Some partial differential equations can be solved exactly in the Wolfram Language using DSolve[eqn, y, x1, x2], and numerically using NDSolve[eqns, y, x, xmin, xmax, t, tmin, tmax].
In general, partial differential equations are much more difficult to solve analytically than are ordinary differential equations. They may sometimes be solved using a Bäcklund transformation, characteristics, Green's function, integral transform, Lax pair, separation of variables, or--when all else fails (which it frequently does)--numerical methods such as finite differences.
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Early training in the elementary techniques of partial differential equations is invaluable to students in engineering and the sciences as well as mathematics. However, to be effective, an undergraduate introduction must be carefully designed to be challenging, yet still reasonable in its demands. Judging from the first edition's popularity, instructors and students agree that despite the subject's complexity, it can be made fairly easy to understand. Revised and updated to reflect the latest version of Mathematica, Partial Differential Equations and Boundary Value Problems with Mathematica, Second Edition meets the needs of mathematics, science, and engineering students even better. While retaining systematic coverage of theory and applications, the authors have made extensive changes that improve the text's accessibility, thoroughness, and practicality.New in this edition:Upgraded and expanded Mathematica sections that include more exercises
An entire chapter on boundary value problems
More on inverse operators, Legendre functions, and Bessel functions
Simplified treatment of Green's functions that make it more accessible to undergraduates
A section on the numerical computation of Green's functions
Mathemcatica codes for solving most of the problems discussed
Boundary value problems from continuum mechanics, particularly on boundary layers and fluctuating flows
Wave propagation and dispersionWith its emphasis firmly on solution methods, this book is ideal for any mathematics curricula. It succeeds not only in preparing readers to meet the challenge of PDEs, but also in imparting the inherent beauty and applicability of the subject.
Interval computations are becoming a well-accepted method of rigorous mathematical proofs. We discuss some possibilities of deploying Mathematica into interval-computation proofs of theorems concerning boundary-value problems for strongly nonlinear differential equations. Alternatively, we also describe development of a highly optimized library supporting special functions from the theory of nonlinear differential equations. For performance and correctness reasons, our library is written in GNU Assembler for Intel Pentium. We use new LibraryLink Mathematica 8 technology for calling user-defined C functions. A C interface is used for accessibility of our library functions from Mathematica. In order to improve performance of our library, we derived various approximate formulas for our special functions using symbolic calculations. The library will be used in proving theorems in the theory of nonlinear differential equations.
We present an overview of an advanced undergraduate mathematics course on partial differential equations where students are given an opportunity to generate and discover rather than to passively receive knowledge. The course material is motivated through projects that involve "real-world" applications where students use technology to solve various problems. We consider several projects: oil flow recovery, aerodynamics, and heat/wave equations. Each project consists of: (a) derivation of differential equations that describe the particular phenomena, (2) mathematical analysis of the problem and development of a numerical method for approximate solving, (3) implementation of the method using various computer algebra systems, and (4) writing a summary of research findings and/or presenting the findings in the class. Students are given a complete set of notes, consisting of book chapters, research papers, and instructor's notes on mathematical formulation of the problem and development of appropriate numerical methods. The instructor also provides a bone structure of the numerical code that students need to modify, incorporating various partial differential equations and initial conditions.
The proposal on introducing real-world projects and technology in courses on differential equations was developed in the fall of 2007, and its implementation was funded by the Curriculum Development Grant at University of Houston-Downtown in the spring of 2008. The projects were implemented in spring 2009, fall 2009, and spring 2011. 2ff7e9595c
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