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partial differential equations

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11: 22.19 Physical Applications
§22.19(iii) Nonlinear ODEs and PDEs
Many nonlinear ordinary and partial differential equations have solutions that may be expressed in terms of Jacobian elliptic functions. …
12: 20.13 Physical Applications
The functions θ j ( z | τ ) , j = 1 , 2 , 3 , 4 , provide periodic solutions of the partial differential equation
13: 16.13 Appell Functions
The following four functions of two real or complex variables x and y cannot be expressed as a product of two F 1 2 functions, in general, but they satisfy partial differential equations that resemble the hypergeometric differential equation (15.10.1): …
14: Bibliography T
  • E. C. Titchmarsh (1958) Eigenfunction Expansions Associated with Second Order Differential Equations, Part 2, Partial Differential Equations. Clarendon Press, Oxford.
  • 15: 3.11 Approximation Techniques
    From the equations S / a k = 0 , k = 0 , 1 , , n , we derive the normal equations
    16: 31.10 Integral Equations and Representations
    and the kernel 𝒦 ( z , t ) is a solution of the partial differential equation
    31.10.8 sin 2 θ ( 2 𝒦 θ 2 + ( ( 1 2 γ ) tan θ + 2 ( δ + ϵ 1 2 ) cot θ ) 𝒦 θ 4 α β 𝒦 ) + 2 𝒦 ϕ 2 + ( ( 1 2 δ ) cot ϕ ( 1 2 ϵ ) tan ϕ ) 𝒦 ϕ = 0 .
    and the kernel 𝒦 ( z ; s , t ) is a solution of the partial differential equation
    31.10.18 2 𝒦 u 2 + 2 𝒦 v 2 + 2 𝒦 w 2 + 2 γ 1 u 𝒦 u + 2 δ 1 v 𝒦 v + 2 ϵ 1 w 𝒦 w = 0 .
    17: 18.38 Mathematical Applications
    This process has been generalized to spectral methods for solving partial differential equations. …
    18: Bibliography S
  • B. I. Schneider, X. Guan, and K. Bartschat (2016) Time propagation of partial differential equations using the short iterative Lanczos method and finite-element discrete variable representation. Adv. Quantum Chem. 72, pp. 95–127.
  • 19: 32.4 Isomonodromy Problems
    Then the equation
    §32.4(ii) First Painlevé Equation
    §32.4(iii) Second Painlevé Equation
    §32.4(iv) Third Painlevé Equation
    20: 36.10 Differential Equations
    36.10.14 3 ( 2 Ψ ( E ) x 2 2 Ψ ( E ) y 2 ) + 2 i z Ψ ( E ) x x Ψ ( E ) = 0 .