Author | Bellman, Richard E. author |
---|---|
Title | Methods in Approximation [electronic resource] : Techniques for Mathematical Modelling / by Richard E. Bellman, Robert S. Roth |
Imprint | Dordrecht : Springer Netherlands, 1986 |
Connect to | http://dx.doi.org/10.1007/978-94-009-4600-2 |
Descript | XV, 224 p. online resource |
Basic Concepts -- Integral Domains, Fields and Vector Spaces -- Subspaces, Bases and Inner Products -- Spaces, Subspaces and Approximation -- The Continuous Function -- Polynomial Subspaces -- Spaces Generated by Differential Equations -- The Piecewise Linear Function -- Discussion -- Bibliograpy and Comments -- Polynomial Approximation -- Piecewise Linear Functions -- Curve Fitting by Straight Lines -- A One Dimensional Process in Dynamic Programming -- The Functional Equation -- The Principle of Optimality -- A Direct Derivation -- Curve Fitting by Segmented Straight Lines -- A Dynamic Programming Approach -- A Computational Procedure -- Three Dimensional Polygonal Approximation -- The Orthogonal Polynomials -- The Approximation Technique -- Discussion -- Bibliography and Comments -- Polynomial Splines -- The Cubic Spline I -- Construction of the Cubic Spline -- Existence and Uniqueness -- A Computational Algorithm โ Potterโs Method -- Splines via Dynamic Programming -- Derivation of Splines by Dynamic Programming -- Equivalence of the Recursive Relations obtained by Dynamic Programming and the Usual results -- Cardinal Splines -- Polynomial Splines -- Generalized Splines -- Mean Square Spline Approximation -- The Cubic Spline II -- The Minimization Procedure -- The Functional Equation -- Recursion Relations -- Bibliography and Comments -- Quasilinearization -- Quasilinearization I -- The Newton Raphson Method -- Quasilinearization II -- Existence -- Convergence -- An Example, Parameter Identification -- Unknown Initial Conditions -- Damped Oscillations -- Segmental Differential Approximation -- Differential Systems with Time Varying Coefficients -- A Method of Solution -- An Interesting Case -- Discussion -- Bibliography and Comments -- Differential Approximation -- Differential Approximation -- Linear Differential Operators -- Degree of Approximation -- Improving the Approximation -- An Example -- Differential-Difference Equations -- A Useful Approximation to g(t) -- Discussion -- An Example -- Functional Differential Equations -- The Nonlinear Spring -- The Van der Pol Equation -- Bibliography and Comments -- Differential Quadrature -- Differential Quadrature -- Determination Of the Weighting Coefficients -- A First Order Problem -- A Nonlinear Wave Equation -- Systems of Nonlinear Partial Differential Equations -- Higher Order Systems -- Long Term Integration -- G(y) Linear -- G(y) Nonlinear -- A Mathematical Problem -- Systems with Partial Information -- Bibliography and Comments -- Exponential Approximation -- Approximation in Function Space -- An Example โ Pharmacokinetics -- Other Physical Processes -- Proneyโs Method -- The Renewal Equation -- The Fredholm Integral Equation -- Bibliography and Comments -- The Riccati Equation -- The Linear Differential Equation -- Differential Inequalities -- Solution of the Riccati Equation in terms of the Maximum Operation -- Upper and Lower Bounds -- Successive Approximations via Quasilinearization -- An Illustrative Example -- Higher Order Approximations -- Multidimensional Riccati Equation -- Variational Problems and the Riccati Equation -- Bibliography and Comments -- Solution of Approximate Equations -- First Order Differential Equations -- The Second Order Differential Equation -- Discussion -- Linear Perturbations -- The Van der Pol Equation I -- The Van der Pol Equation II -- The Riccati Equation -- u? + a(t)u = 0 -- Another Approach -- Discussion -- Bibliography and Comments -- Magnetic Field Determination -- The Theoretical Problem -- Maxwellโs Equations -- A Variational Principle -- The Finite Element Method -- Computational Aspects -- Analytical Considerations -- Boundary Conditions -- Discussion -- Bibliography and Comments