Hi, it seems not usual to solve ODEs using Newton's method. Why do you not consider using Runge-Kutta methods for example. In addition, numerical solvers for ODEs are available in C, FORTRAn, .. but not in symbolic calculation tools as Mathematica or MAPLE.

When using Backward Euler time discretization (which is implicit), you end up with a nonlinear equation of the form F(u_{n+1})=0, where u_{n+1} is the unknown function at (n+1) time step.

For this you can use Newton's method, like FindRoot(F(unew),unew0=uold). According to Mathematica documentation, FindRoot uses Newton's method.

See these links for further reference:

https://scicomp.stackexchange.com/questions/5042/how-to-implement-newtons-method-for-solving-the-algebraic-equations-in-the-back

https://reference.wolfram.com/language/tutorial/UnconstrainedOptimizationNewtonsMethodRoot.html

https://reference.wolfram.com/language/ref/FindRoot.html

@Richard Epenoy

Without doubt one is faster with a numerical ODE solver in C/Fortran.

Implementing a numerical ODE solver in Mathematica might be useful for teaching purposes, for example in a numerical methods course.

Although I would prefer to teach it using Numerical Recipes in C/Fortran.

Strangely enough, RK method is available in MAPLE:

https://www.maplesoft.com/support/help/maple/view.aspx?path=Student%2FNumericalAnalysis%2FRungeKutta

Hi, you are true Maple implements now numerical schemes.

Thank you Richard and Anton.....

Best regards,

Majeed

Hi, in general implicit schemes that require the use of Newton's method are dedicated to stiff ODEs. In case of a classical non stiff system, explicit schemes such as Runge-Kutta, Dormand-Prince etc.. are adapted.

Kindly specify the software that you are interested in. There are various programming languages/code that are used in each software.

Dear O. S. Adesina,

Mathematica any version of it.

Best regards...

Great! Mathematica was an oversight. Now I get. Thanks.

I too what the Mathematica code for Adomain decomposition method for linear as well as nonlinear equations in Fractional partial differential equations.