integrable differential equations
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1: 21.9 Integrable Equations
§21.9 Integrable Equations
►Riemann theta functions arise in the study of integrable differential equations that have applications in many areas, including fluid mechanics (Ablowitz and Segur (1981, Chapter 4)), magnetic monopoles (Ercolani and Sinha (1989)), and string theory (Deligne et al. (1999, Part 3)). … ►2: 33.23 Methods of Computation
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§33.23(iii) Integration of Defining Differential Equations
…3: 9.17 Methods of Computation
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►A comprehensive and powerful approach is to integrate the defining differential equation (9.2.1) by direct numerical methods.
As described in §3.7(ii), to ensure stability the integration path must be chosen in such a way that as we proceed along it the wanted solution grows at least as fast as all other solutions of the differential equation.
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►In the case of the Scorer functions, integration of the differential equation (9.12.1) is more difficult than (9.2.1), because in some regions stable directions of integration do not exist.
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4: 28.34 Methods of Computation
5: 32.17 Methods of Computation
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►The Painlevé equations can be integrated by Runge–Kutta methods for ordinary differential equations; see §3.7(v), Hairer et al. (2000), and Butcher (2003).
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6: 14.32 Methods of Computation
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7: 10.74 Methods of Computation
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►A comprehensive and powerful approach is to integrate the differential equations (10.2.1) and (10.25.1) by direct numerical methods.
As described in §3.7(ii), to insure stability the integration path must be chosen in such a way that as we proceed along it the wanted solution grows in magnitude at least as fast as all other solutions of the differential equation.
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8: 15.19 Methods of Computation
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►A comprehensive and powerful approach is to integrate the hypergeometric differential equation (15.10.1) by direct numerical methods.
…However, since the growth near the singularities of the differential equation is algebraic rather than exponential, the resulting instabilities in the numerical integration might be tolerable in some cases.
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9: 13.29 Methods of Computation
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►A comprehensive and powerful approach is to integrate the differential equations (13.2.1) and (13.14.1) by direct numerical methods.
As described in §3.7(ii), to insure stability the integration path must be chosen in such a way that as we proceed along it the wanted solution grows in magnitude at least as fast as all other solutions of the differential equation.
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