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Submit a ProposalDifferential Equations in the Era of Computational Techniques
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This definitive guide bridges traditional analytical theory with cutting-edge machine learning and quantum computing techniques to master differential equations across modern science, engineering, and finance.
Description
The field of differential equations has long been a cornerstone of mathematical modeling in various scientific and engineering disciplines. Historically, analytical methods for solving differential equations dominated, providing exact solutions to well-defined problems. However, as systems became more complex, particularly with the rise of nonlinear dynamics, high-dimensional systems, and real-world applications, numerical methods gained prominence. In recent years, the advent of machine learning, high-performance computing, and quantum computing has further transformed the landscape of differential equations. This evolution reflects broader trends in computational mathematics and applied science, where interdisciplinary approaches are becoming increasingly vital. This book offers a comprehensive exploration of differential equations, covering a wide array of topics ranging from classical analytical solutions to cutting-edge computational techniques. It provides a foundational overview of differential equations and analytical solutions of both ordinary and partial differential equations, delving into numerical methods for solving these equations, with a focus on techniques such as spectral methods, neural networks, and quantum algorithms. It will also cover domain-specific applications, including the use of differential equations in biology and medicine, finance and economics, engineering, environmental and climate modeling, and quantum computing, making it an essential resource for a wide range of industries.
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The field of differential equations has long been a cornerstone of mathematical modeling in various scientific and engineering disciplines. Historically, analytical methods for solving differential equations dominated, providing exact solutions to well-defined problems. However, as systems became more complex, particularly with the rise of nonlinear dynamics, high-dimensional systems, and real-world applications, numerical methods gained prominence. In recent years, the advent of machine learning, high-performance computing, and quantum computing has further transformed the landscape of differential equations. This evolution reflects broader trends in computational mathematics and applied science, where interdisciplinary approaches are becoming increasingly vital. This book offers a comprehensive exploration of differential equations, covering a wide array of topics ranging from classical analytical solutions to cutting-edge computational techniques. It provides a foundational overview of differential equations and analytical solutions of both ordinary and partial differential equations, delving into numerical methods for solving these equations, with a focus on techniques such as spectral methods, neural networks, and quantum algorithms. It will also cover domain-specific applications, including the use of differential equations in biology and medicine, finance and economics, engineering, environmental and climate modeling, and quantum computing, making it an essential resource for a wide range of industries.
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Author / Editor Details
Mamta Kapoor, PhD is an Associate Professor in the Department of Mathematics at Lovely Professional University, Phagwara, Punjab, India. She has a number of research papers in international journals of repute. Her research focuses on numerical approximation of linear and non-linear partial differential equations, as well as semi-analytical solutions of fractional partial differential equations.
Mamta Kapoor, PhD is an Associate Professor in the Department of Mathematics at Lovely Professional University, Phagwara, Punjab, India. She has a number of research papers in international journals of repute. Her research focuses on numerical approximation of linear and non-linear partial differential equations, as well as semi-analytical solutions of fractional partial differential equations.
Geeta Arora, PhD is a Professor in the Department of Mathematics at Lovely Professional University, Punjab, India, with more than twelve years of teaching experience. She has published more than 55 research papers in national and international journals, 15 book chapters, and two books. Her research focuses on the development of numerical methods and statistics.
