# trapezoidal

trapezoidal, a MATLAB code which solves one or more ordinary differential equations (ODE) using the (implicit) trapezoidal method, using MATLAB's fsolve() to handle the implicit system.

Unless the right hand side of the ODE is linear in the dependent variable, each trapezoidal step requires the solution of an implicit nonlinear equation. This version of the code uses MATLAB's implicit equation solver fsolve().

### Languages:

trapezoidal is available in a FreeFem++ version and a MATLAB version and an Octave version and a Python version and an R version.

### Related Data and codes:

backward_euler, a MATLAB code which solves one or more ordinary differential equations (ODE) using the backward Euler method.

backward_euler_fixed, a MATLAB code which solves one or more ordinary differential equations (ODE) using the (implicit) backward Euler method, using a fixed point iteration for the implicit equation.

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euler, a MATLAB code which solves one or more ordinary differential equations (ODE) using the forward Euler method.

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midpoint_explicit, a MATLAB code which solves one or more ordinary differential equations (ODE) using the (explicit) midpoint method, also called the modified Euler method.

midpoint_fixed, a MATLAB code which solves one or more ordinary differential equations (ODE) using the (implicit) midpoint method, using a simple fixed-point iteration to solve the implicit equation.

rk12, a MATLAB code which implements Runge-Kutta solvers of orders 1 and 2 for a system of ordinary differential equations (ODE).

rk23, a MATLAB code which implements Runge-Kutta ODE solvers of orders 2 and 3.

rk34, a MATLAB code which implements Runge-Kutta ODE solvers of orders 3 and 4.

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theta_method, a MATLAB code which solves one or more ordinary differential equations (ODE) using the theta method.

trapezoidal_explicit, a MATLAB code which solves one or more ordinary differential equations (ODE) using the (explicit) trapezoidal method.

trapezoidal_fixed, a MATLAB code which solves one or more ordinary differential equations (ODE) using the (implicit) trapezoidal method, using the fixed point method to handle the implicit system.

velocity_verlet, a MATLAB code which uses a version of the velocity Verlet method to solve a secord order ordinary differential equation (ODE) of the form y''=f(t,y).

### Source Code:

• trapezoidal.m, a version of the implicit trapezoidal method using fsolve() from the MATLAB Optimization Toolbox.

Last revised on 26 April 2021.