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This text presents numerical differential equations to graduate (doctoral) students. It includes the three standard approaches to numerical PDE, FDM, FEM and CM, and the two most common time stepping techniques, FDM and Runge-Kutta. We present both the numerical technique and the supporting theory.The applied techniques include those that arise in the present literature. The supporting mathematical theory includes the general convergence theory. This material should be readily accessible to students with basic knowledge of mathematical analysis, Lebesgue measure and the basics of Hilbert spaces and Banach spaces. Nevertheless, we have made the book free standing in most respects. Most importantly, the terminology is introduced, explained and developed as needed.The examples presented are taken from multiple vital application areas including finance, aerospace, mathematical biology and fluid mechanics. The text may be used as the basis for several distinct lecture courses or as a reference. For instance, this text will support a general applications course or an FEM course with theory and applications. The presentation of material is empirically-based as more and more is demanded of the reader as we progress through the material. By the end of the text, the level of detail is reminiscent of journal articles. Indeed, it is our intention that this material be used to launch a research career in numerical PDE. [Publisher]

Differential equations --- Finite differences --- Finite element method --- Équations différentielles --- Différences finies. --- Éléments finis, Méthode des. --- Numerical solutions --- Solutions numériques.

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Differential equations, Nonlinear --- Fluid dynamics --- Heat --- Equations différentielles non linéaires --- Fluides, Dynamique des --- Numerical solutions --- Convection --- Mathematical models --- Solutions numériques --- Fluid dynamics. --- Numerical solutions. --- Mathematical models. --- Equations différentielles non linéaires --- Solutions numériques --- Differential equations, Nonlinear - Numerical solutions --- Heat - Convection - Mathematical models

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Harmonic analysis. Fourier analysis --- Generalized spaces --- Harmonic analysis --- Espaces généralisés --- Problèmes aux limites --- Congresses --- Congrès --- Solutions numériques --- 51 --- -Harmonic analysis --- -Analysis (Mathematics) --- Functions, Potential --- Potential functions --- Banach algebras --- Calculus --- Mathematical analysis --- Mathematics --- Bessel functions --- Fourier series --- Harmonic functions --- Time-series analysis --- Geometry of paths --- Minkowski space --- Spaces, Generalized --- Weyl space --- Calculus of tensors --- Geometry, Differential --- Geometry, Non-Euclidean --- Hyperspace --- Relativity (Physics) --- Congresses. --- -Mathematics --- 51 Mathematics --- Espaces généralisés --- Problèmes aux limites --- Congrès --- Solutions numériques --- -Congresses

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Partial differential equations --- Mathematical physics --- Wave equation --- Solitons --- Equations d'onde --- Numerical solutions --- Solutions numériques --- Nonlinear wave equations --- 517.988 --- 517.988 Nonlinear functional analysis and approximation methods --- Nonlinear functional analysis and approximation methods --- Pulses, Solitary wave --- Solitary wave pulses --- Wave pulses, Solitary --- Connections (Mathematics) --- Nonlinear theories --- Wave-motion, Theory of --- Numerical analysis --- Solitons. --- Numerical solutions. --- Solutions numériques --- Ondes --- Waves --- Ondes non linéaires --- Nonlinear waves --- Propagation --- Ondes non linéaires. --- Waves. --- Nonlinear wave equations - Numerical solutions --- Equations aux derivees partielles non lineaires --- Equation des ondes --- Equation --- Physique quantique

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Approximation theory --- Differential equations, Partial --- Théorie de l'approximation --- Equations aux dérivées partielles --- Numerical solutions --- Solutions numériques --- -Approximation theory --- 519.6 --- 681.3 *G18 --- Theory of approximation --- Functional analysis --- Functions --- Polynomials --- Chebyshev systems --- Partial differential equations --- Computational mathematics. Numerical analysis. Computer programming --- Partial differential equations: difference methods elliptic equations finite element methods hyperbolic equations method of lines parabolic equations (Numerical analysis) --- Approximation theory. --- Numerical solutions. --- 681.3 *G18 Partial differential equations: difference methods elliptic equations finite element methods hyperbolic equations method of lines parabolic equations (Numerical analysis) --- 519.6 Computational mathematics. Numerical analysis. Computer programming --- Théorie de l'approximation --- Equations aux dérivées partielles --- Solutions numériques --- 681.3 *G18 Partial differential equations: difference methods; elliptic equations; finite element methods; hyperbolic equations; method of lines; parabolic equations (Numerical analysis) --- Partial differential equations: difference methods; elliptic equations; finite element methods; hyperbolic equations; method of lines; parabolic equations (Numerical analysis) --- Numerical solutions of differential equations