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Table of Contents
Preface
Acknowledgements
1. A Comprehensive Guide to Partial Differential Equations: Theory, Numerical Methods, and Applications
Jaya Gupta and Shubham Mishra
1.1 Introduction
1.2 Ordinary vs. Partial
1.3 Origins and Historical Development
1.4 Numerical Approaches Used to Solve Partial Differential Equations
1.5 Classical Basis Function Methods Used for Solving Partial Differential Equations
1.6 Conclusion
References
2. Analytical Solutions of Various Types of Differential Equations via Variants of the Differential Transform Method
Adarsh Bhardwaj and Geeta Arora
2.1 Introduction
2.2 Methodology
2.2.1 Differential Transform Method
2.2.2 Reduced Differential Transform Method
2.2.3 Fractional Differential Transform Method
2.2.4 Modified Fractional Differential Transform Method
2.3 Numerical Examples
2.3.1 Solving Ordinary Differential Equations
2.3.2 Solving Partial Differential Equations
2.3.3 Solving Fractional Differential Equations
2.4 Conclusion
References
3. Numerical Solution of Burgers’ Equation Using Orthogonal Collocation Method
Shelly Arora, Muskan, Abdul-Majeed Ayiebre and Happy Kumar
3.1 Introduction
3.2 Orthogonal Collocation Method
3.2.1 Collocation Points
3.3 Burgers’ Equation
3.4 Methodology
3.5 Results and Discussion
3.6 Conclusions
References
4. Soliton Solutions and Chaotic Wave Dynamics in the Combined KdV–mKdV Equation
Hamood Ur Rehman, Muhammad Shoaib Saleem and Ifrah Iqbal
List of Abbreviations
4.1 Introduction
4.2 Description of MSSEM
4.3 Simplest Equation Method
4.4 Chaotic Analysis
4.5 Multistability and Sensitivity Analysis
4.6 Results and Discussion
4.7 Novelty and Comparative Analysis
4.8 Conclusion
Acknowledgments
References
5. A Comparative Analysis on the Numerical Evaluation of Multiple Integrals Appearing in Mathematical Physics via Different Orthogonal Polynomial Wavelets
Ashish Rayal, Mohammad Izadi and Sabri T.M. Thabet
List of Abbreviations
5.1 Introduction
5.2 Orthogonal Wavelets
5.2.1 Taylor Wavelets
5.2.2 Muntz Wavelets
5.2.3 Bernstein Wavelets
5.3 Function Approximation via Orthogonal Wavelets
5.4 Wavelets Scheme for Numerous Integrals
5.5 Error Estimation
5.6 Method Implementation
5.7 Conclusion
Acknowledgments
References
6. Analytical and Numerical Solutions of Different Types
of Schrödinger Equations
Keyur Modha and Mamta Kapoor
List of Abbreviations
6.1 Introduction
6.2 Analytical Solutions of the Time-Independent Schrödinger Equation
6.2.1 Separation of Variables and Coordinate Systems
6.2.2 Canonical Solvable Potentials
6.2.3 Special Functions and Symmetry Principles
6.2.4 Analytical Insights vs. Numerical Realities
6.2.5 Outlook
6.3 Analytical Solutions of the Time-Dependent Schrödinger Equation
6.3.1 General Solution Strategies
6.3.2 Free Particle and Wave Packets
6.3.3 Propagator (Green’s Function) Approach
6.3.4 Stationary vs. Nonstationary States
6.3.5 Outlook
6.4 Numerical Approaches—Finite Difference Methods
6.4.1 Why Numerical Methods?
6.4.2 The Finite Difference Idea
6.4.3 Time Evolution: Explicit vs. Implicit Methods
6.4.4 Worked Example: Particle in a 1D Infinite Square Well
6.4.5 Physical Takeaways
6.5 Advanced Numerical Methods: Spectral and Finite Element Approaches
6.5.1 Why Go Beyond Finite Differences?
6.5.2 Spectral Methods
6.5.3 Finite Element Method
6.5.4 Worked Example: Quantum Harmonic Oscillator
6.5.5 Physical Takeaways
6.6 Nonlinear Schrödinger Equation
6.6.1 Basic Properties and Conserved Quantities
6.6.2 Bright Soliton: Derivation of the 1-Soliton Solution (Focusing Case)
6.6.3 Dark Soliton (Defocusing Case)
6.6.4 Integrability and Multisoliton Interactions
6.6.5 Numerical Solution Methods
6.6.5.1 SSFM: The Workhorse
6.6.5.2 Runge–Kutta in Time + Finite Difference/Spectral Spatial Discretization
