1st Edition

# Fractional Calculus with Applications for Nuclear Reactor Dynamics

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*Introduces Novel Applications for Solving Neutron Transport Equations*

While deemed nonessential in the past, fractional calculus is now gaining momentum in the science and engineering community. Various disciplines have discovered that realistic models of physical phenomenon can be achieved with fractional calculus and are using them in numerous ways. Since fractional calculus represents a reactor more closely than classical integer order calculus, **Fractional Calculus with Applications for Nuclear Reactor Dynamics** focuses on the application of fractional calculus to describe the physical behavior of nuclear reactors. It applies fractional calculus to incorporate the mathematical methods used to analyze the diffusion theory model of neutron transport and explains the role of neutron transport in reactor theory.

The author discusses fractional calculus and the numerical solution for fractional neutron point kinetic equation (FNPKE), introduces the technique for efficient and accurate numerical computation for FNPKE with different values of reactivity, and analyzes the fractional neutron point kinetic (FNPK) model for the dynamic behavior of neutron motion. The book begins with an overview of nuclear reactors, explains how nuclear energy is extracted from reactors, and explores the behavior of neutron density using reactivity functions. It also demonstrates the applicability of the Haar wavelet method and introduces the neutron diffusion concept to aid readers in understanding the complex behavior of average neutron motion.

This text:

- Applies the effective analytical and numerical methods to obtain the solution for the NDE
- Determines the numerical solution for one-group delayed neutron FNPKE by the explicit finite difference method
- Provides the numerical solution for classical as well as fractional neutron point kinetic equations
- Proposes the Haar wavelet operational method (HWOM) to obtain the numerical approximate solution of the neutron point kinetic equation, and more

**Fractional Calculus with Applications for Nuclear Reactor Dynamics** thoroughly and systematically presents the concepts of fractional calculus and emphasizes the relevance of its application to the nuclear reactor.

**Mathematical Methods in Nuclear Reactor Physics**Analytical Methods and Numerical Techniques for Solving Deterministic Neutron Diffusion and Kinetic Models

Numerical Methods for Solving Stochastic Point Kinetic Equations

**Neutron Diffusion Equation Model in Dynamical Systems**

Introduction

Outline of the Present Study

Application of the Variational Iteration Method to Obtain the Analytical Solution of the NDE

Application of the Modified Decomposition Method to Obtain the Analytical Solution of NDE

Numerical Results and Discussions for Neutron Diffusion Equations

One-Group NDE in Cylindrical and Hemispherical Reactors

Application of the ADM for One-Group Neutron Diffusion Equations

Conclusion

**Fractional Order Neutron Point Kinetic Model**

Introduction

Brief Description for Fractional Calculus

FNPKE and Its Derivation

Application of Explicit Finite Difference Scheme for FNPKE

Analysis for Stability of Numerical Computation

Numerical Experiments with Change of Reactivity

Conclusion

**Numerical Solution for Deterministic Classical and Fractional Order Neutron Point Kinetic Model**

Introduction

Application of MDTM to Classical Neutron Point Kinetic Equation

Numerical Results and Discussions for Classical Neutron Point Kinetic Model Using Different Reactivity Functions

Mathematical Model for Fractional Neutron Point Kinetic Equation

Fractional Differential Transform Method

Application of MDTM to Fractional Neutron Point Kinetic Equation

Numerical Results and Discussions for Fractional Neutron Point Kinetic Equation

Conclusion

**Classical and Fractional Order Stochastic Neutron Point Kinetic Model**

Introduction

Evolution of Stochastic Neutron Point Kinetic Model

Classical Order Stochastic Neutron Point Kinetic Model

Numerical Solution of the Classical Stochastic Neutron Point Kinetic Equation

Numerical Results and Discussions for the Solution of Stochastic Point Kinetic Model

Application of Explicit Finite Difference Method for Solving Fractional Order Stochastic Neutron Point Kinetic Model

Numerical Results and Discussions for the FSNPK Equations

Analysis for Stability of Numerical Computation for the FSNPK Equations

Conclusion

**Solution for Nonlinear Classical and Fractional Order Neutron Point Kinetic Model with Newtonian Temperature Feedback Reactivity**

Introduction

Classical Order Nonlinear Neutron Point Kinetic Model

Numerical Solution of Nonlinear Neutron Point Kinetic Equation in the Presence of Reactivity Function

Numerical Results and Discussions for the Classical Order Nonlinear Neutron Point Kinetic Equation

Mathematical Model for Nonlinear Fractional Neutron Point Kinetic Equation

Application of EFDM for Solving the Fractional Order Nonlinear Neutron Point Kinetic Model

Numerical Results and Discussions for Fractional Nonlinear Neutron Point Kinetic Equation with Temperature Feedback Reactivity Function

Computational Error Analysis for the Fractional Order Nonlinear Neutron Point Kinetic Equation

Conclusion

**Numerical Simulation Using Haar Wavelet Method for Neutron Point Kinetic Equation Involving Imposed Reactivity Function**

Introduction

Haar Wavelets

Function Approximation and Operational Matrix of the General Order Integration

Application of the HWOM for Solving Neutron Point Kinetic Equation

Numerical Results and Discussions

Convergence Analysis and Error Estimation

Conclusion

**Numerical Solution Using Two- Dimensional Haar Wavelet Method for Stationary Neutron Transport Equation in Homogeneous Isotropic Medium**

Introduction

Formulation of Neutron Transport Equation Model

Mathematical Model of the Stationary Neutron Transport Equation in a Homogeneous Isotropic Medium

Application of the Two-Dimensional Haar Wavelet Collocation Method to Solve the Stationary Neutron Transport Equation

Numerical Results and Discussions for Stationary Integer Order Neutron Transport Equation

Mathematical Model for Fractional Order Stationary Neutron Transport Equation

Application of the Two-Dimensional Haar Wavelet Collocation Method to the Fractional Order Stationary Neutron Transport Equation

Numerical Results and Discussions for Fractional Order Neutron Transport Equation

Convergence Analysis of the Two-Dimensional Haar Wavelet Method

Conclusion

**References**

### Biography

**Dr. Santanu Saha Ray** is an associate professor at the National Institute of Technology, Rourkela, India. He earned a Ph. D. in applied mathematics at Jadavpur University. He is a member of SIAM, the AMS, and the Indian Science Congress Association, and serves as the editor-in-chief for the *International Journal of Applied and Computational Mathematics*. Dr. Saha Ray has done extensive work in the area of fractional calculus and its role in nuclear science and engineering.

—Igor A. Bolotnov, Department of Nuclear Engineering, North Carolina State University"The textbook covers a wide range of models and methods for neutron transport and will be a welcome addition to many undergraduate and graduate courses. Graduate students working in nuclear engineering neutron transport area will appreciate the details of the methods, and will find this book useful to guide their research.""I have read the two sample chapters provided with me. I found no irregulaity in these two chapters. The table of content is absolutely ideal. This book focuses on the application of fractional calculus to describe the physical behavior of nuclear reactors. The sample chapters reflect the concept of the subject fairly.

In my opinion this book will be indispensable in the field of nuclear reactor science and engineering."

—Dr. Rasajit Kumar Bera, M. Sc. (Applied Maths., Gold Medalist), Ph. D.(Sc.), Retired Professor & Head of the Department of Science, National Institute of Technical Teacher’s Training and Research (Eastern Region), Block-FC, Sector-III, Salt Lake City, Kolkata-700106, India