![]() ![]() The variational, finite element, and finite difference methods constitute the very core of engineering analysis, but the associated computations are tedious at best, and often obscure both the ideas and the techniques of the approach. starting from an analytical derivation of exact governing equations, constructing discretizations and analytical formulas of a numerical method, performing numerical procedure, obtaining various visualizations, and comparing the numerical solution obtained with other types of solutions considered in the book, e.g. One of the main points (related to CAS) is based on the implementation of a whole solution method (e.g. ![]() As in the our previous books, we propose the idea to use in parallel both systems, Maple and Mathematica, since in many research problems frequently it is required to compare independent results obtained by using different computer algebra systems, Maple and/or Mathematica, at all stages of the solution process. Among a big number of CAS, we choose the two systems, Maple and Mathematica, that are used worldwide by students, research mathematicians, scientists, and engineers. Numerous comparisons and relationships between various types of solutions, different methods and approaches are provided, the results obtained in Maple and Mathematica, facilitates a deeper understanding of the subject. The reader can learn a wide variety of techniques and solve numerous nonlinear PDEs included and many other differential equations, simplifying and transforming the equations and solutions, arbitrary functions and parameters, presented in the book). The emphasis of the book is given in how to construct different types of solutions (exact, approximate analytical, numerical, graphical) of numerous nonlinear PDEs correctly, easily, and quickly. With respectto our example of computing?, we can mention that recently (in 2002) Y. Therefore the computers can be used to verify the results obtained by humans, to discovery new results, to - provetheresultsthatahumancanobtainwithoutanytechnology. In modern mathematics there exist computers that can perform various mathematical operations for which humans are incapable. ![]() But sometimes, the solution of such problems required such techn- ogy that was not available at that time. These errors could have a large e?ect on results obtained by engineers. Sommerfeld), and the mathematical tables, exact - lutions, and formulas, published in many mathematical textbooks, were not veri?ed rigorously. For the next examples, we can mention the history of computing the ?ne-structure constant ? (that was ?rst discovered by A. Shanks calculated ? with 707 digits (within 15 years), although due to mistakes only the ?rst 527 were correct. Machin (in 1706), who was the ?rst to correctly compute 100 digits of ?. The ?rst formula for computing decimal digits of ? was disc- ered by J. Let us look at some examples, the history of computing the number ? began in Egypt and Babylon about 2000 years BC, since then many mathematicians have calculated ? (e. In the history of mathematics there are many situations in which cal- lations were performed incorrectly for important practical applications. Publisher: Springer Science & Business Media ![]()
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