6.6.5.3 Adaptive and Energy-Preserving Schemes
6.6.6 Worked Numerical Example
6.6.7 Applications and Physical Contexts
6.7 Multidimensional Schrödinger Equation
6.7.1 Introduction
6.7.2 General Form of the Multidimensional Schrödinger Equation
6.7.3 Separation of Variables in Higher Dimensions
6.7.4 2D and 3D Quantum Wells and Quantum Dots
6.7.5 PDE Solvers in Higher Dimensions
6.7.6 Visualization and Physical Insights
6.7.7 Applications
6.7.8 Summary
6.8 Applications in Physics and Optics
6.8.1 Introduction
6.8.2 Applications in Physics
6.8.2.1 Atomic and Molecular Physics
6.8.2.2 Condensed Matter Physics
6.8.2.3 Nuclear and Particle Physics
6.8.3 Applications in Optics
6.8.3.1 Lasers and Coherent Light
6.8.3.2 Quantum Optics
6.8.3.3 Photonic Crystals and Waveguides
6.8.3.4 Nonlinear Optics
6.8.4 Interdisciplinary Applications
6.8.5 Summary
6.9 Conclusions and Future Directions
References
7. A Computational Study of Reaction–Convection–Diffusion Processes through Quintic Hermite Collocation Method
Inderpreet Kaur
7.1 Introduction
7.2 Numerical Approach
7.3 Error Analysis
7.4 Stability Analysis
7.4.1 Discretization in t and x
7.5 Numerical Examples
7.5.1 Problem
7.5.2 Problem
7.5.3 Problem
7.5.4 Problem
7.6 Conclusions
7.7 Future Scope
References
8. Numerical Solution of Generalized Burger–Huxley Equation
Using Cubic Hermite Splines
Shelly Arora, Ishika Mittal, Abdul-Majeed Ayiebre and Happy Kumar
8.1 Introduction
8.2 Generalized Burgers–Huxley Equation
8.3 Hermite Interpolation
8.4 Implementation of Hermite Collocation Technique
8.5 Results and Discussion
8.6 Conclusions
References
9. Evaluation of Runge–Kutta Methods for Time Integration of the Burgers’ Equation Utilizing the Compact Finite Difference Scheme
Geeta Arora
9.1 Introduction
9.2 Sixth-Order Compact Finite Difference Scheme for the First-Order Derivative
9.2.1 Taylor-Series Expansions and Coefficient Determination
9.2.2 Interior Compact Scheme
9.2.3 Boundary Treatment for the First-Order Derivative
9.2.4 Matrix Formulation of the First-Order Derivative
9.3 Compact Finite Difference Approximation for the Second-Order Derivative
9.3.1 Taylor-Series Expansion and Determination of Coefficients
9.3.2 Sixth-Order Compact Scheme for Interior Points
9.3.3 Boundary Treatment for the Second-Order Derivative
9.3.4 Matrix Formulation of the Second-Order Derivative
9.4 Numerical Implementation of the Compact Scheme for Burgers’ Equation
9.5 Runge–Kutta Schemes
9.5.1 General Explicit RK Formulation
9.5.2 Second-Order RK Methods
9.5.2.1 Standard RK2 Method
9.5.2.2 Ralston Method
9.5.2.3 Third-Order RK Methods
9.5.2.4 Fourth-Order RK Methods
9.5.2.5 Rk45
9.5.3 Low-Storage RK Methods (Williamson Form)
9.5.4 Strong Stability-Preserving RK Methods
9.6 Numerical Results
9.7 Conclusion
References
10. Analytical Approximate Solutions for 2D Fractional PDEs: A Conformable Fractional Reduced Differential Transform Method
Maher Jneid and Abir Chaouk
10.1 Introduction
10.2 Analysis of CFRDTM
10.3 Applications and Discussion
10.4 Conclusion
References
11. Semianalytical Solution of Nonlinear Fractional Partial Differential Equations Using Accelerated HPTM
Shabnam Jasrotia and Prince Singh
11.1 Introduction
11.1.1 Klein–Gordon
11.1.2 Korteweg-DeVries
11.1.3 Burgers’ Equation
11.2 Preliminaries
11.2.1 Fractional Calculus
11.2.2 Role of the Laplace Transform in the CF Framework
11.3 AHPTM
11.3.1 He’s Polynomials and Accelerated He’s Polynomials
11.4 Condition of Convergence of AHPTM
11.5 Application
11.6 Conclusion
Acknowledgment
References
12. Solution of Fuzzy Riccati Differential Equation Using
Fuzzy Sumudu Decomposition Method
Asha N., Deepak Kumar Sah, Sreenivasulu Ballem, K. Venkateshwarlu, Kottakkaran Sooppy Nisar, Ramalingaiah Kadari and Devika Dabke
