# Midpoint Method for Solving ODEs

by admin in Differential Equations , Math, Statistics, and Optimization , MATLAB Family on April 27, 2019This code uses Midepoint Method to Solve single Ordinary Differential Equations (ODE). This code compare between the exact solution (if exist) and numerical solution, and presents the results on one chart, it shows the absolute Errors also on a new chart.

### Code Outputs:

- Chart presenting Midpoint method compared with Exact solution (if exist).
- Error chart (the absolute error between exact and Midpoint method).
- Printed Maximum absolute error for Midpoint method.

Max. Error = 0.052424

### Input Requirements:

- The differential equation you intended to calculated.
- Initial and Final Integral Terms.
- Initial Conditions.
- Exact solution if exist.
- Number of iterations.

### About the Method:

In numerical analysis, a branch of applied mathematics, the **midpoint method** is a one-step method for numerically solving the differential equation,

- .

The explicit midpoint method is given by the formula

the implicit midpoint method by

for **n = 0,1,2,…** Here, * h* is the

*step size*— a small positive number,

**tn = t0 + n.h**and

*is the computed approximate value of*

**yn***The explicit midpoint method is also known as the*

**y(tn)****modified Euler method**, the implicit method is the most simple collocation method, and, applied to Hamiltonian dynamics, a symplectic integrator.

The name of the method comes from the fact that in the formula above the function* f* giving the slope of the solution is evaluated at (note : is ) which is the midpoint between at which the value of is known and at which the value of needs to be found.

A geometric interpretation may give a better intuitive understanding of the method. In the basic Euler’s method, the tangent of the curve at is computed using . The next value { is found where the tangent intersects the vertical line . However, if the second derivative is only positive between and , or only negative (as in the diagram), the curve will increasingly veer away from the tangent, leading to larger errors as increases. The diagram illustrates that the tangent at the midpoint (upper, green line segment) would most likely give a more accurate approximation of the curve in that interval. This tangent is estimated by using the original Euler’s method to estimate the value of at the midpoint, then computing the slope of the tangent with. Finally, the improved tangent is used to calculate the value of from . This last step is represented by the red chord in the diagram. Note that the red chord is not exactly parallel to the green segment (the true tangent), due to the error in estimating the value of at the midpoint.

The local error at each step of the midpoint method is of order , giving a global error of order . Thus, while more computationally intensive than Euler’s method, the midpoint method’s error generally decreases faster as .

The methods are examples of a class of higher-order methods known as Runge–Kutta methods.

### Derivation of the Midpoint Method:

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