![]() ![]() ode45, ode23, ode23s, ode23t, and ode23tb all employ single-step algorithms. Runge-Kutta algorithms are all single-step solvers, since each step only depends on the result of the previous step. Theoretically, this numerical solution technique is possible because of the connection between differential equations and integrals provided by the fundamental theorem of calculus: The final result is that the ODE solver returns a vector of time steps $t_0,t_1.,t_f$ as well as the corresponding solution at each time step $y_0,y_1.,y_f$. At the first such step, the initial conditions provide the necessary information that allows the integration to proceed. Starting with the initial conditions $y_0$, and a period of time over which the answer is to be obtained $(t_0,t_f)$, the solution is obtained iteratively by using the results of previous steps according to the solver's algorithm. ![]() ![]() Where $M(t,y)$ is referred to as the mass matrix. The ODE solvers in MATLAB all work on initial value problems of the form, To choose between the solvers, it's first necessary to understand why one solver might be better than another for a given problem. (note that ode15i is left out of this discussion because it solves its own class of initial value problems: fully implicit ODEs of the form $f(t,y,y') = 0$) ![]()
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