logo1

What is Separation of Variables? Eigenvalue Problems for the Laplace Operator

Separation of Variables – Eigenvalues ofthe Laplace Operator

Bernd Schroder

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Eigenvalues of the Laplace Operator

logo1

What is Separation of Variables? Eigenvalue Problems for the Laplace Operator

Separation of Variables1. Solution technique for partial differential equations.

2. If the unknown function u depends on variables x,y,z, t, weassume there is a solution of the form u = f (x,y,z)T(t).

3. The special form of this solution function allows us toreplace the original partial differential equation withseveral ordinary differential equations.

4. Key step: If a(t) = b(x,y,z), then a and b must be constant.5. Solutions of the ordinary differential equations we obtain

must typically be processed some more to give usefulresults for the partial differential equations.

6. Some very powerful and deep theorems can be used toformally justify the approach for many equations involvingthe Laplace operator.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Eigenvalues of the Laplace Operator

logo1

What is Separation of Variables? Eigenvalue Problems for the Laplace Operator

Separation of Variables1. Solution technique for partial differential equations.2. If the unknown function u depends on variables x,y,z, t, we

assume there is a solution of the form u = f (x,y,z)T(t).

3. The special form of this solution function allows us toreplace the original partial differential equation withseveral ordinary differential equations.

4. Key step: If a(t) = b(x,y,z), then a and b must be constant.5. Solutions of the ordinary differential equations we obtain

must typically be processed some more to give usefulresults for the partial differential equations.

6. Some very powerful and deep theorems can be used toformally justify the approach for many equations involvingthe Laplace operator.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Eigenvalues of the Laplace Operator

logo1

What is Separation of Variables? Eigenvalue Problems for the Laplace Operator

Separation of Variables1. Solution technique for partial differential equations.2. If the unknown function u depends on variables x,y,z, t, we

assume there is a solution of the form u = f (x,y,z)T(t).3. The special form of this solution function allows us to

replace the original partial differential equation withseveral ordinary differential equations.

4. Key step: If a(t) = b(x,y,z), then a and b must be constant.5. Solutions of the ordinary differential equations we obtain

must typically be processed some more to give usefulresults for the partial differential equations.

6. Some very powerful and deep theorems can be used toformally justify the approach for many equations involvingthe Laplace operator.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Eigenvalues of the Laplace Operator

logo1

What is Separation of Variables? Eigenvalue Problems for the Laplace Operator

Separation of Variables1. Solution technique for partial differential equations.2. If the unknown function u depends on variables x,y,z, t, we

assume there is a solution of the form u = f (x,y,z)T(t).3. The special form of this solution function allows us to

replace the original partial differential equation withseveral ordinary differential equations.

4. Key step: If a(t) = b(x,y,z), then a and b must be constant.

5. Solutions of the ordinary differential equations we obtainmust typically be processed some more to give usefulresults for the partial differential equations.

6. Some very powerful and deep theorems can be used toformally justify the approach for many equations involvingthe Laplace operator.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Eigenvalues of the Laplace Operator

logo1

What is Separation of Variables? Eigenvalue Problems for the Laplace Operator

Separation of Variables1. Solution technique for partial differential equations.2. If the unknown function u depends on variables x,y,z, t, we

assume there is a solution of the form u = f (x,y,z)T(t).3. The special form of this solution function allows us to

replace the original partial differential equation withseveral ordinary differential equations.

4. Key step: If a(t) = b(x,y,z), then a and b must be constant.5. Solutions of the ordinary differential equations we obtain

must typically be processed some more to give usefulresults for the partial differential equations.

6. Some very powerful and deep theorems can be used toformally justify the approach for many equations involvingthe Laplace operator.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Eigenvalues of the Laplace Operator

logo1

What is Separation of Variables? Eigenvalue Problems for the Laplace Operator

Separation of Variables1. Solution technique for partial differential equations.2. If the unknown function u depends on variables x,y,z, t, we

assume there is a solution of the form u = f (x,y,z)T(t).3. The special form of this solution function allows us to

replace the original partial differential equation withseveral ordinary differential equations.

