Abstract
Direct formal integration (DFI; Payne [1]) is an analytic-numerical method that has been uniformly successful in solving a large number of nonlinear (NL) and linear physical problems. Applications to Prandtl boundary layer, Navier-Stokes, turbulence, cavity flow, aerodynamics, chaos, Tricomi transsonics and Euler problems have been described (cf. Payne [2] through [8]). Other successes include solid state physics, predator-prey systems, and flight and orbital mechanics. The handling of parabolic and elliptic PDEs is straightforward. One hyperbolic PDE (Tricomi) has been treated. DFI has three stages:
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1.
Formally integrate DEs along one or more trajectories thereby converting the DEs to Volterra-type IEs or IDEs of second kind.
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2.
Study all forms (IEs, IDEs, DEs) for new insights into the problem.
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3.
Solve the equations a) analytically by hand near initial points to discover the solution behavior there and b) numerically upon computer for details.
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References
F.R. Payne, Lecture Notes: 1981 AIAA Symposium, Arlington, TX (unpublished).
F.R. Payne, A simple conversion of two-point BVP, in Trends in the Theory and Practice of Nonlinear Analysis, North-Holland, Amsterdam, 1985, 377–385.
F.R. Payne, Direct formal integration (DFI): a global alternative to FDM/FEM, in Integral Methods in Science and Engineering, Hemisphere, Washington, 1986, 62–73.
F.R. Payne, A triad of solutions for 2-D Navier-Stokes: global, semilocal and local, in Integral Methods in Science and Engineering, Hemisphere, New York, 1991, 352–359.
F.R. Payne, Euler and inviscid Burger high-accuracy solutions, in Nonlinear Problems in Aerospace and Aviation, vol. 2, European Conference Publications, Cambridge, 2001, 601–606.
F.R. Payne, Hybrid Laplace and Poisson solvers. Part I: Dirichlet BCs, in Integral Methods in Science and Engineering, Birkhäuser, Boston, 2002, 203–209.
F.R. Payne and K.R. Payne, New facets of DFI, a DE solver for all seasons, in Integral Methods in Science and Engineering, Vol. 2 Pitman Res. Notes Math. Ser. 375, Longman, Harlow, 1997, 176–180.
F.R. Payne and K.R. Payne, Linear and sublinear Tricomi via DFI, in Integral Methods in Science and Engineering, Res. Notes Math. Ser. 418, Chapman & Hall/CRC, Boca Raton, FL, 2000, 268–273.
W.V. Lovitt, Linear Integral Equations, McGraw-Hill, Dover, 1950.
F.G. Tricomi, Integral Equations, Dover Publications, Inc., New York, 1985.
J. Liouville, Sur le dévelopment des fonctions ou parties de fonctions en séries. Sur la théorie des équations différentielles linéairs et les dévelopments des fonctions en séries, J. Math. 2 (1837), 16–22; 3 (1838), 561-614.
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Payne, F.R. (2004). Hybrid Laplace and Poisson Solvers. II: Robin BCs. In: Constanda, C., Largillier, A., Ahues, M. (eds) Integral Methods in Science and Engineering. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-0-8176-8184-5_30
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DOI: https://doi.org/10.1007/978-0-8176-8184-5_30
Publisher Name: Birkhäuser, Boston, MA
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