List of Abbreviations
12.1 Introduction
12.2 Novelty of the Work
12.3 Preliminaries
12.4 Fuzzy Riccati Differential Equation
12.5 Numerical Results
12.6 Conclusion
Acknowledgments
References
13. Chaotic Behavior of Fractional Delay Differential Equations
S. Priyadharsini, E. Kungumaraj, S. Saravanakumar and T. Sathiyaraj
List of Abbreviations
13.1 Introduction
13.1.1 Background: Hutchinson Equation
13.2 Preliminaries
13.2.1 Preliminary Concepts
13.3 Main Result: Method of Steps
13.4 Limitations
13.5 Computational Method
13.6 Example
13.6.1 Analysis of Hutchinson Equation
13.7 Significance of the Proposed Model
13.8 Conclusion
References
14. Physics-Informed Neural Network for Solving Coupled
Ordinary Differential Equations
Priyanka Chandra and Raja Das
List of Abbreviations
14.1 Introduction
14.2 Differential Equations
14.3 Physics-Informed Neural Networks
14.4 Implementation of PINNs in ODEs
14.4.1 Convergence and Stability of PINNs
14.5 Results and Discussion
14.6 Conclusion
Acknowledgments
References
15. Artificial Neural Network–Based Approximation of Nonlinear ODEs with Comparative Analysis of Activation Functions
Hetvi Patel, Dhvani Patel and Yogeshwari F. Patel
List of Abbreviations
15.1 Introduction
15.1.1 Scope of This Chapter
15.2 Theoretical Background of Multilayer Perceptron Neural Network
15.2.1 Methodology
15.3 Illustration
15.4 Conclusion
References
16. AI-Driven Numerical and Machine Learning Approaches for Solving Differential Equations
S. Padmanabhan, Pallavi G., Kavya K. N. and Vijayalakshmi S.
16.1 Introduction and Motivation
16.1.1 Importance of Differential Equations in Computational Science
16.1.2 Applications in Physics, Engineering, Finance, and Biology
16.1.3 Limitations of Conventional Analytical and Numerical Methods
16.1.4 Emergence of AI and ML in Computational Modeling
16.2 Traditional Numerical Approaches
16.2.1 Euler Method
16.2.2 RK Methods
16.2.3 Spectral and Finite Difference Methods
16.3 Machine Learning Approaches for Differential Equations
16.3.1 Neural Network Solvers
16.3.2 Physics-Informed Neural Networks
16.3.3 Deep Operator Networks
16.4 Hybrid AI–Numerical Techniques
16.4.1 Combining Classical Solvers with Machine Learning
16.4.2 Adaptive Time-Stepping with ML Predictions
16.4.3 Data-Driven Reduced-Order Modeling
16.4.4 Hybrid Workflow
16.5 Data Pipeline and High-Dimensionality Handling
16.5.1 Data Generation, Preprocessing, and Validation
16.5.2 Handling High-Dimensional PDEs
16.6 Applications
16.6.1 Climate and Environmental Modeling
16.6.2 Epidemiology and Biology
16.6.3 Quantum and Engineering Applications
16.6.4 Summary of Applications
16.7 Case Studies and Experimental Results
16.7.1 Experimental Setup
16.7.2 Comparative Performance
16.7.3 Error Analysis
16.7.4 Visualization and Performance Plots
16.7.5 Computation Time
16.8 Comparison of Methods
16.9 Challenges and Research Gaps
16.10 Tools and Implementation Details
16.11 Recent Research (2020–2025) and Literature Gaps
16.12 Future Directions and Emerging Trends
16.12.1 High-Performance Computing for Large-Scale AI Solvers
16.12.2 Quantum Machine Learning for Differential Equations
16.12.3 Integration of Symbolic AI with Numerical Solvers
16.12.4 Outlook
16.13 Conclusion
References
17. PDE-Constrained Neural Fusion for Infrared and Visible
Images
Gargi Trivedi
17.1 Introduction
17.2 Mathematical Foundations
17.2.1 Anisotropic Diffusion PDE
