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Nonlinear ones are of particular interest for their commonality in describing real-world systems and how much more difficult they are to solve compared to linear differential equations. This list presents nonlinear ordinary differential equations that have been named, sorted by area of interest.
For nonlinear equations these questions are in general very hard: for example, the hardest part of Yau's solution of the Calabi conjecture was the proof of existence for a Monge–Ampere equation. The open problem of existence (and smoothness) of solutions to the Navier–Stokes equations is one of the seven Millennium Prize problems in ...
In mathematics and science, a nonlinear system (or a non-linear system) is a system in which the change of the output is not proportional to the change of the input. [1] [2] Nonlinear problems are of interest to engineers, biologists, [3] [4] [5] physicists, [6] [7] mathematicians, and many other scientists since most systems are inherently nonlinear in nature. [8]
In complex analysis, the Riccati equation occurs as the first-order nonlinear ODE in the complex plane of the form [4] = (,) = (,) (,), where and are polynomials in and locally analytic functions of , i.e., is a complex rational function. The only equation of this form that is of Painlevé type, is the Riccati equation = + + (), where () are ...
Hence, an example of a linear equation would be: [1] = + () (,) As a note on naming convention: i) u(x) is called the unknown function, ii) f(x) is called a known function, iii) K(x,t) is a function of two variables and often called the Kernel function, and iv) λ is an unknown factor or parameter, which plays the same role as the eigenvalue in ...
Anyway, using the homotopy analysis method or harmonic balance, one can derive a frequency response equation in the following form: [9] [5] [() + ()] =. For the parameters of the Duffing equation, the above algebraic equation gives the steady state oscillation amplitude z {\displaystyle z} at a given excitation frequency.
For example, consider the ordinary differential equation ′ = + The Euler method for solving this equation uses the finite difference quotient (+) ′ to approximate the differential equation by first substituting it for u'(x) then applying a little algebra (multiplying both sides by h, and then adding u(x) to both sides) to get (+) + (() +).
The linear approximation of a function is the first order Taylor expansion around the point of interest. In the study of dynamical systems, linearization is a method for assessing the local stability of an equilibrium point of a system of nonlinear differential equations or discrete dynamical systems. [1]