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

Separation of Variables – Eigenvalues of the Laplace Operator

Bernd Schroder¨

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Separation of Variables – Eigenvalues of the Laplace Operator 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.

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

Separation of Variables 1. Solution technique for partial differential equations.

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Separation of Variables – Eigenvalues of the Laplace Operator 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.

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).

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Separation of Variables – Eigenvalues of the Laplace Operator 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.

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.

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Separation of Variables – Eigenvalues of the Laplace Operator 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.

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.

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Separation of Variables – Eigenvalues of the Laplace Operator 6. Some very powerful and deep theorems can be used to formally justify the approach for many equations involving the Laplace operator.

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.

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Separation of Variables – Eigenvalues of the Laplace Operator 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.

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Separation of Variables – Eigenvalues of the Laplace Operator plus about 200 pages of really awesome functional analysis.

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

How Deep?

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Separation of Variables – Eigenvalues of the Laplace Operator What is Separation of Variables? Eigenvalue Problems for the Laplace Operator

How Deep?

plus about 200 pages of really awesome functional analysis. logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Separation of Variables – Eigenvalues of the Laplace Operator ∂ 2u ∆u = k u(x,y,z,t) = f (x,y,z)T(t) ∂t2 ∂ 2 ∆f (x,y,z)T(t)
= kf (x,y,z)T(t)
∂t2 T∆f = kf T00 T∆f f T00 = k Tf Tf ∆f T00 = k = λ f T λ ∆f = λf T00 − T = 0 k

What is Separation of Variables? Eigenvalue Problems for the Laplace Operator ∂ 2u The Wave Equation ∆u = k ∂t2

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Separation of Variables – Eigenvalues of the Laplace Operator u(x,y,z,t) = f (x,y,z)T(t) ∂ 2 ∆f (x,y,z)T(t)
= kf (x,y,z)T(t)
∂t2 T∆f = kf T00 T∆f f T00 = k Tf Tf ∆f T00 = k = λ f T λ ∆f = λf T00 − T = 0 k

What is Separation of Variables? Eigenvalue Problems for the Laplace Operator ∂ 2u The Wave Equation ∆u = k ∂t2

∂ 2u ∆u = k ∂t2

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Separation of Variables – Eigenvalues of the Laplace Operator ∂ 2 ∆f (x,y,z)T(t)
= kf (x,y,z)T(t)
∂t2 T∆f = kf T00 T∆f f T00 = k Tf Tf ∆f T00 = k = λ f T λ ∆f = λf T00 − T = 0 k

What is Separation of Variables? Eigenvalue Problems for the Laplace Operator ∂ 2u The Wave Equation ∆u = k ∂t2

∂ 2u ∆u = k u(x,y,z,t) = f (x,y,z)T(t) ∂t2

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Separation of Variables – Eigenvalues of the Laplace Operator T∆f = kf T00 T∆f f T00 = k Tf Tf ∆f T00 = k = λ f T λ ∆f = λf T00 − T = 0 k

What is Separation of Variables? Eigenvalue Problems for the Laplace Operator ∂ 2u The Wave Equation ∆u = k ∂t2

∂ 2u ∆u = k u(x,y,z,t) = f (x,y,z)T(t) ∂t2 ∂ 2 ∆f (x,y,z)T(t)
= kf (x,y,z)T(t)
∂t2

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Separation of Variables – Eigenvalues of the Laplace Operator T∆f f T00 = k Tf Tf ∆f T00 = k = λ f T λ ∆f = λf T00 − T = 0 k

What is Separation of Variables? Eigenvalue Problems for the Laplace Operator ∂ 2u The Wave Equation ∆u = k ∂t2

∂ 2u ∆u = k u(x,y,z,t) = f (x,y,z)T(t) ∂t2 ∂ 2 ∆f (x,y,z)T(t)
= kf (x,y,z)T(t)
∂t2 T∆f = kf T00

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Separation of Variables – Eigenvalues of the Laplace Operator ∆f T00 = k = λ f T λ ∆f = λf T00 − T = 0 k

