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
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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
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