100% satisfaction guarantee Immediately available after payment Both online and in PDF No strings attached
logo-home
Lecture 1 CA$14.61
Add to cart

Class notes

Lecture 1

 12 views  1 purchase

Introduction to partial differential equation

Preview 2 out of 6  pages

  • December 10, 2022
  • 6
  • 2020/2021
  • Class notes
  • Justin ko
  • All classes
All documents for this subject (1)
avatar-seller
9kfhgia89h1
May 7, 2020 APM346 – Week 1 Justin Ko


1 Introduction to Partial Differential Equations
Recall that an ordinary differential equation (ODE) of order k is an equation involving a single variable
function u(x), its derivatives, and the variable x,
F (u(k) , u(k−1) , . . . , u, x) = 0.


Definition 1. A partial differential equation (PDE) of order k is an equation involving a multivariable
function u : Rn → R, its partial derivatives, and the independent variable ~x ∈ Rn (one of these variables
might be a time variable t),
F (Dk u, Dk−1 u, . . . , u, ~x) = F ( ux1 ...x1 , . . . , ux1 x1 , ux1 x2 , . . . uxn xn , ux1 , . . . , uxn , . . . , u, ~x) = 0. (1)
| {z }
kth deriv

The order of the PDE is the order of the highest derivative in the equation. The function u is called
a solution if u satisfies (1) in some region in Rn .
Example 1. The following second order PDE
uxy + x = 0
has general solution
yx2
u=−
+ f (x) + g(y)
2
where f and g are arbitrary differentiable functions.

1.1 Boundary and Initial Value Problems
The general solutions of PDEs are usually not unique. We impose some additional conditions such as
a domain, boundary values, or initial values to give solutions some additional nice properties.
Definition 2. A PDE is well-posed if it satisfies:
1. Existence: There is at least one solution u(~x) satisfying the PDE and the additional conditions.
2. Unique: There is at most one solution u(~x) satisfying the PDE and the additional conditions.
3. Stability: The solution depends continuously on the initial data. This means that if the conditions
are changed a little, the corresponding solution changes only a little.
Example 2. The following second order PDE
uxy + x = 0 for x, y > 0,
with boundary conditions
u|y=0 = 0, u|x=0 = 0
has the particular solution
yx2
. u=−
2
If we give too few constraints, then we might not get a unique solution. If we give too many
constraints, then a solution may fail to exist.
Definition 3. We have the following terminology for these different types of constraints:
1. Initial Value Problem (IVP): A constraint on the time variable is put at t = 0, e.g. u|t=0 = f (~x).
2. Boundary Value Problem (BVP): A constraint on the spacial variable is put on the boundary of
the domain Ω, e.g. u|∂Ω = f (~x).
3. Initial Boundary Value Problem (IBVP): A constraint on both the time and space variables.


Page 1 of 6

, May 7, 2020 APM346 – Week 1 Justin Ko


1.2 Classification
Definition 4. An operator L is linear if for every pair of functions u, v and numbers s, t

L[su + tv] = sL[u] + tL[v].

Example 3. The operator L = x2 ∂x + ∂yy is linear because

L[su + tv] = x2 ∂x (su + tv) + ∂yy (su + tv) = sx2 ux + tx2 vx + suyy + tvyy = sL[u] + tL[v].

A PDE L[u] = f (~x) is linear if L is a linear operator. Nonlinear PDE can be classified based on how
close it is to being linear. Let F be a nonlinear function and α = (α1 , . . . , αn ) denote a multi-index.:
1. Linear: A PDE is linear if the coefficients in front of the partial derivative terms are all functions
of the independent variable ~x ∈ Rn ,
X
aα (~x)Dα u = f (~x).
|α|≤k

A linear PDE is homogeneous if there is no term that depends only on the space variables, i.e.
f = 0. Likewise, a linear PDE is inhomogeneous if f 6= 0.
2. Semilinear: A PDE is semilinear if it is nonlinear, but the coefficients in front of the highest
order partial derivative terms are all functions of the independent variable ~x ∈ Rn ,
X
aα (~x)Dα u + F (Dk−1 u, . . . , Du, u, ~x) = 0.
|α|=k


3. Quasilinear: A PDE is quasilinear if it is nonlinear, but the coefficients in front of the highest
order partial derivative terms are all functions of the independent variable ~x ∈ Rn or lower
derivative terms,
X
aα (Dk−1 u, . . . , Du, u, ~x)Dα u + F (Dk−1 u, . . . , Du, u, ~x) = 0.
|α|=k


4. Fully Nonlinear: A PDE is fully nonlinear if it is not of the above 3 forms. That is, the PDE is
fully nonlinear if it depends nonlinearly on the highest order partial derivative terms,

F (Dk u, . . . , Du, u, ~x) = 0.

Example 4. The forms of the various types first order PDE in R2 are:
1. Linear Homogeneous:
a(x, y)ux + b(x, y)uy + c(x, y)u = 0

2. Linear Inhomogeneous:

a(x, y)ux + b(x, y)uy + c(x, y)u = f (x, y)

3. Semilinear:
a(x, y)ux + b(x, y)uy + F (u, x, y) = 0

4. Quasilinear:
a(x, y, u)ux + b(x, y, u)uy + F (u, x, y) = 0

5. Fully Nonlinear:
F (x, y, u, ux , uy ) = 0


Page 2 of 6

The benefits of buying summaries with Stuvia:

Guaranteed quality through customer reviews

Guaranteed quality through customer reviews

Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.

Quick and easy check-out

Quick and easy check-out

You can quickly pay through credit card or Stuvia-credit for the summaries. There is no membership needed.

Focus on what matters

Focus on what matters

Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!

Frequently asked questions

What do I get when I buy this document?

You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.

Satisfaction guarantee: how does it work?

Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.

Who am I buying these notes from?

Stuvia is a marketplace, so you are not buying this document from us, but from seller 9kfhgia89h1. Stuvia facilitates payment to the seller.

Will I be stuck with a subscription?

No, you only buy these notes for CA$14.61. You're not tied to anything after your purchase.

Can Stuvia be trusted?

4.6 stars on Google & Trustpilot (+1000 reviews)

52355 documents were sold in the last 30 days

Founded in 2010, the go-to place to buy study notes for 14 years now

Start selling
CA$14.61  1x  sold
  • (0)
Add to cart
Added