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Airy equation

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1: 9.16 Physical Applications
A quite different application is made in the study of the diffraction of sound pulses by a circular cylinder (Friedlander (1958)). … Again, the quest for asymptotic approximations that are uniformly valid solutions to this equation in the neighborhoods of critical points leads (after choosing solvable equations with similar asymptotic properties) to Airy functions. …An application of Airy functions to the solution of this equation is given in Gramtcheff (1981). … Reference to many of these applications as well as to the theory of elasticity and to the heat equation are given in Vallée and Soares (2010): a book devoted specifically to the Airy and Scorer functions and their applications in physics. … Solutions of the Schrödinger equation involving the Airy functions are given for other potentials in Vallée and Soares (2010). …
2: 9.2 Differential Equation
§9.2(i) Airy’s Equation
§9.2(ii) Initial Values
§9.2(iii) Numerically Satisfactory Pairs of Solutions
Table 9.2.1: Numerically satisfactory pairs of solutions of Airy’s equation.
Pair Interval or Region
§9.2(vi) Riccati Form of Differential Equation
3: 9.13 Generalized Airy Functions
§9.13 Generalized Airy Functions
§9.13(i) Generalizations from the Differential Equation
Each of the functions A k ( z , p ) and B k ( z , p ) satisfies the differential equation …and the difference equation
4: 9.11 Products
§9.11(i) Differential Equation
5: Bibliography S
  • Z. Schulten, D. G. M. Anderson, and R. G. Gordon (1979) An algorithm for the evaluation of the complex Airy functions. J. Comput. Phys. 31 (1), pp. 60–75.
  • D. R. Smith (1990) A Riccati approach to the Airy equation. In Asymptotic and computational analysis (Winnipeg, MB, 1989), R. Wong (Ed.), pp. 403–415.
  • C. A. Swanson and V. B. Headley (1967) An extension of Airy’s equation. SIAM J. Appl. Math. 15 (6), pp. 1400–1412.
  • 6: 9.10 Integrals
    Let w ( z ) be any solution of Airy’s equation (9.2.1). …
    9.10.18 Ai ( z ) = 3 z 5 / 4 e ( 2 / 3 ) z 3 / 2 4 π 0 t 3 / 4 e ( 2 / 3 ) t 3 / 2 Ai ( t ) z 3 / 2 + t 3 / 2 d t , | ph z | < 2 3 π .
    9.10.19 Bi ( x ) = 3 x 5 / 4 e ( 2 / 3 ) x 3 / 2 2 π 0 t 3 / 4 e ( 2 / 3 ) t 3 / 2 Ai ( t ) x 3 / 2 t 3 / 2 d t , x > 0 ,
    7: 36.10 Differential Equations
    K = 1 , fold: (36.10.1) becomes Airy’s equation9.2(i)) …
    8: 9.18 Tables
  • Smirnov (1960) tabulates U 1 ( x , α ) , U 2 ( x , α ) , defined by (9.13.20), (9.13.21), and also U 1 ( x , α ) / x , U 2 ( x , α ) / x , for α = 1 , x = 6 ( .01 ) 10 to 5D or 5S, and also for α = ± 1 4 , ± 1 3 , ± 1 2 , ± 2 3 , ± 3 4 , 5 4 , 4 3 , 3 2 , 5 3 , 7 4 , 2, x = 0 ( .01 ) 6 ; 4D.

  • 9: 9.17 Methods of Computation
    §9.17 Methods of Computation
    §9.17(ii) Differential Equations
    The methods for Ai ( z ) are similar. …
    §9.17(v) Zeros
    For the computation of the zeros of the Scorer functions and their derivatives see Gil et al. (2003c).
    10: 9.12 Scorer Functions
    where A and B are arbitrary constants, w 1 ( z ) and w 2 ( z ) are any two linearly independent solutions of Airy’s equation (9.2.1), and p ( z ) is any particular solution of (9.12.1). …
    9.12.20 Hi ( z ) = 1 π 0 exp ( 1 3 t 3 + z t ) d t ,
    9.12.30 0 z Gi ( t ) d t 1 π ln z + 2 γ + ln 3 3 π 1 π k = 1 ( 3 k 1 ) ! k ! ( 3 z 3 ) k , | ph z | 1 3 π δ .
    9.12.31 0 z Hi ( t ) d t 1 π ln z + 2 γ + ln 3 3 π + 1 π k = 1 ( 1 ) k 1 ( 3 k 1 ) ! k ! ( 3 z 3 ) k , | ph z | 2 3 π δ ,