Fractional Calculus with Applications for Nuclear Reactor Dynamics

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 PhysicsAnalytical Methods and Numerical Techniques for Solving Deterministic Neutron Diffusion and Kinetic ModelsNumerical Methods for Solving Stochastic Point Kinetic EquationsNeutron Diffusion Equation Model in Dynamical SystemsIntroductionOutline of the Present StudyApplication of the Variational Iteration Method to Obtain the Analytical Solution of the NDEApplication of the Modified Decomposition Method to Obtain the Analytical Solution of NDENumerical Results and Discussions for Neutron Diffusion EquationsOne-Group NDE in Cylindrical and Hemispherical ReactorsApplication of the ADM for One-Group Neutron Diffusion EquationsConclusionFractional Order Neutron Point Kinetic ModelIntroductionBrief Description for Fractional CalculusFNPKE and Its DerivationApplication of Explicit Finite Difference Scheme for FNPKEAnalysis for Stability of Numerical ComputationNumerical Experiments with Change of ReactivityConclusionNumerical Solution for Deterministic Classical and Fractional Order Neutron Point Kinetic ModelIntroductionApplication of MDTM to Classical Neutron Point Kinetic EquationNumerical Results and Discussions for Classical Neutron Point Kinetic Model Using Different Reactivity FunctionsMathematical Model for Fractional Neutron Point Kinetic EquationFractional Differential Transform MethodApplication of MDTM to Fractional Neutron Point Kinetic EquationNumerical Results and Discussions for Fractional Neutron Point Kinetic EquationConclusionClassical and Fractional Order Stochastic Neutron Point Kinetic ModelIntroductionEvolution of Stochastic Neutron Point Kinetic ModelClassical Order Stochastic Neutron Point Kinetic ModelNumerical Solution of the Classical Stochastic Neutron Point Kinetic EquationNumerical Results and Discussions for the Solution of Stochastic Point Kinetic ModelApplication of Explicit Finite Difference Method for Solving Fractional Order Stochastic Neutron Point Kinetic ModelNumerical Results and Discussions for the FSNPK EquationsAnalysis for Stability of Numerical Computation for the FSNPK EquationsConclusionSolution for Nonlinear Classical and Fractional Order Neutron Point Kinetic Model with Newtonian Temperature Feedback ReactivityIntroductionClassical Order Nonlinear Neutron Point Kinetic ModelNumerical Solution of Nonlinear Neutron Point Kinetic Equation in the Presence of Reactivity FunctionNumerical Results and Discussions for the Classical Order Nonlinear Neutron Point Kinetic EquationMathematical Model for Nonlinear Fractional Neutron Point Kinetic EquationApplication of EFDM for Solving the Fractional Order Nonlinear Neutron Point Kinetic ModelNumerical Results and Discussions for Fractional Nonlinear Neutron Point Kinetic Equation with Temperature Feedback Reactivity FunctionComputational Error Analysis for the Fractional Order Nonlinear Neutron Point Kinetic EquationConclusionNumerical Simulation Using Haar Wavelet Method for Neutron Point Kinetic Equation Involving Imposed Reactivity FunctionIntroductionHaar WaveletsFunction Approximation and Operational Matrix of the General Order IntegrationApplication of the HWOM for Solving Neutron Point Kinetic EquationNumerical Results and DiscussionsConvergence Analysis and Error EstimationConclusionNumerical Solution Using Two- Dimensional Haar Wavelet Method for Stationary Neutron Transport Equation in Homogeneous Isotropic MediumIntroductionFormulation of Neutron Transport Equation ModelMathematical Model of the Stationary Neutron Transport Equation in a Homogeneous Isotropic MediumApplication of the Two-Dimensional Haar Wavelet Collocation Method to Solve the Stationary Neutron Transport EquationNumerical Results and Discussions for Stationary Integer Order Neutron Transport EquationMathematical Model for Fractional Order Stationary Neutron Transport EquationApplication of the Two-Dimensional Haar Wavelet Collocation Method to the Fractional Order Stationary Neutron Transport EquationNumerical Results and Discussions for Fractional Order Neutron Transport EquationConvergence Analysis of the Two-Dimensional Haar Wavelet MethodConclusionReferences

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.