bessel s function

Bessels Function A Touch of Magic Fayez Karoji 1 Casey Tsai 1 Rachel - PowerPoint PPT Presentation

Introduction Application Properties Bessels Function A Touch of Magic Fayez Karoji 1 Casey Tsai 1 Rachel Weyrens 2 1 Department of Mathematics Louisiana State University 2 Department of Mathematics University of Arkansas SMILE REU Summer


  1. Introduction Application Properties Bessel’s Function A Touch of Magic Fayez Karoji 1 Casey Tsai 1 Rachel Weyrens 2 1 Department of Mathematics Louisiana State University 2 Department of Mathematics University of Arkansas SMILE REU Summer 2010 Karoji, Tsai, Weyrens Bessel Functions

  2. Introduction Application Properties Outline Introduction Bessel Functions Terminology Application Drum Example Properties Orthogonality Karoji, Tsai, Weyrens Bessel Functions

  3. Introduction Bessel Functions Application Terminology Properties General Form Bessel’s differential equation is x 2 y ′′ + xy ′ + ( x 2 − n 2 ) y = 0 Karoji, Tsai, Weyrens Bessel Functions

  4. Introduction Bessel Functions Application Terminology Properties General Form Bessel’s differential equation is x 2 y ′′ + xy ′ + ( x 2 − n 2 ) y = 0 The linearly independent solutions are J n and Y n . Karoji, Tsai, Weyrens Bessel Functions

  5. Introduction Bessel Functions Application Terminology Properties General Form Bessel’s differential equation is x 2 y ′′ + xy ′ + ( x 2 − n 2 ) y = 0 The linearly independent solutions are J n and Y n . The zeros are j n and y n . Karoji, Tsai, Weyrens Bessel Functions

  6. Introduction Bessel Functions Application Terminology Properties Bessel Functions of Order Zero /Bessel/j0.pdf Figure: Bessel Function of the First Kind, J 0 Karoji, Tsai, Weyrens Bessel Functions

  7. Introduction Bessel Functions Application Terminology Properties Bessel Functions of Order Zero /Bessel/y0.pdf Figure: Bessel Function of the Second Kind, Y 0 Karoji, Tsai, Weyrens Bessel Functions

  8. Introduction Bessel Functions Application Terminology Properties Key Terms ▶ Separation of variables Karoji, Tsai, Weyrens Bessel Functions

  9. Introduction Bessel Functions Application Terminology Properties Key Terms ▶ Separation of variables ▶ Regular singular Karoji, Tsai, Weyrens Bessel Functions

  10. Introduction Bessel Functions Application Terminology Properties Key Terms ▶ Separation of variables ▶ Regular singular ▶ Superposition Karoji, Tsai, Weyrens Bessel Functions

  11. Introduction Application Drum Example Properties Physical Description ▶ Radially symmetric Karoji, Tsai, Weyrens Bessel Functions

  12. Introduction Application Drum Example Properties Physical Description ▶ Radially symmetric ▶ Radius r = 1 Karoji, Tsai, Weyrens Bessel Functions

  13. Introduction Application Drum Example Properties Physical Description ▶ Radially symmetric ▶ Radius r = 1 ▶ Beginning at rest Karoji, Tsai, Weyrens Bessel Functions

  14. Introduction Application Drum Example Properties Physical Description ▶ Radially symmetric ▶ Radius r = 1 ▶ Beginning at rest ▶ Edges fixed Karoji, Tsai, Weyrens Bessel Functions

  15. Introduction Application Drum Example Properties Boundary Valued Problem This physical problem can be represented by the following boundary valued problem: Karoji, Tsai, Weyrens Bessel Functions

  16. Introduction Application Drum Example Properties Boundary Valued Problem This physical problem can be represented by the following boundary valued problem: ▶ u tt = u rr + 1 r u r Karoji, Tsai, Weyrens Bessel Functions

  17. Introduction Application Drum Example Properties Boundary Valued Problem This physical problem can be represented by the following boundary valued problem: ▶ u tt = u rr + 1 r u r ▶ u ( r , 0 ) = f ( r ) Karoji, Tsai, Weyrens Bessel Functions

  18. Introduction Application Drum Example Properties Boundary Valued Problem This physical problem can be represented by the following boundary valued problem: ▶ u tt = u rr + 1 r u r ▶ u ( r , 0 ) = f ( r ) ▶ u t ( r , 0 ) = 0 Karoji, Tsai, Weyrens Bessel Functions

  19. Introduction Application Drum Example Properties Boundary Valued Problem This physical problem can be represented by the following boundary valued problem: ▶ u tt = u rr + 1 r u r ▶ u ( r , 0 ) = f ( r ) ▶ u t ( r , 0 ) = 0 ▶ u ( 1 , t ) = 0 Karoji, Tsai, Weyrens Bessel Functions

  20. Introduction Application Drum Example Properties Separation of Variables We have u ( r , t ) = R ( r ) T ( t ) ▶ T ′′ + 휇 T = 0 ▶ R ′′ + 1 r R ′ + 휇 R = 0 Karoji, Tsai, Weyrens Bessel Functions

  21. Introduction Application Drum Example Properties Separation of Variables We have u ( r , t ) = R ( r ) T ( t ) ▶ T ′′ + 휇 T = 0 ▶ R ′′ + 1 r R ′ + 휇 R = 0 ▶ 휇 = 훼 2 > 0 Karoji, Tsai, Weyrens Bessel Functions

  22. Introduction Application Drum Example Properties Solutions The solutions of the given ODE’s are Karoji, Tsai, Weyrens Bessel Functions

