Bessel Functions Bernd Schr oder logo1 Bernd Schr oder - PowerPoint PPT Presentation

Overview Solving the Bessel Equation Bessel Functions Application Bessel Functions Bernd Schr¨ oder logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Bessel Functions

Overview Solving the Bessel Equation Bessel Functions Application Why are Bessel Functions Important? logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Bessel Functions

Overview Solving the Bessel Equation Bessel Functions Application Why are Bessel Functions Important? 1. Parametric Bessel equations x 2 y ′′ + xy ′ + � λ 2 x 2 − ν 2 � y = 0 ∂ t and ∆ u = k ∂ 2 u arise when the equations ∆ u = k ∂ u ∂ t 2 are solved with separation of variables in polar or cylindrical coordinates. The function y ( r ) describes the radial part of the solution. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Bessel Functions

Overview Solving the Bessel Equation Bessel Functions Application Why are Bessel Functions Important? 1. Parametric Bessel equations x 2 y ′′ + xy ′ + � λ 2 x 2 − ν 2 � y = 0 ∂ t and ∆ u = k ∂ 2 u arise when the equations ∆ u = k ∂ u ∂ t 2 are solved with separation of variables in polar or cylindrical coordinates. The function y ( r ) describes the radial part of the solution. 2. Because 0 is a regular singular point of the equation, it is natural to attempt a solution using the method of Frobenius. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Bessel Functions

Overview Solving the Bessel Equation Bessel Functions Application Frobenius Solution for x 2 y ′′ + xy ′ + λ 2 x 2 − ν 2 � � y = 0 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Bessel Functions

Overview Solving the Bessel Equation Bessel Functions Application Frobenius Solution for x 2 y ′′ + xy ′ + λ 2 x 2 − ν 2 � � y = 0 x 2 y ′′ + xy ′ + λ 2 x 2 − ν 2 � � y = 0 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Bessel Functions

Overview Solving the Bessel Equation Bessel Functions Application Frobenius Solution for x 2 y ′′ + xy ′ + λ 2 x 2 − ν 2 � � y = 0 x 2 y ′′ + xy ′ + λ 2 x 2 − ν 2 � � y = 0 ∞ c n ( n + r )( n + r − 1 ) x n + r − 2 x 2 ∑ n = 0 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Bessel Functions

Overview Solving the Bessel Equation Bessel Functions Application Frobenius Solution for x 2 y ′′ + xy ′ + λ 2 x 2 − ν 2 � � y = 0 x 2 y ′′ + xy ′ + λ 2 x 2 − ν 2 � � y = 0 ∞ ∞ c n ( n + r )( n + r − 1 ) x n + r − 2 + x c n ( n + r ) x n + r − 1 x 2 ∑ ∑ n = 0 n = 0 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Bessel Functions

Overview Solving the Bessel Equation Bessel Functions Application Frobenius Solution for x 2 y ′′ + xy ′ + λ 2 x 2 − ν 2 � � y = 0 x 2 y ′′ + xy ′ + λ 2 x 2 − ν 2 � � y = 0 ∞ ∞ λ 2 x 2 − ν 2 � ∞ c n ( n + r )( n + r − 1 ) x n + r − 2 + x c n ( n + r ) x n + r − 1 + c n x n + r x 2 ∑ ∑ � ∑ n = 0 n = 0 n = 0 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Bessel Functions

Overview Solving the Bessel Equation Bessel Functions Application Frobenius Solution for x 2 y ′′ + xy ′ + λ 2 x 2 − ν 2 � � y = 0 x 2 y ′′ + xy ′ + λ 2 x 2 − ν 2 � � y = 0 ∞ ∞ λ 2 x 2 − ν 2 � ∞ c n x n + r = 0 c n ( n + r )( n + r − 1 ) x n + r − 2 + x c n ( n + r ) x n + r − 1 + x 2 ∑ ∑ � ∑ n = 0 n = 0 n = 0 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Bessel Functions

Overview Solving the Bessel Equation Bessel Functions Application Frobenius Solution for x 2 y ′′ + xy ′ + λ 2 x 2 − ν 2 � � y = 0 x 2 y ′′ + xy ′ + λ 2 x 2 − ν 2 � � y = 0 ∞ ∞ λ 2 x 2 − ν 2 � ∞ c n x n + r = 0 c n ( n + r )( n + r − 1 ) x n + r − 2 + x c n ( n + r ) x n + r − 1 + x 2 ∑ ∑ � ∑ n = 0 n = 0 n = 0 ∞ ( n + r )( n + r − 1 ) c n x n + r ∑ n = 0 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Bessel Functions

Overview Solving the Bessel Equation Bessel Functions Application Frobenius Solution for x 2 y ′′ + xy ′ + λ 2 x 2 − ν 2 � � y = 0 x 2 y ′′ + xy ′ + λ 2 x 2 − ν 2 � � y = 0 ∞ ∞ λ 2 x 2 − ν 2 � ∞ c n x n + r = 0 c n ( n + r )( n + r − 1 ) x n + r − 2 + x c n ( n + r ) x n + r − 1 + x 2 ∑ ∑ � ∑ n = 0 n = 0 n = 0 ∞ ∞ ( n + r )( n + r − 1 ) c n x n + r + ( n + r ) c n x n + r ∑ ∑ n = 0 n = 0 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Bessel Functions

