I am searching for applications of first or second-order non-linear ordinary differential equations.
The following are three excellent books on the subject:
1) Polyanin, A. D., V. F. Zaitsev and A. Moussiaux (2002), Handbook of First Order Partial Differential Equations, Taylor & Francis.
2) Polyanin, A. D. and V. F. Zaitsev (2004), Handbook of Nonlinear Partial Differential Equations, Chapman and Hall/CRC Press.
3) Polyanin, A. D. and A. V. Manzhirov (2007), Handbook of Mathematics For Engineers and Scientists, Chapman and Hall/CRC Press.
Regards,
Graham W Griffiths
Article Nonlinear Differential Equations in Physics and Their Geomet...
Schrödinger wave equation in physics, Riccati differential equation, Able's equation, Emden Fawler equations are the most common nonlinear differential equations.
1. S. Ghosh, A. Roy, and D. Roy, “An adaptation of Adomian decomposition for numeric–analytic integration of strongly nonlinear and chaotic oscillators,” Computer Methods in Applied Mechanics and Engineering, vol. 196, pp. 1133–1153, Jan 2007.
2. M. Arya and A. Ujlayan, “Solution of Riccati Differential Equation With Green’s Function and Padè Approximation Technique,” Adv. Differ. Equations Control Process., vol. 21, no. 1, pp. 31–52, 2019.
3. M. Dehghan and A. Taleei, “A compact split-step finite difference method for solving the nonlinear Schrödinger equations with constant and variable coefficients,” Comput. Phys. Commun., vol. 181, no. 1, pp. 43–51, Jan. 2010.
4. R. Singh, J. Kumar, and G. Nelakanti, “Approximate series solution of nonlinear singular boundary value problems arising in physiology,” The Scientific World Journal, vol. 2014,
@Bahram Agheli
Second order see mathematical physics book .Controlled and uncontrolled vibrations by B.Rath .
LC,LCR ,SHO are simple examples . Heat flow etc . Hope it will be helpful to you .
Biswanath Rath.
The following are three excellent books on the subject:
1) Polyanin, A. D., V. F. Zaitsev and A. Moussiaux (2002), Handbook of First Order Partial Differential Equations, Taylor & Francis.
2) Polyanin, A. D. and V. F. Zaitsev (2004), Handbook of Nonlinear Partial Differential Equations, Chapman and Hall/CRC Press.
3) Polyanin, A. D. and A. V. Manzhirov (2007), Handbook of Mathematics For Engineers and Scientists, Chapman and Hall/CRC Press.
Regards,
Graham W Griffiths
The well-known Duffing and Mathieu equations, wit a lot of applications in both issues. See "The Duffing equation: Nonlinear oscillations and their behavior", Ivana Kovacic, M.J Brennan
There are many - some already given. The Van der Pol equation is one of the more interesting in that it not only pops up in the physical sciences, Van der Pol Oscillator, but also in biology (model for action potentials of neurons) and other disciplines.
https://en.wikipedia.org/wiki/Van_der_Pol_oscillator
https://www.cs.bham.ac.uk/~szh/teaching/matlabmodeling/Lecture10_body.pdf
The equation originates from Dutch electrical engineer Balhasar van der Pol to describe oscillations in circuits using vacuum tubes in the 1920. It has been a wealth of examples of the phenomena of behavior that have been discovered about nonlinear systems. Cartwright in a 1952 paper described behavior of the forced van der Pol equation that later became known as Chaotic behavior. The bifurcation properties of the forced van der Pol equation are extremely rich and interesting.
http://math.colgate.edu/~wweckesser/pubs/FVDPI.pdf
The Riccati equation mentioned above has a rich interaction the asymptotic integration of linear Hamiltonian systems and of course flows in the Symplectic Group Sp(n), equivalently to properties of the Lie Algebra of vector fields over the Sp(n).
See
Article Boundary value problems for second order, ordinary different...
for the scalar case and
Article Singular boundary value problems for linear Hamiltonian syst...
for the case of a liner system.
In both cases the properties of the corresponding Riccati equation played a critical role in determining the properties of the behavior of the Hamiltonian system - both one dimensional and n-dimensional.
The Riccati equation has been generalized to normed division algebras, i.e., quaternions, octonions where it plays a key role in Quaternionic quantum field theory.
Again, like the van der Pol equation, Duffing equation, the Riccati equation is a simple looking equation that keeps popping up in many a diverse places.
Nonlinear difference equations for a modified Laguerre polynomials.
Non linear difference equations arising from a deformation of the q-Laguerre weight
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