What is Separation of Variables? Eigenvalue Problems for the Laplace Operator ∂ 2u The Wave Equation ∆u = k ∂t2

∂ 2u ∆u = k u(x,y,z,t) = f (x,y,z)T(t) ∂t2 ∂ 2 ∆f (x,y,z)T(t)
= kf (x,y,z)T(t)
∂t2 T∆f = kf T00 T∆f f T00 = k Tf Tf

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Separation of Variables – Eigenvalues of the Laplace Operator = λ λ ∆f = λf T00 − T = 0 k

What is Separation of Variables? Eigenvalue Problems for the Laplace Operator ∂ 2u The Wave Equation ∆u = k ∂t2

∂ 2u ∆u = k u(x,y,z,t) = f (x,y,z)T(t) ∂t2 ∂ 2 ∆f (x,y,z)T(t)
= kf (x,y,z)T(t)
∂t2 T∆f = kf T00 T∆f f T00 = k Tf Tf ∆f T00 = k f T

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Separation of Variables – Eigenvalues of the Laplace Operator λ ∆f = λf T00 − T = 0 k

What is Separation of Variables? Eigenvalue Problems for the Laplace Operator ∂ 2u The Wave Equation ∆u = k ∂t2

∂ 2u ∆u = k u(x,y,z,t) = f (x,y,z)T(t) ∂t2 ∂ 2 ∆f (x,y,z)T(t)
= kf (x,y,z)T(t)
∂t2 T∆f = kf T00 T∆f f T00 = k Tf Tf ∆f T00 = k = λ f T

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Separation of Variables – Eigenvalues of the Laplace Operator λ T00 − T = 0 k

What is Separation of Variables? Eigenvalue Problems for the Laplace Operator ∂ 2u The Wave Equation ∆u = k ∂t2

∂ 2u ∆u = k u(x,y,z,t) = f (x,y,z)T(t) ∂t2 ∂ 2 ∆f (x,y,z)T(t)
= kf (x,y,z)T(t)
∂t2 T∆f = kf T00 T∆f f T00 = k Tf Tf ∆f T00 = k = λ f T

∆f = λf

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Separation of Variables – Eigenvalues of the Laplace Operator What is Separation of Variables? Eigenvalue Problems for the Laplace Operator ∂ 2u The Wave Equation ∆u = k ∂t2

∂ 2u ∆u = k u(x,y,z,t) = f (x,y,z)T(t) ∂t2 ∂ 2 ∆f (x,y,z)T(t)
= kf (x,y,z)T(t)
∂t2 T∆f = kf T00 T∆f f T00 = k Tf Tf ∆f T00 = k = λ f T λ ∆f = λf T00 − T = 0 k logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Separation of Variables – Eigenvalues of the Laplace Operator ∂u ∆u = k u(x,y,z,t) = f (x,y,z)T(t) ∂t ∂ ∆f (x,y,z)T(t)
= kf (x,y,z)T(t)
∂t T∆f = kf T0 T∆f f T0 = k Tf Tf ∆f T0 = k = λ f T λ ∆f = λf T0 = T k

What is Separation of Variables? Eigenvalue Problems for the Laplace Operator ∂u The Heat Equation ∆u = k ∂t

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Separation of Variables – Eigenvalues of the Laplace Operator u(x,y,z,t) = f (x,y,z)T(t) ∂ ∆f (x,y,z)T(t)
= kf (x,y,z)T(t)
∂t T∆f = kf T0 T∆f f T0 = k Tf Tf ∆f T0 = k = λ f T λ ∆f = λf T0 = T k

What is Separation of Variables? Eigenvalue Problems for the Laplace Operator ∂u The Heat Equation ∆u = k ∂t ∂u ∆u = k ∂t

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Separation of Variables – Eigenvalues of the Laplace Operator ∂ ∆f (x,y,z)T(t)
= kf (x,y,z)T(t)
∂t T∆f = kf T0 T∆f f T0 = k Tf Tf ∆f T0 = k = λ f T λ ∆f = λf T0 = T k

