Hi, Dear Hossein

you can use Runge-Kutta Method to solve nonlinear differential equation.

Dear Hossein

The unplesent fact about nonlinear differential equations of even second order

is that there is no unifying general theory  for them. And what do you mean by  solution ? And do you consider equation which are in normal form i.e

y''=f(t,y,y')  in general equations g(t,y',y'')=0 ?

1) If you want to solve the equation with subject to some initial condition y(t0)=t0 and  y'(t0)=y'0 you can of course resort to some numerical solution method. What method you should use then depends about the equation i.e is it stiff etc...

2) Do you want to know if the solution can be represented with elementary functions in closed form. Usally this can not be done and in the age of modern computers this question has almost totally lost its practical value...

Series solutions are usally not very practical for nonlinear DE:s, but if you want to

consider for example perturbation methods or asymptotic expansions there are

special books for those too.  

If you want to consider numerics a really detailed look to different numerical methods can be found from two books of Hairer and Wanner "Solving Ordinary Differential Equations 1&2"

a) http://www.amazon.com/Solving-Ordinary-Differential-Equations-Computational/dp/3642051634

b) http://www.amazon.com/Solving-Ordinary-Differential-Equations-Computational/dp/3642051634

If you absolutely need to consider 2) there are actually some approaches: For example finding Lie-symmetries and infinidesimal generators for  the equation.

These techniques are presented in quite elementary level in the book by Nail. H Ibragimoc "A Practical Course in Differential Equations and Mathematical Modelling" .

http://www.amazon.com/Practical-Differential-Equations-Mathematical-Modelling/dp/9814291943

In general this book might be a good book to start "self-study" of DE:s. Its preliminary expectations are not so high and yet it covers some quite modern methods of DE-analysis.

3) If you are interested about the DE:s in "dynamical point of view" then there are

good introdoctory level books for dynamical systems as well. A good book to start might be"Differential Equations, Dynamical Systems, and an Introduction to Chaos" by Hirch, Smale and Devaney

a) http://www.amazon.com/Differential-Equations-Dynamical-Systems-Introduction/dp/0123820103

Well here are just some quick suggestions...

regards

Samuli

Dear Hossein

The unplesent fact about nonlinear differential equations of even second order

is that there is no unifying general theory for them. And what do you mean by solution ? And do you consider equation which are in normal form i.e

y''=f(t,y,y') or in general equations g(t,y',y'')=0 ?

1) If you want to solve the equation with subject to some initial condition y(t0)=t0 and y'(t0)=y'0 you can of course resort to some numerical solution method. What method you should use then depends about the equation i.e is it stiff etc...

2) Do you want to know if the solution can be represented with elementary functions in closed form. Usally this can not be done and in the age of modern computers this question has almost totally lost its practical value...

Series solutions are usally not very practical for nonlinear DE:s, but if you want to

consider for example perturbation methods or asymptotic expansions there are

special books for those too.

If you want to consider numerics a really detailed look to different numerical methods can be found from two books of Hairer and Wanner "Solving Ordinary Differential Equations 1&2"

a) http://www.amazon.com/Solving-Ordinary-Differential-Equations-Computational/dp/3642051634

b) http://www.amazon.com/Solving-Ordinary-Differential-Equations-Computational/dp/3642051634

If you absolutely need to consider 2) there are actually some approaches: For example finding Lie-symmetries and infinidesimal generators for the equation.

These techniques are presented in quite elementary level in the book by Nail. H Ibragimov  "A Practical Course in Differential Equations and Mathematical Modelling" 

http://www.amazon.com/Practical-Differential-Equations-Mathematical-Modelling/dp/9814291943

In general this book might be a good book to start "self-study" of DE:s. Its preliminary expectations are not so high and yet it covers some quite modern methods of DE-analysis.

3) If you are interested about the DE:s in "dynamical point of view" then there are

good introdoctory level books for dynamical systems as well. A good book to start might be "Differential Equations, Dynamical Systems, and an Introduction to Chaos" by Hirch, Smale and Devaney

a) http://www.amazon.com/Differential-Equations-Dynamical-Systems-Introduction/dp/0123820103

Well here are just some quick suggestions...

regards

Samuli

What is the specific DE?

Hi, Dear Hossein

you can use Runge-Kutta Method to solve nonlinear differential equation.

Dear Hosein

If you want to simulate equation by using MATLAB, you can use 17 season of the following book. 

