Is it stiff? If stiff use backward Euler.  If not stiff, transform into a system of two first order differential equations and then use 4th or 5th order Runge Kutta.

Dear @Mohammad, I do not know which is the best method for, but maybe You are going to find some answers in the paper Modified Taylor series method for solving nonlinear differential equations with mixed boundary conditions defined on finite intervals with many examples and error analysis based on simulations where different method were applied!

http://www.ncbi.nlm.nih.gov/pmc/articles/PMC4000591/

Is it stiff? If stiff use backward Euler.  If not stiff, transform into a system of two first order differential equations and then use 4th or 5th order Runge Kutta.

Also is this an initial value problem or a boundary value problem? BVP changes things and I recommend collocation.

Nyström methods are efficient Runge-Kutta-type integration schemes, tailored for 2nd order ordinary differential equations. See for example Section II.14 of the "Solving Ordinary Differential Equations I - Nonstiff Problems". However, transforming the 2nd order ODE into a system of first-order ODEs as proposed by Diana should also work well.

RK4 should be the first choice if you don't know what to expect in terms of the behavior of the iterations. Then analyse what happens and it will enlighten you on what method will be the best. BTW: even if the problem is stiff, you can tailor your stoppage criteria and play with the steps in such a way to produce the convergence you require - sometimes this will become evidently cumbersome, of course. Implicit schemes are great but they come with a price... ;)

Hi, Dear Professor Maria Thomas 

This is an initial value problem.

Dear Mohammad , have you searched all possibilities of an analytical solution by using series metods? Otherwise the class of RK4 methods is a suitable suggestion.

If we consider only 2nd-order ODE, then there are many, which can be classified and an analytic solution could be found correspondingly. However, in general, the numerical methods should be attained; in such cases, a suitable method depends on the degree and or complexity of the non-linearity. The MATLAB toolbox offers a variety of methods.     

Thanks for helpful suggestions.

You should first try to transform the higher order differential equations to several first order equations and then apply the known iterative methods as suggested above. Runge Kutta 4th order iteration can help.

There is no such thing as best. It only depends upon the demand of the situation.

Sometimes lesser order can provide acceptable result. Stiffness of the equation is also a parameter. So try using MATLAB's ODE solver and experiment with different solvers. The one that gives you the acceptable result, you should go for it. You can also go for fixed and variable steps iteration and compare your results.

please download this paper

www.math.umn.edu/~olver/am_/odz.pdf

regards

Then all you need is Runge-Kutta. You can change your second order system into a system of two first order ODEs by making the change of variables u=y and v=y'.  Then u'=v and v'= y" = Sub in your equation with all the y's transformed to v and all the y's transformed to u.  Then apply this algorithm: http://www.phy.davidson.edu/FacHome/dmb/py200/RungeKuttaMethod.htm.  If you have Matlab, Maple, or Mathematica you can just use their solver.

you may use rung kutta for solving differential equations.

I myself use rung kutta method for this purpose in Matlab Software !!!

If you want to study deeper the whole class of Runge-Kutta like methods, read here:

http://www.scholarpedia.org/article/Runge-Kutta_methods

Read the "Butcher tableau" the definition of stages for a method.

Maybe next link will further help you:

http://www.iact.ugr-csic.es/personal/julyan_cartwright/papers/rkpaper/node2.html

And don't miss this:

http://en.wikipedia.org/wiki/Runge%E2%80%93Kutta_methods

Visit the website

http://www.iact.ugr-csic.es/personal/julyan_cartwright/papers/rkpaper/node2.html

you can use Runge kutta methods

Perhaps it's related to the intial valuse and step size

variation of parameter could also help depending on the nature of your differential equation.

hi

http://www.phy.davidson.edu/FacHome/dmb/py200/RungeKuttaMethod.htm.

There are many methods to deal with such type of problems. One can go through modified Euler method and R K method of fourth order for simultaneous two first order differential equation, as mentioned by experts above. Apart from this, you can firstly integrate the given equation (if possible) once and then apply R K method of fourth order, it will results in better accuracy.

Before we search for a numerical method to solve the second-order nonlinear DE, there are many types of these equations have an analytic exact solution. If it is happen that there is no such exact solution, then the next step is to analyze the nonlinearity of the equation to select suitable numerical method.

You will have a chapter on this in Schetzen book (The Volterra and Wiener Theories of Nonlinear Systems).

WKB Method works equally well for solving this types of ODEs although a bit lengthy but works in all cases.

Depending on the nature of your equations .. so more details needs to be given .. but Range - Kutta and Euler methods would be first things that come to mind

Best of luck

Range - Kutta and Euler methods could help depending on the nature of your differential equation.

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,

http://people.uncw.edu/hermanr/mat361/Simulink/ODE_Simulink.pdf

this will help

This depends on a problem solved, especially on their stiffness (determined by equation coefficients). Different problems may be solved optimally by use different methods and/or algorithms.