What is Separation of Variables? Eigenvalue Problems for the Laplace Operator ∂u The Heat Equation ∆u = k ∂t ∂u ∆u = k u(x,y,z,t) = f (x,y,z)T(t) ∂t

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Separation of Variables – Eigenvalues of the Laplace Operator T∆f = kf T0 T∆f f T0 = k Tf Tf ∆f T0 = k = λ f T λ ∆f = λf T0 = T k

What is Separation of Variables? Eigenvalue Problems for the Laplace Operator ∂u The Heat Equation ∆u = k ∂t ∂u ∆u = k u(x,y,z,t) = f (x,y,z)T(t) ∂t ∂ ∆f (x,y,z)T(t)
= kf (x,y,z)T(t)
∂t

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Separation of Variables – Eigenvalues of the Laplace Operator T∆f f T0 = k Tf Tf ∆f T0 = k = λ f T λ ∆f = λf T0 = T k

What is Separation of Variables? Eigenvalue Problems for the Laplace Operator ∂u The Heat Equation ∆u = k ∂t ∂u ∆u = k u(x,y,z,t) = f (x,y,z)T(t) ∂t ∂ ∆f (x,y,z)T(t)
= kf (x,y,z)T(t)
∂t T∆f = kf T0

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Separation of Variables – Eigenvalues of the Laplace Operator ∆f T0 = k = λ f T λ ∆f = λf T0 = T k

What is Separation of Variables? Eigenvalue Problems for the Laplace Operator ∂u The Heat Equation ∆u = k ∂t ∂u ∆u = k u(x,y,z,t) = f (x,y,z)T(t) ∂t ∂ ∆f (x,y,z)T(t)
= kf (x,y,z)T(t)
∂t T∆f = kf T0 T∆f f T0 = k Tf Tf

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Separation of Variables – Eigenvalues of the Laplace Operator = λ λ ∆f = λf T0 = T k

What is Separation of Variables? Eigenvalue Problems for the Laplace Operator ∂u The Heat Equation ∆u = k ∂t ∂u ∆u = k u(x,y,z,t) = f (x,y,z)T(t) ∂t ∂ ∆f (x,y,z)T(t)
= kf (x,y,z)T(t)
∂t T∆f = kf T0 T∆f f T0 = k Tf Tf ∆f T0 = k f T

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Separation of Variables – Eigenvalues of the Laplace Operator λ ∆f = λf T0 = T k

What is Separation of Variables? Eigenvalue Problems for the Laplace Operator ∂u The Heat Equation ∆u = k ∂t ∂u ∆u = k u(x,y,z,t) = f (x,y,z)T(t) ∂t ∂ ∆f (x,y,z)T(t)
= kf (x,y,z)T(t)
∂t T∆f = kf T0 T∆f f T0 = k Tf Tf ∆f T0 = k = λ f T

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Separation of Variables – Eigenvalues of the Laplace Operator λ T0 = T k

What is Separation of Variables? Eigenvalue Problems for the Laplace Operator ∂u The Heat Equation ∆u = k ∂t ∂u ∆u = k u(x,y,z,t) = f (x,y,z)T(t) ∂t ∂ ∆f (x,y,z)T(t)
= kf (x,y,z)T(t)
∂t T∆f = kf T0 T∆f f T0 = k Tf Tf ∆f T0 = k = λ f T

∆f = λf

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Separation of Variables – Eigenvalues of the Laplace Operator What is Separation of Variables? Eigenvalue Problems for the Laplace Operator ∂u The Heat Equation ∆u = k ∂t ∂u ∆u = k u(x,y,z,t) = f (x,y,z)T(t) ∂t ∂ ∆f (x,y,z)T(t)
= kf (x,y,z)T(t)
∂t T∆f = kf T0 T∆f f T0 = k Tf Tf ∆f T0 = k = λ f T λ ∆f = λf T0 = T k

logo1 Bernd Schroder¨ Louisiana Tech University, College of Engineering and Science Separation of Variables – Eigenvalues of the Laplace Operator