Handbook of Nonlinear Partial Differential Equations

By Andrei D. Polyanin, V. Zaitsev. 2003

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On the Identifiability of the Post-Nonlinear Causal Model (2009, neutral): “…of the functions mentioned above is constantly zero, it can be shown that the functional equation of the form (12) has three solutions, which are (Polyanin & Zaitsev, 2004) Solution 1: Φ1 = A1Φ3 + A2Φ4, Φ2 = A3Φ3 + A4Φ4, Ψ3 = −A1Ψ1 − A3Ψ2, Ψ4 = −A2Ψ1 − A4Ψ2, (14) Solution 2: Φ1 = B1Φ3, Φ2 =…”
Application of homotopy-perturbation and variational iteration methods to nonlinear heat transfer and porous media equations (2007, background): “This equation often occurs in nonlinear problems of heat and mass transfer, combustion theory and flows in porous media [23].”
On Causal Discovery with Cyclic Additive Noise Models (2011, methodology): “Inspired by the identiﬁability proof in [13], we adopt the solution method from [14, Supplement S.4.3] that gives a general method for solving functional-differential equations of the form where the functionals Φ i ( x ) and Ψ i ( y ) depend only on x and y , respectively: The idea behind the…”