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In multivariable calculus, an initial value problem [a] (IVP) is an ordinary differential equation together with an initial condition which specifies the value of the unknown function at a given point in the domain. Modeling a system in physics or other sciences frequently amounts to
Simulation-based methods for time-based network analysis solve a circuit that is posed as an initial value problem (IVP). That is, the values of the components with memories (for example, the voltages on capacitors and currents through inductors) are given at an initial point of time t 0 , and the analysis is done for the time t 0 ≤ t ≤ t f ...
Ordinary differential equations occur in many scientific disciplines, including physics, chemistry, biology, and economics. [1] In addition, some methods in numerical partial differential equations convert the partial differential equation into an ordinary differential equation, which must then be solved.
A standard proof relies on transforming the differential equation into an integral equation, then applying the Banach fixed-point theorem to prove the existence of a solution, and then applying Grönwall's lemma to prove the uniqueness of the solution.
Let be an open subset of with : a continuous function and ′ = (, ()) a continuous, explicit first-order differential equation defined on D, then every initial value problem () = for f with (,) has a local solution : where is a neighbourhood of in , such that ′ = (, ()) for all .
Boundary value problems are similar to initial value problems.A boundary value problem has conditions specified at the extremes ("boundaries") of the independent variable in the equation whereas an initial value problem has all of the conditions specified at the same value of the independent variable (and that value is at the lower boundary of the domain, thus the term "initial" value).
Applied mathematics is the application of mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and industry. Thus, applied mathematics is a combination of mathematical science and specialized knowledge.
Note that bold text indicates that the quantity is a vector. Latin characters Symbol ... The Physics Hypertextbook; NIST (16 August 2023). "SI Units". www ...