Numerical Treatment and Analysis of Time-Fractional Evolution Equations

This book discusses numerical methods for solving time-fractional evolution equations, focusing on stability, error analysis, implementation, and qualitative properties. It is recommended for graduate students and researchers in applied and computational mathematics.

Format: Hardback
Length: 427 pages
Publication date: 28 February 2023
Publisher: Springer International Publishing AG

This comprehensive book delves into the realm of numerical methods for solving time-fractional evolution equations, offering a robust approach based on a combination of techniques. Firstly, the spatial variables are discretized using the Galerkin finite element method, employing piecewise linear trial functions. Subsequently, appropriate time stepping schemes, such as convolution quadrature or finite difference, are employed to advance the solutions. The primary focus lies in ensuring stability and error analysis of approximate solutions, along with the efficient implementation and qualitative properties, considering various regularity assumptions on the problem data. Drawing upon semigroup theory and Laplace transform, the book provides a thorough survey of the current ideas and methods in analysis, encompassing the most significant topics within this active and rapidly evolving field of research. It is highly recommended for graduate students and researchers in applied and computational mathematics, with a particular emphasis on numerical analysis.

Introduction:
The study of time-fractional evolution equations has gained significant attention in recent years due to their wide applications in various fields, including physics, engineering, and mathematics. These equations model the evolution of physical systems over time, taking into account the effects of time dilation, memory, and non-linearities. Numerical methods play a crucial role in solving these equations, as they allow for the accurate representation of complex phenomena and the computation of solutions in a computationally efficient manner.
Approach:
The approach outlined in this book revolves around the discretization of the spatial variables using the Galerkin finite element method. This method involves approximating the solution by a combination of piecewise linear functions, referred to as trial functions. The choice of trial functions is important, as they must satisfy certain conditions to ensure the stability and accuracy of the numerical solution. Once the spatial variables are discretized, suitable time stepping schemes are employed to advance the solutions in time. These schemes can be either convolution quadrature or finite difference, depending on the specific problem at hand.
Stability and Error Analysis:
Stability is a critical aspect of any numerical method, as it ensures the convergence of the solution to the exact solution as the time step size increases. Error analysis is also crucial in assessing the accuracy and reliability of the numerical solution. The book provides detailed discussions on stability and error analysis, using tools from semigroup theory and Laplace transform. These tools allow for the analysis of the behavior of the numerical solution, including the growth rate of errors and the stability of the scheme.
Efficient Implementation:
Efficient implementation is another important consideration in numerical methods. The book discusses various techniques for optimizing the computational efficiency of the numerical solution, such as matrix decomposition, parallel computing, and optimization algorithms. These techniques help reduce the computational time required for solving large-scale problems, making the method more practical for real-world applications.
Qualitative Properties:
In addition to stability and error analysis, the book also explores qualitative properties of the numerical solution. These properties include the behavior of solutions near singularities, the accuracy of approximations, and the stability of the scheme in the presence of disturbances. Understanding these qualitative properties is essential for the validation of the numerical solution and the development of reliable models.
Applications:
The book showcases a wide range of applications of time-fractional evolution equations in various fields, including fluid dynamics, solid mechanics, biology, and finance. The examples provided demonstrate the practical significance of the numerical methods discussed in the book and highlight their potential for solving complex problems.
Conclusion:
In conclusion, this book provides a comprehensive survey on the present ideas and methods of analysis for solving time-fractional evolution equations. It covers a wide range of topics, including stability and error analysis, efficient implementation, qualitative properties, and applications. The book is written in a clear and concise manner, making it accessible to graduate students and researchers in applied and computational mathematics, particularly numerical analysis. By leveraging the techniques discussed in this book, researchers can develop more accurate and reliable models for a wide range of real-world applications.

Weight: 828g
Dimension: 235 x 155 (mm)
ISBN-13: 9783031210495
Edition number: 1st ed. 2023