4. Key step: If a(t) = b(x,y,z), then a and b must be constant.5. Solutions of the ordinary differential equations we obtain

must typically be processed some more to give usefulresults for the partial differential equations.

6. Some very powerful and deep theorems can be used toformally justify the approach for many equations involvingthe Laplace operator.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Eigenvalues of the Laplace Operator

logo1

What is Separation of Variables? Eigenvalue Problems for the Laplace Operator

How Deep?

plus about 200 pages of reallyawesome functional analysis.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Eigenvalues of the Laplace Operator

logo1

What is Separation of Variables? Eigenvalue Problems for the Laplace Operator

How Deep?

plus about 200 pages of reallyawesome functional analysis.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Eigenvalues of the Laplace Operator

logo1

What is Separation of Variables? Eigenvalue Problems for the Laplace Operator

The Wave Equation ∆u = k∂ 2u∂ t2

∆u = k∂ 2u∂ t2

u(x,y,z, t) = f (x,y,z)T(t)

∆(f (x,y,z)T(t)

)=

∂ 2

∂ t2(kf (x,y,z)T(t)

)T∆f = kf T ′′

T∆fTf

= kf T ′′

Tf∆ff

= kT ′′

T= λ

∆f = λ f T ′′− λ

kT = 0

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Eigenvalues of the Laplace Operator

logo1

What is Separation of Variables? Eigenvalue Problems for the Laplace Operator

The Wave Equation ∆u = k∂ 2u∂ t2

∆u = k∂ 2u∂ t2

u(x,y,z, t) = f (x,y,z)T(t)

∆(f (x,y,z)T(t)

)=

∂ 2

∂ t2(kf (x,y,z)T(t)

)T∆f = kf T ′′

T∆fTf

= kf T ′′

Tf∆ff

= kT ′′

T= λ

∆f = λ f T ′′− λ

kT = 0

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Eigenvalues of the Laplace Operator

logo1

What is Separation of Variables? Eigenvalue Problems for the Laplace Operator

The Wave Equation ∆u = k∂ 2u∂ t2

∆u = k∂ 2u∂ t2

u(x,y,z, t) = f (x,y,z)T(t)

∆(f (x,y,z)T(t)

)=

∂ 2

∂ t2(kf (x,y,z)T(t)

)T∆f = kf T ′′

T∆fTf

= kf T ′′

Tf∆ff

= kT ′′

T= λ

∆f = λ f T ′′− λ

kT = 0

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Eigenvalues of the Laplace Operator

logo1

What is Separation of Variables? Eigenvalue Problems for the Laplace Operator

The Wave Equation ∆u = k∂ 2u∂ t2

∆u = k∂ 2u∂ t2

u(x,y,z, t) = f (x,y,z)T(t)

∆(f (x,y,z)T(t)

)=

∂ 2

∂ t2(kf (x,y,z)T(t)

)

T∆f = kf T ′′

T∆fTf

= kf T ′′

Tf∆ff

= kT ′′

T= λ

∆f = λ f T ′′− λ

kT = 0

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Eigenvalues of the Laplace Operator

logo1

What is Separation of Variables? Eigenvalue Problems for the Laplace Operator

The Wave Equation ∆u = k∂ 2u∂ t2

∆u = k∂ 2u∂ t2

u(x,y,z, t) = f (x,y,z)T(t)

∆(f (x,y,z)T(t)

)=

∂ 2

∂ t2(kf (x,y,z)T(t)

)T∆f = kf T ′′

T∆fTf

= kf T ′′

Tf∆ff

= kT ′′

T= λ

∆f = λ f T ′′− λ

kT = 0

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Eigenvalues of the Laplace Operator

logo1

What is Separation of Variables? Eigenvalue Problems for the Laplace Operator

The Wave Equation ∆u = k∂ 2u∂ t2

∆u = k∂ 2u∂ t2

u(x,y,z, t) = f (x,y,z)T(t)

∆(f (x,y,z)T(t)