  23. Introduction Application Drum Example Properties Solutions The solutions of the given ODE’s are ▶ T ( t ) = c 1 cos ( 훼 t ) + c 2 sin ( 훼 t ) Karoji, Tsai, Weyrens Bessel Functions

  24. Introduction Application Drum Example Properties Solutions The solutions of the given ODE’s are ▶ T ( t ) = c 1 cos ( 훼 t ) + c 2 sin ( 훼 t ) ▶ R ( r ) = c 3 J 0 ( 훼 r ) + c 4 Y 0 ( 훼 r ) Karoji, Tsai, Weyrens Bessel Functions

  25. Introduction Application Drum Example Properties Evaluation Using initial and boundary conditions, we have u n ( r , t ) = A n J 0 ( j n r ) cos ( j n t ) Karoji, Tsai, Weyrens Bessel Functions

  26. Introduction Application Drum Example Properties Evaluation Using initial and boundary conditions, we have u n ( r , t ) = A n J 0 ( j n r ) cos ( j n t ) General solution Karoji, Tsai, Weyrens Bessel Functions

  27. Introduction Application Drum Example Properties Evaluation Using initial and boundary conditions, we have u n ( r , t ) = A n J 0 ( j n r ) cos ( j n t ) General solution ∞ ∑ u ( r , t ) = A n J 0 ( j n r ) cos ( j n t ) n = 1 Karoji, Tsai, Weyrens Bessel Functions

  28. Introduction Application Drum Example Properties Amplitude The amplitude of displacement, from u ( r , 0 ) = f ( r ) is: Karoji, Tsai, Weyrens Bessel Functions

  29. Introduction Application Drum Example Properties Amplitude The amplitude of displacement, from u ( r , 0 ) = f ( r ) is: ∫ 1 0 rJ 0 ( j n r ) f ( r ) dr A n = ∫ 1 0 rJ 0 ( j n r ) J 0 ( j n r ) dr Karoji, Tsai, Weyrens Bessel Functions

  30. Introduction Application Drum Example Properties Frequencies j 1 Fundamental pitch 2 휋 j 2 First overtone 2 휋 j 3 Second overtone 2 휋 Karoji, Tsai, Weyrens Bessel Functions

  31. Introduction Application Orthogonality Properties Orthogonality Property of Bessel Functions /Bessel/jnspdf.pdf Figure: Bessel Functions of the First Kind Karoji, Tsai, Weyrens Bessel Functions

  32. Introduction Application Orthogonality Properties Problems in Mathematical Physics ▶ PDE’s model physical phenomena. ▶ Example: Steady Temperatures in Circular Cylinder (Laplacian in Cylindrical Coordinates). ▶ Example: The Vibrating Drumhead (Wave Equation in Polar Coordinates). Karoji, Tsai, Weyrens Bessel Functions

  33. Introduction Application Orthogonality Properties Methods of Solution ▶ PDE’s are difficult to solve. ▶ Fourier’s Method: Linear and homogeneous PDE’s with homogeneous boundary conditions. ▶ Also known as Separation of Variables. Karoji, Tsai, Weyrens Bessel Functions

  34. Introduction Application Orthogonality Properties Fourier’s Method: PDE − → ODE’s ▶ PDE: Wave Equation in Polar Coordinates ▶ Apply Fourier’s Method ▶ Two second order ODE’s ▶ Simple Harmonic Motion T ′′ + 휇 T = 0 ▶ Bessel’s Equation R ′′ + 1 r R ′ + R = 0 Karoji, Tsai, Weyrens Bessel Functions

  35. Introduction Application Orthogonality Properties Orthogonal Functions ▶ Analysis of solutions to ODE’s ▶ Underlying Theme: Orthogonal Functions ▶ Examples: ▶ Sine and Cosine Functions ▶ Legendre Polynomials (Special Function) ▶ Bessel Functions (A "Very" Special Function) Karoji, Tsai, Weyrens Bessel Functions

  36. Introduction Application Orthogonality Properties What is Orthogonality? ▶ Dot Product or Inner Product in ℝ n Karoji, Tsai, Weyrens Bessel Functions

  37. Introduction Application Orthogonality Properties What is Orthogonality? ▶ Dot Product or Inner Product in ℝ n ▶ Given x , y ∈ ℝ n Karoji, Tsai, Weyrens Bessel Functions

  38. Introduction Application Orthogonality Properties What is Orthogonality? ▶ Dot Product or Inner Product in ℝ n ▶ Given x , y ∈ ℝ n ▶ Define x ⋅ y = ∑ n i = 1 x i y i Karoji, Tsai, Weyrens Bessel Functions

  39. Introduction Application Orthogonality Properties What is Orthogonality? ▶ Dot Product or Inner Product in ℝ n ▶ Given x , y ∈ ℝ n ▶ Define x ⋅ y = ∑ n i = 1 x i y i ▶ x and y are orthogonal when ∑ n i = 1 x i y i = 0 Karoji, Tsai, Weyrens Bessel Functions

  40. Introduction Application Orthogonality Properties Generalize Orthogonality ▶ Inner Product in ℛ [ a , b ] Karoji, Tsai, Weyrens Bessel Functions

  41. Introduction Application Orthogonality Properties Generalize Orthogonality ▶ Inner Product in ℛ [ a , b ] ▶ Given f , g ∈ ℛ [ a , b ] Karoji, Tsai, Weyrens Bessel Functions

  42. Introduction Application Orthogonality Properties Generalize Orthogonality ▶ Inner Product in ℛ [ a , b ] ▶ Given f , g ∈ ℛ [ a , b ] ∫ b ▶ Define ⟨ f , g ⟩ = a f ( x ) g ( x ) dx Karoji, Tsai, Weyrens Bessel Functions

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