Separation of Variables Eigenvalues of the Laplace Operator Bernd - - PowerPoint PPT Presentation

logo1 What is Separation of Variables? Eigenvalue Problems for the Laplace Operator

Separation of Variables – Eigenvalues of the Laplace Operator

Bernd Schr¨

• der

Bernd Schr¨

• der

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 Variables

• 1. Solution technique for partial differential equations.

Bernd Schr¨

• der

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 Variables

• 1. 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).

Bernd Schr¨

• der

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 Variables

• 1. 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 with several ordinary differential equations.

Bernd Schr¨

• der

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 Variables

• 1. 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 with several ordinary differential equations.

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

Bernd Schr¨

• der

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 Variables

• 1. 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 with several 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 useful results for the partial differential equations.

Bernd Schr¨

• der

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 Variables

• 1. 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 with several 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 useful results for the partial differential equations.

• 6. Some very powerful and deep theorems can be used to

formally justify the approach for many equations involving the Laplace operator.

Bernd Schr¨

• der

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?

Bernd Schr¨

• der

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 really awesome functional analysis.

Bernd Schr¨

• der

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

Bernd Schr¨

• der

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

Bernd Schr¨

• der

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)

Bernd Schr¨

• der

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)
• Bernd Schr¨
• der

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′′

Bernd Schr¨

• der

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∆f Tf = k f T′′ Tf

Bernd Schr¨

• der

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∆f Tf = k f T′′ Tf ∆f f = kT′′ T

Bernd Schr¨

• der

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∆f Tf = k f T′′ Tf ∆f f = kT′′ T = λ

Bernd Schr¨

• der

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∆f Tf = k f T′′ Tf ∆f f = kT′′ T = λ ∆f = λf

Bernd Schr¨

• der

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∆f Tf = k f T′′ Tf ∆f f = kT′′ T = λ ∆f = λf T′′ − λ k T = 0

Bernd Schr¨

• der

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

Bernd Schr¨

• der

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

Bernd Schr¨

• der

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)

Bernd Schr¨

• der

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)
• Bernd Schr¨
• der

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′

Bernd Schr¨

• der

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∆f Tf = k f T′ Tf

Bernd Schr¨

• der

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∆f Tf = k f T′ Tf ∆f f = kT′ T

Bernd Schr¨

• der

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∆f Tf = k f T′ Tf ∆f f = kT′ T = λ

Bernd Schr¨

• der

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∆f Tf = k f T′ Tf ∆f f = kT′ T = λ ∆f = λf

Bernd Schr¨

• der

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∆f Tf = k f T′ Tf ∆f f = kT′ T = λ ∆f = λf T′ = λ k T

Bernd Schr¨

• der

Louisiana Tech University, College of Engineering and Science Separation of Variables – Eigenvalues of the Laplace Operator