"Essential MATLAB for Engineers and Scientists"

If this is an initial Value problem, that is you have,  y(o) and y'(0), you can substitute w(t)=y'(t) and you now have a first order ODE in the vector (y,w). 

Then apply a numerical solver.  If you know very little about the system, you can use LSODA from ODEPACK.  It will shift between stiff and non-stiff methods as needed, and will vary the order of the method to achieve the accuracy with minimum computational effort.

in addtion to all the answers, I may propose you a open-source projet that allows you to solve this kind of problem.

http://openopt.org/interalg

http://vlab.infotech.monash.edu.au/simulations/non-linear/

This two projets are free and still have some update.

  I should have mentioned that ODEPACK is available via the website

http://www.netlib.org 

which is a repository of very good free scientific/mathematical  software.

You can use a lot of numerical or symbolic software. Several of them are for free, as open softwares. At my homepage you can find some of them:

http://ensino.univates.br/~chaet   english version, software page

If you are interested in closed form solutions, the most comprehensive method is probably Lie's symmetry analysis. It is described in full detail in my book "Algorithmic Lie Theorie for Solving Ordinary Differential Equations", published by Chapman & Hall/CRC in 2007. It emphasizes algorithmic methods which are explained by numerous examples.

You can use a semi-analytical method , say differential transform method (DTM), Adomian Decomposition Method (ADM), ...

Analytically, you can import new variable as y=x', so that the second order ODE becomes a two-dimensional system only with the first order derivative.   If the system is analytically solvable, then you can solve it directly or compute one or two first integrals for the system, which may help you to solve the system.  Also you can compute the equilibriums of the system, and then analyze the stability of these equilibria.   As for the other solutions, including periodic solutions, it needs more delicate or tough analysis and calculations. 

Numerically, once you have the two-dimensional system, you can directly use the Euler or the Runge-Kutta Schemes to solve this system.   All the schemes could be found as some compiled codes in Matlab/Maple/Mathematica.

ِA transformation like Laplace transformation is so important in solving DE 

Numericly, you can use SCILAB (opensource), MatLab or Mathematica.

For a formal solution (if it exists) You can use Maxima (opensource) or Maple

USE NUMERICAL METHOD TO SOLVE IT.YOU SHOULD HAVE POSED OR DISPLAYED AN  OKAFOREXAMPLE.

UCHENWA LINUS

You can convert your 2nd order nonlinear equation to the system of 1st order nonlinear equation, and then use an optimization methods to solve your system. for example use Jacobian matrix to linearize your system.

You can use similarity of variables

Go to www.civilized.com, download and install MLAB, run MLAB

and type-in your ODE as, for example:

fct c't(t) = k1*(a0-c)*(b0-c) -k2*c -k3*c*c

fct d't(t) = k3*c*c

initial c(0)=0

initial d(0)=0

a0=1; b0 =1.5

k1=.1; k2=.012; k3=.2

mc=points(c, 0:15!200); 

md= points(d, 0:15!200)

draw mc, color red

draw md

view

- You can also fit your model to data to estimate

unknown parameters like k1, k2, k3.

By the way,  to specify a second order ODE,  you

type:

fct y''t(t) = f(y, y', t)

initial y't(0) = yp0

initial y(0) = y0

and then solve and draw the graphs.

It makes a big difference (even restricting to scalar equations) whether the data is given as initial data (say, specifying both y and y' at t=0) or as a 2-point boundary value problem (say, specifying one thing at 0 and also one thing at t=1, as giving y(0) and y(1)).  Sometimes one can find a "general solution" (with 2 "arbitrary constants") analytically, but usually not.  Most important is that these problems do not always have solutions (even for first order: say, looking on [0,1] for a solution to y'=1+y^2 with y(0)=1 or linear second order: say looking for a solution of y''+y=0 with y(0)=1 and y(\pi)=0).)

The best possible answer for solving a second-order nonlinear ordinary differential equation is an expression in closed form form involving two constants, i.e. finding the general solution.  If such a solution does exist, the best procedure for achieving this goal is Lie's symmetry analysis. It is explained in full detail  with particular emphasis on algorithmic procedures, in my book "Algorithmic Lie Theory for Solving Ordinary Differential Equations" that has been published by Chapman & Hall/CRC in 2007.