)=

∂ 2

∂ t2(kf (x,y,z)T(t)

)T∆f = kf T ′′

T∆fTf

= kf T ′′

Tf

∆ff

= kT ′′

T= λ

∆f = λ f T ′′− λ

kT = 0

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Eigenvalues of the Laplace Operator

logo1

What is Separation of Variables? Eigenvalue Problems for the Laplace Operator

The Wave Equation ∆u = k∂ 2u∂ t2

∆u = k∂ 2u∂ t2

u(x,y,z, t) = f (x,y,z)T(t)

∆(f (x,y,z)T(t)

)=

∂ 2

∂ t2(kf (x,y,z)T(t)

)T∆f = kf T ′′

T∆fTf

= kf T ′′

Tf∆ff

= kT ′′

T

= λ

∆f = λ f T ′′− λ

kT = 0

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Eigenvalues of the Laplace Operator

logo1

What is Separation of Variables? Eigenvalue Problems for the Laplace Operator

The Wave Equation ∆u = k∂ 2u∂ t2

∆u = k∂ 2u∂ t2

u(x,y,z, t) = f (x,y,z)T(t)

∆(f (x,y,z)T(t)

)=

∂ 2

∂ t2(kf (x,y,z)T(t)

)T∆f = kf T ′′

T∆fTf

= kf T ′′

Tf∆ff

= kT ′′

T= λ

∆f = λ f T ′′− λ

kT = 0

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Eigenvalues of the Laplace Operator

logo1

What is Separation of Variables? Eigenvalue Problems for the Laplace Operator

The Wave Equation ∆u = k∂ 2u∂ t2

∆u = k∂ 2u∂ t2

u(x,y,z, t) = f (x,y,z)T(t)

∆(f (x,y,z)T(t)

)=

∂ 2

∂ t2(kf (x,y,z)T(t)

)T∆f = kf T ′′

T∆fTf

= kf T ′′

Tf∆ff

= kT ′′

T= λ

∆f = λ f

T ′′− λ

kT = 0

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Eigenvalues of the Laplace Operator

logo1

What is Separation of Variables? Eigenvalue Problems for the Laplace Operator

The Wave Equation ∆u = k∂ 2u∂ t2

∆u = k∂ 2u∂ t2

u(x,y,z, t) = f (x,y,z)T(t)

∆(f (x,y,z)T(t)

)=

∂ 2

∂ t2(kf (x,y,z)T(t)

)T∆f = kf T ′′

T∆fTf

= kf T ′′

Tf∆ff

= kT ′′

T= λ

∆f = λ f T ′′− λ

kT = 0

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Eigenvalues of the Laplace Operator

logo1

What is Separation of Variables? Eigenvalue Problems for the Laplace Operator

The Heat Equation ∆u = k∂u∂ t

∆u = k∂u∂ t

u(x,y,z, t) = f (x,y,z)T(t)

∆(f (x,y,z)T(t)

)=

∂

∂ t

(kf (x,y,z)T(t)

)T∆f = kf T ′

T∆fTf

= kf T ′

Tf∆ff

= kT ′

T= λ

∆f = λ f T ′ =λ

kT

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Eigenvalues of the Laplace Operator

logo1

What is Separation of Variables? Eigenvalue Problems for the Laplace Operator

The Heat Equation ∆u = k∂u∂ t

∆u = k∂u∂ t

u(x,y,z, t) = f (x,y,z)T(t)

∆(f (x,y,z)T(t)

)=

∂

∂ t

(kf (x,y,z)T(t)

)T∆f = kf T ′

T∆fTf

= kf T ′

Tf∆ff

= kT ′

T= λ

∆f = λ f T ′ =λ

kT

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Eigenvalues of the Laplace Operator

logo1

What is Separation of Variables? Eigenvalue Problems for the Laplace Operator

The Heat Equation ∆u = k∂u∂ t

∆u = k∂u∂ t

u(x,y,z, t) = f (x,y,z)T(t)

∆(f (x,y,z)T(t)

)=

∂

∂ t

(kf (x,y,z)T(t)

