for the h(y) general solution, then the solution of g(x) will be limited to those values.

In other words, with K already determined, the other ODE is also equal to this value.

Question 23

What are the relationships between the hyperbolic functions sinh, cosh and tanh to exponentials?

Answer 23

Question 24

If you reach this during the solving a PDE, what can you do?

Answer 24