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Explicit methods calculate the state of a system at a later time from the state of the system at the current time, while implicit methods find a solution by solving an equation involving both the current state of the system and the later one.
The method of lines most often refers to the construction or analysis of numerical methods for partial differential equations that proceeds by first discretizing the spatial derivatives only and leaving the time variable continuous. This leads to a system of ordinary differential equations to which a numerical method for initial value ordinary ...
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.
Deterministic system (mathematics) Linear system; Partial differential equation; Dynamical systems and chaos theory; Chaos theory. Chaos argument; Butterfly effect; 0-1 test for chaos; Bifurcation diagram; Feigenbaum constant; Sharkovskii's theorem; Attractor. Strange nonchaotic attractor; Stability theory. Mechanical equilibrium; Astable ...
These equations define the peculiarity of the state variables, the environment factors so to speak, of a system. These parameters of a system change in a continuous way and thus change the state of the entire system. [6] The set of differential equations can be formulated in a conceptual model representing the system on an abstract level. In ...
A differential system is a means of studying a system of partial differential equations using geometric ideas such as differential forms and vector fields. For example, the compatibility conditions of an overdetermined system of differential equations can be succinctly stated in terms of differential forms (i.e., for a form to be exact, it ...
A solution to a discretized partial differential equation, obtained with the finite element method. In applied mathematics, discretization is the process of transferring continuous functions, models, variables, and equations into discrete counterparts. This process is usually carried out as a first step toward making them suitable for numerical ...
Here, a differential equation can be thought of as a formula by which the slope of the tangent line to the curve can be computed at any point on the curve, once the position of that point has been calculated.