17.2.2 Neural Network Approximation
17.2.3 APCNF Architecture
17.2.4 Convergence Theorem
17.3 Methodology
17.3.1 Input Processing
17.3.2 Transformer Attention
17.3.3 PDE Constraint
17.3.4 Fusion
17.4 Training Algorithm and Experimental Setup
17.4.1 Dataset
17.4.2 Hyperparameters
17.5 Results and Analysis
17.5.1 Visual Analysis
17.5.2 Quantitative Evaluation
17.6 Conclusion
Acknowledgments
References
18. Numerical Methods in Biological and Medical Modeling—
Diverse Applications
Ravneet Kaur and Mamta Kapoor
18.1 Introduction
18.2 Simulation of Population Dynamics Using Differential Equation Solvers
18.2.1 Classical Population Models: Logistic Growth, Predator–Prey (Lotka–Volterra)
18.2.2 Systems of ODEs for Interacting Species
18.3 Modeling Diffusion–Reaction Processes in Biology and Medicine
18.4 Numerical Methods for Epidemiology
18.5 High-Performance and Data-Driven Methods
18.6 Comparative Analysis of Numerical Approaches
18.7 Conclusion and Future Perspectives
Bibliography
19. Time-Delayed Modeling of Diabetic Neuropathy Using Delay Differential Equations
Dipesh, Ankita Tiwari, Gunal Malhotra and Pankaj Kumar
19.1 Introduction
19.2 Mathematical Model
19.2.1 Boundedness
19.2.2 Positivity of the Solutions
19.2.3 Equilibrium Points
19.2.4 Biological Stability of ADDI∗∗∗∗(),,12 and Hopf Bifurcation
19.3 Numerical Example
19.3.1 Results and Discussion
19.4 Conclusion
19.5 Future Scope
References
20. Computational Modeling and Numerical Simulation of Hepatitis B Dynamics with Nonlinear Incidence
Mustaq Ahmad, Archana Singh Bhadauria and Mahmood Khalid Jasim
20.1 Introduction
20.2 Model Formulation
20.2.1 Model Assumptions
20.3 Parameter Values
20.4 Analytical Analysis
20.4.1 Positivity and Boundedness of the Model
20.4.2 Equilibrium Points
20.4.3 Basic Reproduction Number 0
20.4.4 Stability Analysis
20.5 Numerical Methods
20.5.1 Runge–Kutta Fourth-Order Method
20.5.2 Nonstandard Finite Difference Scheme
20.5.3 Numerical Simulation and Discussion
20.5.4 Comparison of Methods
20.6 Summary
20.7 Conclusion
References
21. Mathematical Analysis of a Plankton–Fish Interaction System in the Presence of Environmental Toxins and Additional Food
Amit Sharma and R.P. Kaur
21.1 Introduction
21.2 Model System
21.3 Dynamical Properties of Model System
21.4 Stability Analysis
21.5 Existence of Various Equilibria
21.6 Stability of the System Around Different Equilibrium States
21.7 Global Stability
21.8 Hopf-Bifurcation Analysis
21.9 Direction of Bifurcating Solutions
21.10 Numerical Simulation
21.11 Role of Available Additional Food (b1 and c1) to Zooplankton and Fish
21.12 Impact of e, b1, and c1 in Termination of Planktonic Blooms
21.13 Discussion and Conclusion
References
Appendix
Index
Back to Top
Preface
Acknowledgements
1. A Comprehensive Guide to Partial Differential Equations: Theory, Numerical Methods, and Applications
Jaya Gupta and Shubham Mishra
1.1 Introduction
1.2 Ordinary vs. Partial
1.3 Origins and Historical Development
1.4 Numerical Approaches Used to Solve Partial Differential Equations
1.5 Classical Basis Function Methods Used for Solving Partial Differential Equations
1.6 Conclusion
References
2. Analytical Solutions of Various Types of Differential Equations via Variants of the Differential Transform Method
Adarsh Bhardwaj and Geeta Arora
2.1 Introduction
2.2 Methodology
2.2.1 Differential Transform Method
2.2.2 Reduced Differential Transform Method
