Lecture 2.7: Bessels equation Matthew Macauley Department of - PowerPoint PPT Presentation

Lecture 2.7: Bessel’s equation Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 4340, Advanced Engineering Mathematics M. Macauley (Clemson) Lecture 2.7: Bessel’s equation Advanced Engineering Mathematics 1 / 6

Bessel’s equation The following ODE will arise when we solve the wave equation in polar coordinates: x 2 y ′′ + xy ′ + ( x 2 − ν 2 ) y = 0 , ν ∈ Z ≥ 0 . M. Macauley (Clemson) Lecture 2.7: Bessel’s equation Advanced Engineering Mathematics 2 / 6

Bessel’s equation: x 2 y ′′ + xy ′ + ( x 2 − ν 2 ) y = 0 ∞ � We assumed a generalized power series solution y ( x ) = x r a n x n , a 0 � = 0, and derived n =0 ( r 2 − ν 2 ) a 0 = 0 , [( r + 1) 2 − ν 2 ] a 1 = 0 , [( n + r ) 2 − ν 2 ] a n + a n − 2 = 0 , for n ≥ 2 . M. Macauley (Clemson) Lecture 2.7: Bessel’s equation Advanced Engineering Mathematics 3 / 6

Bessel functions of the first kind ∞ � x ( − 1) m � 2 m + ν � J ν ( x ) = . m !( ν + m )! 2 m =0 M. Macauley (Clemson) Lecture 2.7: Bessel’s equation Advanced Engineering Mathematics 4 / 6

Summary so far We solved Bessel’s equation: x 2 y ′′ + xy ′ + ( x 2 − ν 2 ) y = 0, using the Frobenius method, and found two generalized power series solutions: ∞ ∞ � a n x n , y 2 ( x ) = x − ν � a n x n . y 1 ( x ) = x ν n =0 n =0 Unfortuntely, if ν ∈ Z , these are not linearly independent. � 1 Since the Wronskian is W ( y 1 , y 2 ) = e − x = c x , both solutions can’t be bounded as x → 0. We called this first solution a Bessel function of the first kind. For each fixed ν , it is ∞ � x ( − 1) m � 2 m + ν � J ν ( x ) = . m !( ν + m )! 2 m =0 To find a second solution, we need to use variation of parameters: assume y 2 ( x ) = v ( x ) J ν ( x ) , and solve for v ( x ). Once normalized, this solution Y ν ( x ) is called a Bessel function of the second kind, and satisfies J α ( x ) cos( απ ) − J − α ( x ) Y ν ( x ) = lim . sin( απ ) α → ν M. Macauley (Clemson) Lecture 2.7: Bessel’s equation Advanced Engineering Mathematics 5 / 6

Bessel functions of the second kind ∞ � x ( − 1) m � 2 m + ν J α ( x ) cos( απ ) − J − α ( x ) � J ν ( x ) = Y ν ( x ) = lim , . m !( ν + m )! 2 sin( απ ) α → ν m =0 M. Macauley (Clemson) Lecture 2.7: Bessel’s equation Advanced Engineering Mathematics 6 / 6