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Definitions and Concepts
Published in Daniel Zwillinger, Vladimir Dobrushkin, Handbook of Differential Equations, 2021
Daniel Zwillinger, Vladimir Dobrushkin
Initial value problem (IVP) An ordinary differential equation with all of the data given at one point is an initial value problem. For example, the equation y′′+y=0 with the data y(0)=1, y′(0)=1 is an initial value problem.
Spatial Models Using Partial Differential Equations
Published in Sandip Banerjee, Mathematical Modeling, 2021
The general solution of the partial differential equation involves as many arbitrary functions as the order of the equation (order of the highest partial differential coefficient in the equation). Certain conditions are required in order to find these arbitrary functions. The standard notation for the space variables in applications are x, y, z, etc., and a solution may be required in some region Ω of space. In such a case, there will be some conditions to be satisfied on the boundary ∂Ω, which are called boundary conditions (BCs). Similarly, in the case of the independent variables, one of them is generally taken as time (say, t), then there will be some initial conditions (ICs) to be satisfied. The conditions of partial differential equations are classified into two categories: Initial value problem (IVP): A partial differential equation with initial conditions, that is, dependent variable and an appropriate number of its derivatives are prescribed at the initial point of domain is called an initial value problem.Boundary value problem (BVP): A partial differential with boundary conditions, that is, dependent variable and an appropriate number of its derivatives are prescribed at the boundary of domain is called a boundary value problem.
Traditional First-Order Differential Equations
Published in Stephen A. Wirkus, Randall J. Swift, Ryan S. Szypowski, A Course in Differential Equations with Boundary-Value Problems, 2017
Stephen A. Wirkus, Randall J. Swift, Ryan S. Szypowski
In contrast to an initial-value problem, a boundary-value problem consists of a differential equation and a set of conditions at different x-values that the solution y(x) must satisfy. Although any number of conditions (≥2) may be specified, usually only two are given. Rather than specifying the initial state of the system, we can think of a boundaryvalue problem as specifying the state of the system at two different physical locations, say x0 = a, x1 = b, a ≠ b.
Existence and uniqueness of the solution to a second-order nonlinear dynamical system model with an unbounded variable and a discontinuous input
Published in Applied Mathematics in Science and Engineering, 2023
Jing Xu, Qiuguo Zhu, Jun Wu, Rong Xiong
Many practical dynamical systems can be modeled by nonlinear ordinary differential equations [1]. In these dynamical system models, the independent variable is time t and the dependent variables are the n state variables that make up a state vector (state for short) . The dimension n of represents the system order. For a dynamical system, the initial value problem (IVP) of an ordinary differential equation (ODE) model can be regarded as the problem of finding the state trajectory of the system when the initial time , initial state (i.e. the state at ), and the control function are known. Obviously, the existence and uniqueness of the solution to the IVP (EUSIVP) are important when analyzing dynamical systems.
Gyrotactic mixed bioconvection flow of a nanofluid over a stretching wedge embedded in a porous media in the presence of binary chemical reaction and activation energy
Published in International Journal of Ambient Energy, 2022
Runge–Kutta–Felhberg (RKF) method is a well-known method for solving a nonlinear first-order ordinary differential equation with its proper boundary conditions. This method is applicable for solving an initial value problem (IVP). The system of higher order differential Equations (12)–(15) with boundary conditions (16) is a Boundary value problem. So we first convert this higher order differential equation into a system of first-order differential equations and the boundary value problem to initial value problem by employing multiple shooting method. Equations (12)–(15) are transformed into nine first-order ordinary differential equation by introducing the following variables, which are given below: Now, these first-order ordinary differential equations can be written in a matrix form as given below: