Could there be nonlinear ODEs that could not even be solved numerically?
I am working on a third-order nonlinear problem which is giving accurate results for a given set of initial conditions (ICs) but for other ICs the matlab ODE solver (checked 4-5 algorithms) do not converge.
If your equation/system of equations is prone to stiffness or strongly nonlinear the ode 45 solver may not be adapted for some ICs because it uses a fully explicit numerical scheme. This page can help you to chose better adapted solvers: https://www.mathworks.com/help/matlab/math/choose-an-ode-solver.html
If none of these solvers works then you may need to write yourself a numerical scheme adapted to your case, probably an implicit or semi-implicit scheme (more stable than explicit schemes) if matlab solvers don't work.
@ Guillaume. tried all of those. No solutions. They give good solutions only when 2x2 = 4 initial conditions are of equal values. I am solving a system of coupled third order nonlinear ODEs.
Then you may need to write your own scheme. You may try to write an implicit scheme (and linearise it if needed), but I can't help you more without seing the system
Have a look here: https://www.mathworks.com/matlabcentral/answers/430969-ode-becomes-unsolveable-unstable-when-transformed-from-second-order-to-third-order-for-most-cases
Your system is VERY nonlinear, I doubt any generic solver can solve it for non special IC's. Did you run some tests to check which element(s) of the equations induces the instability? You could then approximate this element only and keep your code to solve the system as you do now. Otherwise the only solution that I think of is to write a fully implicit finite differences scheme and then linearise it, and making it converge by using an iterative method (such as Picard or Newton iterative method) at each time step. There are also spectral elements schemes that can deal well with nonlinearities in most cases, but it is not my cup of tea!
You can write a stability criterion for your scheme, but it could be easier to write your own (implicit) scheme. Then you will be able to 'debug' your scheme in detail.
I too have been asking and searching for such a ODE problem. Stiff ODEs are the only problem I have found to date. For example, delta = xxx..6 - xxx..2 = 000..4 . Delta consisted of only 1 digit! This gave the numerical methods nothing but trouble. 1 digit suggests that we need larger words ... 120, 240, ??? bits per word! for some math problems.
What is your 3rd order ODE(s) that you are trying to solve? Maybe then I can help answer more on your question.
See my Casebook titled "Engineering Design Optimization using Calculus Level Methods- A Casebook Approach", http://fortrancalculus.info/textbooks/ . Their are some 20+ Engineering and Science problems showing their math problems and Optimal solution. Plus a little write-up / intro to each problem.
Try the FortranCalculus (FC) language that got its start by NASA's Apollo Space program that got us to the moon!
FC-Compiler™ application is a (free) Calculus-level Compiler that simplifies Tweaking parameters in ones math model. FC solves Algebraic through Ordinary Differential Equations Equations; Laplace transforms; etc. FC is based on Automatic Differentiation that simplifies computer code to an absolute minimum; i.e., a mathematical model, constraints, and the objective (function) definition.
Most math Modeling, Simulations, & Optimizations can be easily solved with a Calculus-level language/compiler. A half-hour per problem should be enough time to code and execute most math problems. Solves Algebraic Equations through Ordinary Differential Equations, for more visit http://fortrancalculus.info/apps/fc-compiler.html . Equations may be implicit, non-linear, Boundary Value Problems (BVP), Initial Value Problems (IVP), etc. Download, install, & execute some demo problems. Simple and its Freeware! (PDE solvers are in the works!)
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