)T∆f = kf T ′

T∆fTf

= kf T ′

Tf∆ff

= kT ′

T= λ

∆f = λ f T ′ =λ

kT

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Eigenvalues of the Laplace Operator

logo1

What is Separation of Variables? Eigenvalue Problems for the Laplace Operator

The Heat Equation ∆u = k∂u∂ t

∆u = k∂u∂ t

u(x,y,z, t) = f (x,y,z)T(t)

∆(f (x,y,z)T(t)

)=

∂

∂ t

(kf (x,y,z)T(t)

)

T∆f = kf T ′

T∆fTf

= kf T ′

Tf∆ff

= kT ′

T= λ

∆f = λ f T ′ =λ

kT

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Eigenvalues of the Laplace Operator

logo1

What is Separation of Variables? Eigenvalue Problems for the Laplace Operator

The Heat Equation ∆u = k∂u∂ t

∆u = k∂u∂ t

u(x,y,z, t) = f (x,y,z)T(t)

∆(f (x,y,z)T(t)

)=

∂

∂ t

(kf (x,y,z)T(t)

)T∆f = kf T ′

T∆fTf

= kf T ′

Tf∆ff

= kT ′

T= λ

∆f = λ f T ′ =λ

kT

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Eigenvalues of the Laplace Operator

logo1

What is Separation of Variables? Eigenvalue Problems for the Laplace Operator

The Heat Equation ∆u = k∂u∂ t

∆u = k∂u∂ t

u(x,y,z, t) = f (x,y,z)T(t)

∆(f (x,y,z)T(t)

)=

∂

∂ t

(kf (x,y,z)T(t)

)T∆f = kf T ′

T∆fTf

= kf T ′

Tf

∆ff

= kT ′

T= λ

∆f = λ f T ′ =λ

kT

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Eigenvalues of the Laplace Operator

logo1

What is Separation of Variables? Eigenvalue Problems for the Laplace Operator

The Heat Equation ∆u = k∂u∂ t

∆u = k∂u∂ t

u(x,y,z, t) = f (x,y,z)T(t)

∆(f (x,y,z)T(t)

)=

∂

∂ t

(kf (x,y,z)T(t)

)T∆f = kf T ′

T∆fTf

= kf T ′

Tf∆ff

= kT ′

T

= λ

∆f = λ f T ′ =λ

kT

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Eigenvalues of the Laplace Operator

logo1

What is Separation of Variables? Eigenvalue Problems for the Laplace Operator

The Heat Equation ∆u = k∂u∂ t

∆u = k∂u∂ t

u(x,y,z, t) = f (x,y,z)T(t)

∆(f (x,y,z)T(t)

)=

∂

∂ t

(kf (x,y,z)T(t)

)T∆f = kf T ′

T∆fTf

= kf T ′

Tf∆ff

= kT ′

T= λ

∆f = λ f T ′ =λ

kT

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Eigenvalues of the Laplace Operator

logo1

What is Separation of Variables? Eigenvalue Problems for the Laplace Operator

The Heat Equation ∆u = k∂u∂ t

∆u = k∂u∂ t

u(x,y,z, t) = f (x,y,z)T(t)

∆(f (x,y,z)T(t)

)=

∂

∂ t

(kf (x,y,z)T(t)

)T∆f = kf T ′

T∆fTf

= kf T ′

Tf∆ff

= kT ′

T= λ

∆f = λ f

T ′ =λ

kT

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Eigenvalues of the Laplace Operator

logo1

What is Separation of Variables? Eigenvalue Problems for the Laplace Operator

The Heat Equation ∆u = k∂u∂ t

∆u = k∂u∂ t

u(x,y,z, t) = f (x,y,z)T(t)

∆(f (x,y,z)T(t)

)=

∂

∂ t

(kf (x,y,z)T(t)

)T∆f = kf T ′

T∆fTf

= kf T ′

Tf∆ff

= kT ′

T= λ

∆f = λ f T ′ =λ

kT

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Separation of Variables – Eigenvalues of the Laplace Operator