2.2.3 Fractional Differential Transform Method
2.2.4 Modified Fractional Differential Transform Method
2.3 Numerical Examples
2.3.1 Solving Ordinary Differential Equations
2.3.2 Solving Partial Differential Equations
2.3.3 Solving Fractional Differential Equations
2.4 Conclusion
References
3. Numerical Solution of Burgers’ Equation Using Orthogonal Collocation Method
Shelly Arora, Muskan, Abdul-Majeed Ayiebre and Happy Kumar
3.1 Introduction
3.2 Orthogonal Collocation Method
3.2.1 Collocation Points
3.3 Burgers’ Equation
3.4 Methodology
3.5 Results and Discussion
3.6 Conclusions
References
4. Soliton Solutions and Chaotic Wave Dynamics in the Combined KdV–mKdV Equation
Hamood Ur Rehman, Muhammad Shoaib Saleem and Ifrah Iqbal
List of Abbreviations
4.1 Introduction
4.2 Description of MSSEM
4.3 Simplest Equation Method
4.4 Chaotic Analysis
4.5 Multistability and Sensitivity Analysis
4.6 Results and Discussion
4.7 Novelty and Comparative Analysis
4.8 Conclusion
Acknowledgments
References
5. A Comparative Analysis on the Numerical Evaluation of Multiple Integrals Appearing in Mathematical Physics via Different Orthogonal Polynomial Wavelets
Ashish Rayal, Mohammad Izadi and Sabri T.M. Thabet
List of Abbreviations
5.1 Introduction
5.2 Orthogonal Wavelets
5.2.1 Taylor Wavelets
5.2.2 Muntz Wavelets
5.2.3 Bernstein Wavelets
5.3 Function Approximation via Orthogonal Wavelets
5.4 Wavelets Scheme for Numerous Integrals
5.5 Error Estimation
5.6 Method Implementation
5.7 Conclusion
Acknowledgments
References
6. Analytical and Numerical Solutions of Different Types
of Schrödinger Equations
Keyur Modha and Mamta Kapoor
List of Abbreviations
6.1 Introduction
6.2 Analytical Solutions of the Time-Independent Schrödinger Equation
6.2.1 Separation of Variables and Coordinate Systems
6.2.2 Canonical Solvable Potentials
6.2.3 Special Functions and Symmetry Principles
6.2.4 Analytical Insights vs. Numerical Realities
6.2.5 Outlook
6.3 Analytical Solutions of the Time-Dependent Schrödinger Equation
6.3.1 General Solution Strategies
6.3.2 Free Particle and Wave Packets
6.3.3 Propagator (Green’s Function) Approach
6.3.4 Stationary vs. Nonstationary States
6.3.5 Outlook
6.4 Numerical Approaches—Finite Difference Methods
6.4.1 Why Numerical Methods?
6.4.2 The Finite Difference Idea
6.4.3 Time Evolution: Explicit vs. Implicit Methods
6.4.4 Worked Example: Particle in a 1D Infinite Square Well
6.4.5 Physical Takeaways
6.5 Advanced Numerical Methods: Spectral and Finite Element Approaches
6.5.1 Why Go Beyond Finite Differences?
6.5.2 Spectral Methods
6.5.3 Finite Element Method
6.5.4 Worked Example: Quantum Harmonic Oscillator
6.5.5 Physical Takeaways
6.6 Nonlinear Schrödinger Equation
6.6.1 Basic Properties and Conserved Quantities
6.6.2 Bright Soliton: Derivation of the 1-Soliton Solution (Focusing Case)
6.6.3 Dark Soliton (Defocusing Case)
6.6.4 Integrability and Multisoliton Interactions
6.6.5 Numerical Solution Methods
6.6.5.1 SSFM: The Workhorse
6.6.5.2 Runge–Kutta in Time + Finite Difference/Spectral Spatial Discretization
6.6.5.3 Adaptive and Energy-Preserving Schemes
6.6.6 Worked Numerical Example
6.6.7 Applications and Physical Contexts
6.7 Multidimensional Schrödinger Equation
6.7.1 Introduction
6.7.2 General Form of the Multidimensional Schrödinger Equation
6.7.3 Separation of Variables in Higher Dimensions
6.7.4 2D and 3D Quantum Wells and Quantum Dots
