Boundary Value Problem
A boundary value problem in engineering refers to a type of mathematical problem that involves finding a solution to a differential equation subject to specified boundary conditions. These conditions are typically defined at the boundaries of the problem domain. Boundary value problems are important in engineering as they help in modeling and solving real-world physical phenomena, such as heat conduction and fluid flow.
3 Key excerpts on "Boundary Value Problem"
eBook - ePubMATLAB® Essentials
A First Course for Engineers and Scientists
- William Bober(Author)
- 2017(Publication Date)
- CRC Press(Publisher)
...11 Boundary Value Problems of Ordinary Differential Equations 11.1 Introduction When an ordinary differential equation involves boundary conditions instead of initial conditions, then a numerical approach is most often used to solve the problem. In a Boundary Value Problem, we essentially need to fit a solution into the known boundary conditions as opposed to simply integrating from the initial conditions. An example of this type of problem is the temperature of a bar subjected to known different temperatures at the ends as it looses heat along the bar by natural convection. Another example would be the deflection of a beam due to an applied load along the beam and where the boundary conditions at both ends of the beam are specified. Another example of this type of problem is the determination of the electric field between the plates of a capacitor with a known charge density between the plates and a fixed voltage across the plates. In these three examples, a solution is found by numerically solving a second-order, nonhomogeneous ordinary differential equation using finite difference methods. 11.2 Difference Formulas To numerically solve a Boundary Value Problem involving an ordinary, linear, and differential equation, we will need the difference formulas obtained by Taylor series expansion using just a few terms. The finite difference method first involves subdividing the independent variable domain into N subdivisions. The finite difference formulas that will be used are tabulated in Table 11.1. In the following table, y i is the value of y at position x i and Δ x = x i + 1 − x i. TABLE 11.1 Summary of Finite Difference Formulas for Boundary Value Problems y ′ i = y i + 1 − y i Δ x First-order forward difference formula. Usually used for a y ' boundary condition at the beginning of domain. y ′ i = y i − y i − 1 Δ x First-order backward difference formula...
eBook - ePub- O. C. Zienkiewicz, K. Morgan(Authors)
- 2013(Publication Date)
- Dover Publications(Publisher)
...CHAPTER ONE Continuum Boundary Value Problems and the Need for Numerical Discretization. Finite Difference Methods 1.1. INTRODUCTION While searching for a quantitative description of physical phenomena, the engineer or the physicist establishes generally a system of ordinary or partial differential equations valid in a certain region (or domain) and imposes on this system suitable boundary and initial conditions. At this stage the mathematical model is complete, and for practical applications “merely” a solution for a particular set of numerical data is needed. Here, however, come the major difficulties, as only the very simplest forms of equations, within geometrically trivial boundaries, are capable of being solved exactly with available mathematical methods. Ordinary differential equations with constant coefficients are one of the few examples for which standard solution procedures are available—and even here, with a large number of dependent variables, considerable difficulties are encountered. To overcome such difficulties and to enlist the aid of the most powerful tool developed in this century—the digital computer—it is necessary to recast the problem in a purely algebraic form, involving only the basic arithmetic operations. To achieve this, various forms of discretization of the continuum problem defined by the differential equations can be used. In such a discretization the infinite set of numbers representing the unknown function or functions is replaced by a finite number of unknown parameters, and this process, in general, requires some form of approximation. Of the various forms of discretization which are possible, one of the simplest is the finite difference process. In this chapter we describe some of the essentials of this process to set the stage, but the remainder of this book is concerned with various trial function approximations falling under the general classification of finite element methods...
eBook - ePubUltra Wideband Antennas
Design, Methodologies, and Performance
- Giselle M. Galvan-Tejada, Marco Antonio Peyrot-Solis, Hildeberto Jardón Aguilar(Authors)
- 2017(Publication Date)
- CRC Press(Publisher)
...If a particular solution is specified at a single point, let us say (x 0, y 0), then y (x 0) = y 0 (B.9) is referred to as an initial condition, and together with Equation (B.8) forms what is known as an initial value problem. It is worth mentioning that the term initial conditions usually refers to an initial time, i.e., time is the independent variable. In contrast, when the differential equation that describes a certain problem in a region R is subject to several conditions over the boundary of R, these are called boundary value conditions, and therefore there is a Boundary Value Problem. These conditions describe a specific physical condition at the boundary of the body under analysis. In the case of the Maxwell’s equation systems, for a thin layer surface between two regions R 1 and R 2 the boundary conditions. are: n × (E 1 − E 2) = 0 (B.10) n × (H 1 − H 2) = K (B.11) n ⋅ ε 0 (E 1 − E 2) = ρ s (B.12) n ⋅ μ 0 (H 1 − H 2) = 0 (B.13) n ⋅ (J 1 − J 2) + ∇ S ⋅ K = − ∂ ρ[--=PLGO-SEPARATO. R=--]s ∂ t (B.14) which relate the electric and magnetic fields both within a structure and to its surroundings. For (B.10) – (B.14) n represents the component normal to the surface, K the surface current density, and subscripts 1 and 2 are for the two regions. B.5 Existence and Uniqueness The first question that arises is if the differential equation has a solution. Then we are talking about the existence of the solution of that equation. This aspect comes from the mathematical formulation of a physical problem...
