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Introduction ∆ in a Rotationally Symmetric 2d Geometry Separating Polar Coordinates Separation of Variables – Bessel Equations Bernd Schr¨ oder logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Separation of Variables – Bessel Equations

Introduction ∆ in a Rotationally Symmetric 2d Geometry Separating Polar Coordinates Separation of Variables 1. Solution technique for partial differential equations. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Separation of Variables – Bessel Equations

Introduction ∆ in a Rotationally Symmetric 2d Geometry Separating Polar Coordinates Separation of Variables 1. Solution technique for partial differential equations. 2. If the unknown function u depends on variables r , θ , t , we assume there is a solution of the form u = R ( r ) D ( θ ) T ( t ) . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Separation of Variables – Bessel Equations

Introduction ∆ in a Rotationally Symmetric 2d Geometry Separating Polar Coordinates Separation of Variables 1. Solution technique for partial differential equations. 2. If the unknown function u depends on variables r , θ , t , we assume there is a solution of the form u = R ( r ) D ( θ ) 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 Schr¨ oder Louisiana Tech University, College of Engineering and Science Separation of Variables – Bessel Equations

Introduction ∆ in a Rotationally Symmetric 2d Geometry Separating Polar Coordinates Separation of Variables 1. Solution technique for partial differential equations. 2. If the unknown function u depends on variables r , θ , t , we assume there is a solution of the form u = R ( r ) D ( θ ) 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 f ( t ) = g ( r , θ ) , then f and g must be constant. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Separation of Variables – Bessel Equations

Introduction ∆ in a Rotationally Symmetric 2d Geometry Separating Polar Coordinates Separation of Variables 1. Solution technique for partial differential equations. 2. If the unknown function u depends on variables r , θ , t , we assume there is a solution of the form u = R ( r ) D ( θ ) 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 f ( t ) = g ( r , θ ) , then f and g 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 Schr¨ oder Louisiana Tech University, College of Engineering and Science Separation of Variables – Bessel Equations

Introduction ∆ in a Rotationally Symmetric 2d Geometry Separating Polar Coordinates Separation of Variables 1. Solution technique for partial differential equations. 2. If the unknown function u depends on variables r , θ , t , we assume there is a solution of the form u = R ( r ) D ( θ ) 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 f ( t ) = g ( r , θ ) , then f and g 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 Schr¨ oder Louisiana Tech University, College of Engineering and Science Separation of Variables – Bessel Equations

Introduction ∆ in a Rotationally Symmetric 2d Geometry Separating Polar Coordinates How Deep? logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Separation of Variables – Bessel Equations

Introduction ∆ in a Rotationally Symmetric 2d Geometry Separating Polar Coordinates How Deep? plus about 200 pages of really awesome functional analysis. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Separation of Variables – Bessel Equations

Introduction ∆ in a Rotationally Symmetric 2d Geometry Separating Polar Coordinates The Equation ∆ u = k ∂ u ∂ t logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Separation of Variables – Bessel Equations

Introduction ∆ in a Rotationally Symmetric 2d Geometry Separating Polar Coordinates The Equation ∆ u = k ∂ u ∂ t 1. It’s the heat equation. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Separation of Variables – Bessel Equations

Introduction ∆ in a Rotationally Symmetric 2d Geometry Separating Polar Coordinates The Equation ∆ u = k ∂ u ∂ t 1. It’s the heat equation. 2. Consideration in two dimensions may mean we analyze heat transfer in a thin sheet of metal. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Separation of Variables – Bessel Equations

Introduction ∆ in a Rotationally Symmetric 2d Geometry Separating Polar Coordinates The Equation ∆ u = k ∂ u ∂ t 1. It’s the heat equation. 2. Consideration in two dimensions may mean we analyze heat transfer in a thin sheet of metal. 3. It may also mean that we are working with a cylindrical geometry in which there is no variation in the z -direction. (Heating a metal cylinder in a water bath.) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Separation of Variables – Bessel Equations

Introduction ∆ in a Rotationally Symmetric 2d Geometry Separating Polar Coordinates Separating the Equation ∆ u = k ∂ u ∂ t (Temporal Part) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Separation of Variables – Bessel Equations

Introduction ∆ in a Rotationally Symmetric 2d Geometry Separating Polar Coordinates Separating the Equation ∆ u = k ∂ u ∂ t (Temporal Part) ∂ 2 u ∂ 2 u ∂ u k ∂ u ∂ r 2 + 1 ∂ r + 1 = r 2 ∂θ 2 ∂ t r logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Separation of Variables – Bessel Equations

Introduction ∆ in a Rotationally Symmetric 2d Geometry Separating Polar Coordinates Separating the Equation ∆ u = k ∂ u ∂ t (Temporal Part) ∂ 2 u ∂ 2 u ∂ u k ∂ u ∂ r 2 + 1 ∂ r + 1 = r 2 ∂θ 2 ∂ t r u ( r , θ , t ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Separation of Variables – Bessel Equations

Introduction ∆ in a Rotationally Symmetric 2d Geometry Separating Polar Coordinates Separating the Equation ∆ u = k ∂ u ∂ t (Temporal Part) ∂ 2 u ∂ 2 u ∂ u k ∂ u ∂ r 2 + 1 ∂ r + 1 = r 2 ∂θ 2 ∂ t r u ( r , θ , t ) = R ( r ) D ( Θ ) T ( t ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Separation of Variables – Bessel Equations

Introduction ∆ in a Rotationally Symmetric 2d Geometry Separating Polar Coordinates Separating the Equation ∆ u = k ∂ u ∂ t (Temporal Part) ∂ 2 u ∂ 2 u ∂ u k ∂ u ∂ r 2 + 1 ∂ r + 1 = r 2 ∂θ 2 ∂ t r u ( r , θ , t ) = R ( r ) D ( Θ ) T ( t ) = R · D · T logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Separation of Variables – Bessel Equations

Introduction ∆ in a Rotationally Symmetric 2d Geometry Separating Polar Coordinates Separating the Equation ∆ u = k ∂ u ∂ t (Temporal Part) ∂ 2 u ∂ 2 u ∂ u k ∂ u ∂ r 2 + 1 ∂ r + 1 = r 2 ∂θ 2 ∂ t r u ( r , θ , t ) = R ( r ) D ( Θ ) T ( t ) = R · D · T ∂ 2 ∂ 2 ∂ k ∂ ∂ r 2 RDT + 1 ∂ rRDT + 1 = ∂θ 2 RDT ∂ tRDT r 2 r logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Separation of Variables – Bessel Equations

Introduction ∆ in a Rotationally Symmetric 2d Geometry Separating Polar Coordinates Separating the Equation ∆ u = k ∂ u ∂ t (Temporal Part) ∂ 2 u ∂ 2 u ∂ u k ∂ u ∂ r 2 + 1 ∂ r + 1 = r 2 ∂θ 2 ∂ t r u ( r , θ , t ) = R ( r ) D ( Θ ) T ( t ) = R · D · T ∂ 2 ∂ 2 ∂ k ∂ ∂ r 2 RDT + 1 ∂ rRDT + 1 = ∂θ 2 RDT ∂ tRDT r 2 r R ′′ DT logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Separation of Variables – Bessel Equations

Introduction ∆ in a Rotationally Symmetric 2d Geometry Separating Polar Coordinates Separating the Equation ∆ u = k ∂ u ∂ t (Temporal Part) ∂ 2 u ∂ 2 u ∂ u k ∂ u ∂ r 2 + 1 ∂ r + 1 = r 2 ∂θ 2 ∂ t r u ( r , θ , t ) = R ( r ) D ( Θ ) T ( t ) = R · D · T ∂ 2 ∂ 2 ∂ k ∂ ∂ r 2 RDT + 1 ∂ rRDT + 1 = ∂θ 2 RDT ∂ tRDT r 2 r R ′′ DT + 1 r R ′ DT logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Separation of Variables – Bessel Equations