6.7.5 PDE Solvers in Higher Dimensions
6.7.6 Visualization and Physical Insights
6.7.7 Applications
6.7.8 Summary
6.8 Applications in Physics and Optics
6.8.1 Introduction
6.8.2 Applications in Physics
6.8.2.1 Atomic and Molecular Physics
6.8.2.2 Condensed Matter Physics
6.8.2.3 Nuclear and Particle Physics
6.8.3 Applications in Optics
6.8.3.1 Lasers and Coherent Light
6.8.3.2 Quantum Optics
6.8.3.3 Photonic Crystals and Waveguides
6.8.3.4 Nonlinear Optics
6.8.4 Interdisciplinary Applications
6.8.5 Summary
6.9 Conclusions and Future Directions
References
7. A Computational Study of Reaction–Convection–Diffusion Processes through Quintic Hermite Collocation Method
Inderpreet Kaur
7.1 Introduction
7.2 Numerical Approach
7.3 Error Analysis
7.4 Stability Analysis
7.4.1 Discretization in t and x
7.5 Numerical Examples
7.5.1 Problem
7.5.2 Problem
7.5.3 Problem
7.5.4 Problem
7.6 Conclusions
7.7 Future Scope
References
8. Numerical Solution of Generalized Burger–Huxley Equation
Using Cubic Hermite Splines
Shelly Arora, Ishika Mittal, Abdul-Majeed Ayiebre and Happy Kumar
8.1 Introduction
8.2 Generalized Burgers–Huxley Equation
8.3 Hermite Interpolation
8.4 Implementation of Hermite Collocation Technique
8.5 Results and Discussion
8.6 Conclusions
References
9. Evaluation of Runge–Kutta Methods for Time Integration of the Burgers’ Equation Utilizing the Compact Finite Difference Scheme
Geeta Arora
9.1 Introduction
9.2 Sixth-Order Compact Finite Difference Scheme for the First-Order Derivative
9.2.1 Taylor-Series Expansions and Coefficient Determination
9.2.2 Interior Compact Scheme
9.2.3 Boundary Treatment for the First-Order Derivative
9.2.4 Matrix Formulation of the First-Order Derivative
9.3 Compact Finite Difference Approximation for the Second-Order Derivative
9.3.1 Taylor-Series Expansion and Determination of Coefficients
9.3.2 Sixth-Order Compact Scheme for Interior Points
9.3.3 Boundary Treatment for the Second-Order Derivative
9.3.4 Matrix Formulation of the Second-Order Derivative
9.4 Numerical Implementation of the Compact Scheme for Burgers’ Equation
9.5 Runge–Kutta Schemes
9.5.1 General Explicit RK Formulation
9.5.2 Second-Order RK Methods
9.5.2.1 Standard RK2 Method
9.5.2.2 Ralston Method
9.5.2.3 Third-Order RK Methods
9.5.2.4 Fourth-Order RK Methods
9.5.2.5 Rk45
9.5.3 Low-Storage RK Methods (Williamson Form)
9.5.4 Strong Stability-Preserving RK Methods
9.6 Numerical Results
9.7 Conclusion
References
10. Analytical Approximate Solutions for 2D Fractional PDEs: A Conformable Fractional Reduced Differential Transform Method
Maher Jneid and Abir Chaouk
10.1 Introduction
10.2 Analysis of CFRDTM
10.3 Applications and Discussion
10.4 Conclusion
References
11. Semianalytical Solution of Nonlinear Fractional Partial Differential Equations Using Accelerated HPTM
Shabnam Jasrotia and Prince Singh
11.1 Introduction
11.1.1 Klein–Gordon
11.1.2 Korteweg-DeVries
11.1.3 Burgers’ Equation
11.2 Preliminaries
11.2.1 Fractional Calculus
11.2.2 Role of the Laplace Transform in the CF Framework
11.3 AHPTM
11.3.1 He’s Polynomials and Accelerated He’s Polynomials
11.4 Condition of Convergence of AHPTM
11.5 Application
11.6 Conclusion
Acknowledgment
References
12. Solution of Fuzzy Riccati Differential Equation Using
Fuzzy Sumudu Decomposition Method
Asha N., Deepak Kumar Sah, Sreenivasulu Ballem, K. Venkateshwarlu, Kottakkaran Sooppy Nisar, Ramalingaiah Kadari and Devika Dabke
