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Ebook Matlab - Advanced mathematics and mechanics applications using MATLAB (3/E): Part 2

Part 2 book “Matlab - Advanced mathematics and mechanics applications using MATLAB” has contents: Boundary value problem s for partial differential equations, boundary value problem s for partial differential equations, bending analysis of beams of general cross section, and other contents. | Chapter 9 Boundary Value Problems for Partial Differential Equations 9.1 Several Important Partial Differential Equations Many physical phenomena are characterized by linear partial differential equations. Such equations are attractive to study because (a) principles of superposition apply in the sense that linear combinations of component solutions can often be used to build more general solutions and (b) Þnite difference or Þnite element approximations lead to systems of linear equations amenable to solution by matrix methods. The accompanying table lists several frequently encountered equations and some applications. We only show one- or two-dimensional forms, although some of these equations have relevant applications in three dimensions. In most practical applications the differential equations must be solved within a Þnite region of space while simultaneously prescribing boundary conditions on the function and its derivatives. Furthermore, initial conditions may exist. In dealing with the initial value problem, we are trying to predict future system behavior when initial conditions, boundary conditions, and a governing physical process are known. Solutions to such problems are seldom obtainable in a closed Þnite form. Even when series solutions are developed, an inÞnite number of terms may be needed to provide generality. For example, the problem of transient heat conduction in a circular cylinder leads to an inÞnite series of Bessel functions employing characteristic values which can only be computed approximately. Hence, the notion of an “exact” solution expressed as an inÞnite series of transcendental functions is deceiving. At best, we can hope to produce results containing insigniÞcantly small computation errors. The present chapter applies eigenfunction series to solve nine problems. Examples involving the Laplace, wave, beam, and heat equations are given. Nonhomogeneous boundary conditions are dealt with in several instances. Animation is also .