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
Introduction Application Properties Outline Introduction Bessel Functions Terminology Application Drum Example Properties Orthogonality Karoji, Tsai, Weyrens Bessel Functions
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
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
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
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
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
Introduction Bessel Functions Application Terminology Properties Key Terms ▶ Separation of variables Karoji, Tsai, Weyrens Bessel Functions
Introduction Bessel Functions Application Terminology Properties Key Terms ▶ Separation of variables ▶ Regular singular Karoji, Tsai, Weyrens Bessel Functions
Introduction Bessel Functions Application Terminology Properties Key Terms ▶ Separation of variables ▶ Regular singular ▶ Superposition Karoji, Tsai, Weyrens Bessel Functions
Introduction Application Drum Example Properties Physical Description ▶ Radially symmetric Karoji, Tsai, Weyrens Bessel Functions
Introduction Application Drum Example Properties Physical Description ▶ Radially symmetric ▶ Radius r = 1 Karoji, Tsai, Weyrens Bessel Functions
Introduction Application Drum Example Properties Physical Description ▶ Radially symmetric ▶ Radius r = 1 ▶ Beginning at rest Karoji, Tsai, Weyrens Bessel Functions
Introduction Application Drum Example Properties Physical Description ▶ Radially symmetric ▶ Radius r = 1 ▶ Beginning at rest ▶ Edges fixed Karoji, Tsai, Weyrens Bessel Functions
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
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
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
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
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
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
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
Introduction Application Drum Example Properties Solutions The solutions of the given ODE’s are Karoji, Tsai, Weyrens Bessel Functions
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
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
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
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
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
Introduction Application Drum Example Properties Amplitude The amplitude of displacement, from u ( r , 0 ) = f ( r ) is: Karoji, Tsai, Weyrens Bessel Functions
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
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
Introduction Application Orthogonality Properties Orthogonality Property of Bessel Functions /Bessel/jnspdf.pdf Figure: Bessel Functions of the First Kind Karoji, Tsai, Weyrens Bessel Functions
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
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
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
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
Introduction Application Orthogonality Properties What is Orthogonality? ▶ Dot Product or Inner Product in ℝ n Karoji, Tsai, Weyrens Bessel Functions
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
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
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
Introduction Application Orthogonality Properties Generalize Orthogonality ▶ Inner Product in ℛ [ a , b ] Karoji, Tsai, Weyrens Bessel Functions
Introduction Application Orthogonality Properties Generalize Orthogonality ▶ Inner Product in ℛ [ a , b ] ▶ Given f , g ∈ ℛ [ a , b ] Karoji, Tsai, Weyrens Bessel Functions
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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