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Table of Contents
Intro
Preface
Organization
Contents
Computing
An Auxiliary Inequality Based Method for Stabilization and Mesh-Adaptation of Steady and Time-Dependent Differential Equations
1 Introduction
2 The Auxiliary Entropy Inequality
2.1 Physical Motivation
2.2 A Simple Prototype
2.3 Extensions to General Differential Equations
3 Results
3.1 Diagnosis
3.2 Stabilization
3.3 Numerical Results
4 Conclusions
References
Existence and Uniqueness of Time-Fractional Diffusion Equation on a Metric Star Graph
1 Introduction
2 Preliminaries
3 Existence and Uniqueness Results of a Weak Solution
4 Conclusion and Future Work
References
Learning Numerical Viscosity Using Artificial Neural Regression Network
1 Introduction
2 Finite Difference Discretization
3 Shock Capturing Schemes
3.1 Local Lax-Friedrich Scheme
3.2 Lax-Wendroff Scheme
3.3 Entropy Stable TVD Scheme
4 Artificial Neural Networks (ANNs)
4.1 Network Training
5 Viscosity Network
5.1 Network Architecture
5.2 Training and Validation Data Creation
6 Numerical Results
6.1 Network Loss Plots
6.2 Solution of Scalar Conservation Law
7 Concluding Remarks
References
Computational Study of Some Numerical Methods for the Generalized Burgers-Huxley Equation
1 Introduction
2 Nonstandard Finite Difference Scheme (NSFD)
2.1 NSFD1
2.2 NSFD1-
2.3 NSFD2
2.4 NSFD2-
3 Explicit Exponential Finite Difference Method (EEFDM)
3.1 EEFDM-
4 Conclusion
References
A Local Meshless Method for a Multi-term Time Fractional Non-linear Diffusion Equation
1 Introduction
2 The Time-Semi Discretization Scheme
2.1 Convergence and Stability Analysis
3 Spatial Discretization
4 Numerical Experiments
5 Conclusion
References
On Space-Fractional Diffusion Equations with Conformable Derivative
1 Introduction
2 Preliminary Results
3 Maximum-Minimum Principle
3.1 Uniqueness and Stability of the Solution
4 Construction of Solutions
References
Compact Finite Difference Method for Pricing European and American Options Under Jump-Diffusion Models
1 Introduction
2 The Mathematical Model
3 Compact Finite Difference Method
3.1 Compact Finite Difference Approximations for First and Second Derivatives
3.2 Localization to Bounded Domain
3.3 Temporal Semi-discretization
3.4 The Fully Discrete Problem
3.5 Consistency
3.6 Stability
4 Numerical Results
5 Conclusion and Future Work
References
New Exact Solutions for Double Sine-Gordon Equation
1 Introduction
2 Exact Solutions
2.1 Sin-Cos Ansatz
2.2 Sinh-Cosh Ansatz
2.3 Sec-Csc Ansatz
2.4 Csch-Sech Ansatz
2.5 A Rational Function Ansatz
3 Discussion
References
Numerical Study of Mixed Convection in Single and Double Lid Driven Cavity Using LBM
1 Introduction
2 Problem and Solution Methodology
Preface
Organization
Contents
Computing
An Auxiliary Inequality Based Method for Stabilization and Mesh-Adaptation of Steady and Time-Dependent Differential Equations
1 Introduction
2 The Auxiliary Entropy Inequality
2.1 Physical Motivation
2.2 A Simple Prototype
2.3 Extensions to General Differential Equations
3 Results
3.1 Diagnosis
3.2 Stabilization
3.3 Numerical Results
4 Conclusions
References
Existence and Uniqueness of Time-Fractional Diffusion Equation on a Metric Star Graph
1 Introduction
2 Preliminaries
3 Existence and Uniqueness Results of a Weak Solution
4 Conclusion and Future Work
References
Learning Numerical Viscosity Using Artificial Neural Regression Network
1 Introduction
2 Finite Difference Discretization
3 Shock Capturing Schemes
3.1 Local Lax-Friedrich Scheme
3.2 Lax-Wendroff Scheme
3.3 Entropy Stable TVD Scheme
4 Artificial Neural Networks (ANNs)
4.1 Network Training
5 Viscosity Network
5.1 Network Architecture
5.2 Training and Validation Data Creation
6 Numerical Results
6.1 Network Loss Plots
6.2 Solution of Scalar Conservation Law
7 Concluding Remarks
References
Computational Study of Some Numerical Methods for the Generalized Burgers-Huxley Equation
1 Introduction
2 Nonstandard Finite Difference Scheme (NSFD)
2.1 NSFD1
2.2 NSFD1-
2.3 NSFD2
2.4 NSFD2-
3 Explicit Exponential Finite Difference Method (EEFDM)
3.1 EEFDM-
4 Conclusion
References
A Local Meshless Method for a Multi-term Time Fractional Non-linear Diffusion Equation
1 Introduction
2 The Time-Semi Discretization Scheme
2.1 Convergence and Stability Analysis
3 Spatial Discretization
4 Numerical Experiments
5 Conclusion
References
On Space-Fractional Diffusion Equations with Conformable Derivative
1 Introduction
2 Preliminary Results
3 Maximum-Minimum Principle
3.1 Uniqueness and Stability of the Solution
4 Construction of Solutions
References
Compact Finite Difference Method for Pricing European and American Options Under Jump-Diffusion Models
1 Introduction
2 The Mathematical Model
3 Compact Finite Difference Method
3.1 Compact Finite Difference Approximations for First and Second Derivatives
3.2 Localization to Bounded Domain
3.3 Temporal Semi-discretization
3.4 The Fully Discrete Problem
3.5 Consistency
3.6 Stability
4 Numerical Results
5 Conclusion and Future Work
References
New Exact Solutions for Double Sine-Gordon Equation
1 Introduction
2 Exact Solutions
2.1 Sin-Cos Ansatz
2.2 Sinh-Cosh Ansatz
2.3 Sec-Csc Ansatz
2.4 Csch-Sech Ansatz
2.5 A Rational Function Ansatz
3 Discussion
References
Numerical Study of Mixed Convection in Single and Double Lid Driven Cavity Using LBM
1 Introduction
2 Problem and Solution Methodology