Overview Solving the Bessel Equation Bessel Functions Application Frobenius Solution for x 2 y ′′ + xy ′ + λ 2 x 2 − ν 2 � � y = 0 x 2 y ′′ + xy ′ + λ 2 x 2 − ν 2 � � y = 0 ∞ ∞ λ 2 x 2 − ν 2 � ∞ c n x n + r = 0 c n ( n + r )( n + r − 1 ) x n + r − 2 + x c n ( n + r ) x n + r − 1 + x 2 ∑ ∑ � ∑ n = 0 n = 0 n = 0 ∞ ∞ ∞ ( n + r )( n + r − 1 ) c n x n + r + ( n + r ) c n x n + r + λ 2 c n x n + r + 2 ∑ ∑ ∑ n = 0 n = 0 n = 0 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Bessel Functions

Overview Solving the Bessel Equation Bessel Functions Application Frobenius Solution for x 2 y ′′ + xy ′ + λ 2 x 2 − ν 2 � � y = 0 x 2 y ′′ + xy ′ + λ 2 x 2 − ν 2 � � y = 0 ∞ ∞ λ 2 x 2 − ν 2 � ∞ c n x n + r = 0 c n ( n + r )( n + r − 1 ) x n + r − 2 + x c n ( n + r ) x n + r − 1 + x 2 ∑ ∑ � ∑ n = 0 n = 0 n = 0 ∞ ∞ ∞ ∞ ( n + r )( n + r − 1 ) c n x n + r + ( n + r ) c n x n + r + λ 2 c n x n + r + 2 − ν 2 c n x n + r ∑ ∑ ∑ ∑ n = 0 n = 0 n = 0 n = 0 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Bessel Functions

Overview Solving the Bessel Equation Bessel Functions Application Frobenius Solution for x 2 y ′′ + xy ′ + λ 2 x 2 − ν 2 � � y = 0 x 2 y ′′ + xy ′ + λ 2 x 2 − ν 2 � � y = 0 ∞ ∞ λ 2 x 2 − ν 2 � ∞ c n x n + r = 0 c n ( n + r )( n + r − 1 ) x n + r − 2 + x c n ( n + r ) x n + r − 1 + x 2 ∑ ∑ � ∑ n = 0 n = 0 n = 0 ∞ ∞ ∞ ∞ ν 2 c n x n + r = 0 ( n + r )( n + r − 1 ) c n x n + r + ( n + r ) c n x n + r + λ 2 c n x n + r + 2 − ∑ ∑ ∑ ∑ n = 0 n = 0 n = 0 n = 0 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Bessel Functions

Overview Solving the Bessel Equation Bessel Functions Application Frobenius Solution for x 2 y ′′ + xy ′ + λ 2 x 2 − ν 2 � � y = 0 x 2 y ′′ + xy ′ + λ 2 x 2 − ν 2 � � y = 0 ∞ ∞ λ 2 x 2 − ν 2 � ∞ c n x n + r = 0 c n ( n + r )( n + r − 1 ) x n + r − 2 + x c n ( n + r ) x n + r − 1 + x 2 ∑ ∑ � ∑ n = 0 n = 0 n = 0 ∞ ∞ ∞ ∞ ν 2 c n x n + r = 0 ( n + r )( n + r − 1 ) c n x n + r + ( n + r ) c n x n + r + λ 2 c n x n + r + 2 − ∑ ∑ ∑ ∑ n = 0 n = 0 n = 0 n = 0 ∞ ( k + r )( k + r − 1 ) c k x k + r ∑ k = 0 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Bessel Functions

Overview Solving the Bessel Equation Bessel Functions Application Frobenius Solution for x 2 y ′′ + xy ′ + λ 2 x 2 − ν 2 � � y = 0 x 2 y ′′ + xy ′ + λ 2 x 2 − ν 2 � � y = 0 ∞ ∞ λ 2 x 2 − ν 2 � ∞ c n x n + r = 0 c n ( n + r )( n + r − 1 ) x n + r − 2 + x c n ( n + r ) x n + r − 1 + x 2 ∑ ∑ � ∑ n = 0 n = 0 n = 0 ∞ ∞ ∞ ∞ ν 2 c n x n + r = 0 ( n + r )( n + r − 1 ) c n x n + r + ( n + r ) c n x n + r + λ 2 c n x n + r + 2 − ∑ ∑ ∑ ∑ n = 0 n = 0 n = 0 n = 0 ∞ ∞ ( k + r )( k + r − 1 ) c k x k + r + ( k + r ) c k x k + r ∑ ∑ k = 0 k = 0 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Bessel Functions

Overview Solving the Bessel Equation Bessel Functions Application Frobenius Solution for x 2 y ′′ + xy ′ + λ 2 x 2 − ν 2 � � y = 0 x 2 y ′′ + xy ′ + λ 2 x 2 − ν 2 � � y = 0 ∞ ∞ λ 2 x 2 − ν 2 � ∞ c n x n + r = 0 c n ( n + r )( n + r − 1 ) x n + r − 2 + x c n ( n + r ) x n + r − 1 + x 2 ∑ ∑ � ∑ n = 0 n = 0 n = 0 ∞ ∞ ∞ ∞ ν 2 c n x n + r = 0 ( n + r )( n + r − 1 ) c n x n + r + ( n + r ) c n x n + r + λ 2 c n x n + r + 2 − ∑ ∑ ∑ ∑ n = 0 n = 0 n = 0 n = 0 ∞ ∞ ∞ ( k + r )( k + r − 1 ) c k x k + r + ( k + r ) c k x k + r + λ 2 c k − 2 x k + r ∑ ∑ ∑ k = 0 k = 0 k = 2 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Bessel Functions