(1) If the equation is in the form an initial value problem (and the equation is not partial), a practical way is to use a step by step method like the Runge-Kutta, as its implemented in linear problems, and meanwhile consider some iterative solution method, e.g. modified Newton Raphson, and its parameters, e.g. nonlinearity tolerance and the maximum number of iterations. Then, starting from the initial conditions you can proceed throughout the independent variable of the problem and at the end of each step check the nonlinearity and compare the residuals of the differential equation (or other parameters) with the tolerance, and try to satisfy the equations (or the other parameters the best in view of the nonlinearity parameters, and then continue to the next step, till the end of the interval under consideration; as a suggestion even if at a step the tolerance is not satisfied you would rather continue to the next step after the maximum effort; with appropriately set nonlinearity parameters, issues like numerical stability or even stiffness can mostly be tackled like linear problems.

(2) In the other case, maybe using finite elements to simplify the problem to an algebraic problem is a reasonable suggestion; many details however exist regarding the formulation depending on the problem, type of nonlinearity, and more; some adequate references are the books of Prof. K.J.Bathe, the book of Prof. P.Wriggers, or that by the late Prof. T.Belytschko and his colleagues.

In you are interested insome existence results, then I suggest to you the papers by Jean Mawhin. In particula the folliwing one:

Mawhin, J. Boundary value problems for nonlinear second-order vector differential equations. J. Differential Equations 16 (1974), 257–269.

Dear Hossein

I suggest that you use the Runge Kutta method to obtain an discrete solution ( see ODE45 in MATLAB software). Also you can utilize the spectral method to obtain an analytical approximate solution (see my paper in "nonlinear dynamics" journal, published by  springer)

you should convert it to linear formula first then solve it :)

                                 Dear Hossein Harimi,

My Answer. I suggest the analytical method for solving a nonlinear second order ODE. Suppose that the ODE is in normal form, i.e. y "= f (x, y, y ') and the function in the right side is an analytical function concerning its 3 arguments. Then formally substitute in the differential equation: y(x), y'(x) and y"(x) and f(x,y,y') in the series. Equating coefficients to equal degrees of x and possibly seek recurrent relationship between them. Prove the convergence of the resulting power series (of Macloren). Please, visit the page http://www.sosmath.com/diffeq/series/series03/series03.html. Your sincerely, Anna Tomova

Generally, we could import a new variable and rewrite the equation as a two-dimensional system.   For some specific 2-D systems, we could find their analytical solutions through the method of the first-ingteral.  However, for general 2-D systems, we usually use numerical schemes, such as the Euler scheme and the Runge-Kutta scheme, to simulate the dynamical behaviors of the solutions.

In some cases the ode has periodic solutions. This is the case for some autonomous odes of the form y''=F(y), where F(y) is a polynomial w.r.t. y. This is a very important class of nonlinear ODES.

We may try to obtain either exact or quasi exact solution in the form y=Q(u), where Q is either a polynomial or  a rational function of u and u is some periodic or double periodic function (for example, the Jacobi Elliptic Functions, The Weierstrass P function, the Pi-Elliptic functions and others).

You can use Matlab built-in solvers such as: “ode45″ or “bvp4c” to optimize the coupled nonlinear differential equations. In your case, you can use "ode45" solver. First, try to linearize the equation. You can visit Matlab central or MathWorks for the links (for both analytical and numerical solutions).

A recent article in the "Bulletin of Mathematical Sciences" entitled "Decomposition of Ordinary Differential Equations" generalizes decomposition to nonlinear ordinary

differential equations; in many cases it is superior to Lie's symmetry methods.

There is a new method called Piecewise Analytic Method PAM. It gives excellent results see the discussion in the URL:

https://www.researchgate.net/post/What_do_you_know_about_Piecewise_Analytic_Method_PAM-Published_January_2012_by_T_Abassy

Best regards,

Depending on the type of NDE:

(A) If it is nonlinear ODE of the form u''=f(x,u,u'), then the substitution w=u' can work sometimes.

(B) If it is nonlinear PDE, (elliptic, hyperbolic, or parabolic), then asymptotic analysis is one of the methods that gives an good approximation solution.

Numerical methods include multi-scale, Fourier-spectral, or time-splitting pseudo-spectral method for equations defined on bounded domains.

References:

Kamke, E. Differentialgleichungen: Losungsmethoden und Losungen. Akademische Verlagsgesellschaft, 1959

Yu. N. Kosovtsov, The general solutions of some nonlinear second-order PDEs. 2008. arXiv.

Andrei D. Polyanin, and Valentin F. Zaitsev, Handbook of Nonlinear Partial Differential Equations. 2016.

This is probably the best method:

Article Decomposition of ordinary differential equations