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Advanced numerical methods with Matlab 2 : resolution of nonlinear, differential and partial differential equations
The purpose of this book is to introduce and study numerical methods basic and advanced ones for scientific computing. This last refers to the implementation of appropriate approaches to the treatment of a scientific problem arising from physics (meteorology, pollution, etc.) or of engineering (mechanics of structures, mechanics of fluids, treatment signal, etc.). Each chapter of this book recalls the essence of the different methods resolution and presents several applications in the field of engineering as well as programs developed under Matlab software.
Book
Numerical methods for ordinary differential equations
A new edition of this classic work, comprehensively revised to present exciting new developments in this important subject The study of numerical methods for solving ordinary differential equations is constantly developing and regenerating, and this third edition of a popular classic volume, written by one of the world's leading experts.
eBook
Global Smooth Solutions for the Inviscid SQG Equation
In this paper, we show the existence of the first non trivial family of classical global solutions of the inviscid surface
quasi-geostrophic equation.
eBook
A Proof that Artificial Neural Networks Overcome the Curse of Dimensionality in the Numerical Approximation of Black–Scholes Partial Differential Equations
Artificial neural networks (ANNs) have very successfully been used in numerical simulations for a series of computational problems
ranging from image classification/image recognition, speech recognition, time series analysis, game intelligence, and computational
advertising to numerical approximations of partial differential equations (PDEs). Such numerical simulations suggest that ANNs have the
capacity to very efficiently approximate high-dimensional functions and, especially, indicate that ANNs seem to admit the fundamental
power to overcome the curse of dimensionality when approximating the high-dimensional functions appearing in the above named
computational problems. There are a series of rigorous mathematical approximation results for ANNs in the scientific literature. Some of
them prove convergence without convergence rates and some of these mathematical results even rigorously establish convergence rates but
there are only a few special cases where mathematical results can rigorously explain the empirical success of ANNs when approximating
high-dimensional functions. The key contribution of this article is to disclose that ANNs can efficiently approximate high-dimensional
functions in the case of numerical approximations of Black-Scholes PDEs. More precisely, this work reveals that the number of required
parameters of an ANN to approximate the solution of the Black-Scholes PDE grows at most polynomially in both the reciprocal of the
prescribed approximation accuracy
eBook
Quasi-Periodic Standing Wave Solutions of Gravity-Capillary Water Waves
The authors prove the existence and the linear stability of small amplitude time quasi-periodic standing wave solutions (i.e. periodic and even in the space variable x) of a 2-dimensional ocean with infinite depth under the action of gravity and surface tension. Such an existence result is obtained for all the values of the surface tension belonging to a Borel set of asymptotically full Lebesgue measure.
eBook
Continuous-time Random Walks for the Numerical Solution of Stochastic Differential Equations
This paper introduces time-continuous numerical schemes to simulate stochastic differential equations (SDEs) arising in mathematical
finance, population dynamics, chemical kinetics, epidemiology, biophysics, and polymeric fluids. These schemes are obtained by spatially
discretizing the Kolmogorov equation associated with the SDE in such a way that the resulting semi-discrete equation generates a Markov
jump process that can be realized exactly using a Monte Carlo method. In this construction the jump size of the approximation can be
bounded uniformly in space, which often guarantees that the schemes are numerically stable for both finite and long time simulation of
SDEs. By directly analyzing the infinitesimal generator of the approximation, we prove that the approximation has a sharp stochastic
Lyapunov function when applied to an SDE with a drift field that is locally Lipschitz continuous and weakly dissipative. We use this
stochastic Lyapunov function to extend a local semimartingale representation of the approximation. This extension makes it possible to
quantify the computational cost of the approximation. Using a stochastic representation of the global error, we show that the
approximation is (weakly) accurate in representing finite and infinite-time expected values, with an order of accuracy identical to the
order of accuracy of the infinitesimal generator of the approximation. The proofs are carried out in the context of both fixed and
variable spatial step sizes. Theoretical and numerical studies confirm these statements, and provide evidence that these schemes have
several advantages over standard methods based on time-discretization. In particular, they are accurate, eliminate nonphysical moves in
simulating SDEs with boundaries (or confined domains), prevent exploding trajectories from occurring when simulating stiff SDEs, and
solve first exit problems without time-interpolation errors.
eBook
Advances in the applications of nonstandard finite difference schemes
This volume provides a concise introduction to the methodology of nonstandard finite difference (NSFD) schemes construction and shows how they can be applied to the numerical integration of differential equations occurring in the natural, biomedical, and engineering sciences. These methods had their genesis in the work of Mickens in the 1990's and are now beginning to be widely studied and applied by other researchers. The importance of the book derives from its clear and direct explanation of NSFD in the introductory chapter along with a broad discussion of the future directions needed to advance the topic.
eBook