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This book presents the various algebraic techniques for solving partial differential equations to yield exact solutions, techniques developed by the author in recent years and with emphasis on physical equations such as: the Maxwell equations, the Dirac equations, the KdV equation,  the KP equation,  the nonlinear Schrodinger equation,  the Davey and Stewartson equations, the Boussinesq equations in geophysics,  the Navier-Stokes equations and the boundary layer problems.  In order to solve them, I have employed the grading technique, matrix differential operators, stable-range of nonlinear terms, moving frames, asymmetric assumptions,  symmetry transformations,  linearization techniques  and  special functions. The book is self-contained and requires only a minimal understanding of calculus and linear algebra, making it accessible to a broad audience in the fields of mathematics, the sciences and engineering. Readers may find the exact solutions and mathematical skills needed in their own research.

Differential equations, Hyperbolic -- Numerical solutions. --- Differential equations, Partial -- Numerical solutions. --- Differential equations, Partial. --- Mathematics. --- Mathematics --- Physical Sciences & Mathematics --- Calculus --- Differential equations. --- Differential-algebraic equations. --- Algebraic-differential equations --- Differential-algebraic systems --- Equations, Algebraic-differential --- Equations, Differential-algebraic --- Systems, Differential-algebraic --- 517.91 Differential equations --- Differential equations --- Partial differential equations. --- Applied mathematics. --- Engineering mathematics. --- Mathematical physics. --- Partial Differential Equations. --- Mathematical Physics. --- Applications of Mathematics. --- Differential equations, partial. --- Partial differential equations --- Math --- Science --- Engineering --- Engineering analysis --- Mathematical analysis --- Physical mathematics --- Physics --- Differential equations, Partial --- Numerical solutions.

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• Reference Manager
• EndNote
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Choose an application

• Reference Manager
• EndNote
• RefWorks (Direct export to RefWorks)

This book provides explicit representations of finite-dimensional simple Lie algebras, related partial differential equations, linear orthogonal algebraic codes, combinatorics and algebraic varieties, summarizing the author’s works and his joint works with his former students.  Further, it presents various oscillator generalizations of the classical representation theorem on harmonic polynomials, and highlights new functors from the representation category of a simple Lie algebra to that of another simple Lie algebra. Partial differential equations play a key role in solving certain representation problems. The weight matrices of the minimal and adjoint representations over the simple Lie algebras of types E and F are proved to generate ternary orthogonal linear codes with large minimal distances. New multi-variable hypergeometric functions related to the root systems of simple Lie algebras are introduced in connection with quantum many-body systems in one dimension. In addition, the book identifies certain equivalent combinatorial properties on representation formulas, and the irreducibility of representations is proved directly related to algebraic varieties. The book offers a valuable reference guide for mathematicians and scientists alike. As it is largely self-contained – readers need only a minimal background in calculus and linear algebra – it can also be used as a textbook.

Algebra --- Algebraic geometry --- Partial differential equations --- Differential equations --- Mathematics --- Computer science --- differentiaalvergelijkingen --- algebra --- functies (wiskunde) --- wiskunde --- algoritmen