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Differential equations are prominent in many scientific areas. Nonlinear ones are of particular interest for their commonality in describing real-world systems and how much more difficult they are to solve compared to linear differential equations.
In mathematics, an ordinary differential equation (ODE) is a differential equation (DE) dependent on only a single independent variable.As with any other DE, its unknown(s) consists of one (or more) function(s) and involves the derivatives of those functions. [1]
If a set of ODEs has a particular form, then the Picard method can be used to find their solution in the form of a power series. If the ODEs do not have the required form, it is nearly always possible to find an expanded set of equations that do have the required form, such that a subset of the solution is a solution of the original ODEs.
Numerical methods for ordinary differential equations are methods used to find numerical approximations to the solutions of ordinary differential equations (ODEs). Their use is also known as "numerical integration", although this term can also refer to the computation of integrals. Many differential equations cannot be solved exactly.
In a large class of singularly perturbed problems, the domain may be divided into two or more subdomains. In one of these, often the largest, the solution is accurately approximated by an asymptotic series [2] found by treating the problem as a regular perturbation (i.e. by setting a relatively small parameter to zero).
GNU Octave is a scientific programming language for scientific computing and numerical computation.Octave helps in solving linear and nonlinear problems numerically, and for performing other numerical experiments using a language that is mostly compatible with MATLAB.
The following pseudocode in MATLAB style demonstrates Richardson extrapolation to help solve the ODE ′ =, () = with the Trapezoidal method. In this example we halve the step size h {\displaystyle h} each iteration and so in the discussion above we'd have that t = 2 {\displaystyle t=2} .
where y and f may denote vectors (in which case this equation represents a system of coupled ODEs in several variables). We are given the function f(t,y) and the initial conditions (a, y a), and we are interested in finding the solution at t = b. Let y(b) denote the exact solution at b, and let y b denote the solution that we compute.