List of Abbreviations
12.1 Introduction
12.2 Novelty of the Work
12.3 Preliminaries
12.4 Fuzzy Riccati Differential Equation
12.5 Numerical Results
12.6 Conclusion
Acknowledgments
References
13. Chaotic Behavior of Fractional Delay Differential Equations
S. Priyadharsini, E. Kungumaraj, S. Saravanakumar and T. Sathiyaraj
List of Abbreviations
13.1 Introduction
13.1.1 Background: Hutchinson Equation
13.2 Preliminaries
13.2.1 Preliminary Concepts
13.3 Main Result: Method of Steps
13.4 Limitations
13.5 Computational Method
13.6 Example
13.6.1 Analysis of Hutchinson Equation
13.7 Significance of the Proposed Model
13.8 Conclusion
References
14. Physics-Informed Neural Network for Solving Coupled
Ordinary Differential Equations
Priyanka Chandra and Raja Das
List of Abbreviations
14.1 Introduction
14.2 Differential Equations
14.3 Physics-Informed Neural Networks
14.4 Implementation of PINNs in ODEs
14.4.1 Convergence and Stability of PINNs
14.5 Results and Discussion
14.6 Conclusion
Acknowledgments
References
15. Artificial Neural Network–Based Approximation of Nonlinear ODEs with Comparative Analysis of Activation Functions
Hetvi Patel, Dhvani Patel and Yogeshwari F. Patel
List of Abbreviations
15.1 Introduction
15.1.1 Scope of This Chapter
15.2 Theoretical Background of Multilayer Perceptron Neural Network
15.2.1 Methodology
15.3 Illustration
15.4 Conclusion
References
16. AI-Driven Numerical and Machine Learning Approaches for Solving Differential Equations
S. Padmanabhan, Pallavi G., Kavya K. N. and Vijayalakshmi S.
16.1 Introduction and Motivation
16.1.1 Importance of Differential Equations in Computational Science
16.1.2 Applications in Physics, Engineering, Finance, and Biology
16.1.3 Limitations of Conventional Analytical and Numerical Methods
16.1.4 Emergence of AI and ML in Computational Modeling
16.2 Traditional Numerical Approaches
16.2.1 Euler Method
16.2.2 RK Methods
16.2.3 Spectral and Finite Difference Methods
16.3 Machine Learning Approaches for Differential Equations
16.3.1 Neural Network Solvers
16.3.2 Physics-Informed Neural Networks
16.3.3 Deep Operator Networks
16.4 Hybrid AI–Numerical Techniques
16.4.1 Combining Classical Solvers with Machine Learning
16.4.2 Adaptive Time-Stepping with ML Predictions
16.4.3 Data-Driven Reduced-Order Modeling
16.4.4 Hybrid Workflow
16.5 Data Pipeline and High-Dimensionality Handling
16.5.1 Data Generation, Preprocessing, and Validation
16.5.2 Handling High-Dimensional PDEs
16.6 Applications
16.6.1 Climate and Environmental Modeling
16.6.2 Epidemiology and Biology
16.6.3 Quantum and Engineering Applications
16.6.4 Summary of Applications
16.7 Case Studies and Experimental Results
16.7.1 Experimental Setup
16.7.2 Comparative Performance
16.7.3 Error Analysis
16.7.4 Visualization and Performance Plots
16.7.5 Computation Time
16.8 Comparison of Methods
16.9 Challenges and Research Gaps
16.10 Tools and Implementation Details
16.11 Recent Research (2020–2025) and Literature Gaps
16.12 Future Directions and Emerging Trends
16.12.1 High-Performance Computing for Large-Scale AI Solvers
16.12.2 Quantum Machine Learning for Differential Equations
16.12.3 Integration of Symbolic AI with Numerical Solvers
16.12.4 Outlook
16.13 Conclusion
References
17. PDE-Constrained Neural Fusion for Infrared and Visible
Images
Gargi Trivedi
17.1 Introduction
17.2 Mathematical Foundations
17.2.1 Anisotropic Diffusion PDE
17.2.2 Neural Network Approximation
17.2.3 APCNF Architecture
17.2.4 Convergence Theorem
17.3 Methodology
17.3.1 Input Processing
17.3.2 Transformer Attention
17.3.3 PDE Constraint
17.3.4 Fusion
17.4 Training Algorithm and Experimental Setup
17.4.1 Dataset
17.4.2 Hyperparameters
17.5 Results and Analysis
17.5.1 Visual Analysis
17.5.2 Quantitative Evaluation
17.6 Conclusion
Acknowledgments
References
18. Numerical Methods in Biological and Medical Modeling—
Diverse Applications
Ravneet Kaur and Mamta Kapoor
18.1 Introduction
18.2 Simulation of Population Dynamics Using Differential Equation Solvers
18.2.1 Classical Population Models: Logistic Growth, Predator–Prey (Lotka–Volterra)
18.2.2 Systems of ODEs for Interacting Species
18.3 Modeling Diffusion–Reaction Processes in Biology and Medicine
18.4 Numerical Methods for Epidemiology
18.5 High-Performance and Data-Driven Methods
18.6 Comparative Analysis of Numerical Approaches
18.7 Conclusion and Future Perspectives
Bibliography
19. Time-Delayed Modeling of Diabetic Neuropathy Using Delay Differential Equations
Dipesh, Ankita Tiwari, Gunal Malhotra and Pankaj Kumar
19.1 Introduction
19.2 Mathematical Model
19.2.1 Boundedness
19.2.2 Positivity of the Solutions
19.2.3 Equilibrium Points
19.2.4 Biological Stability of ADDI∗∗∗∗(),,12 and Hopf Bifurcation
19.3 Numerical Example
19.3.1 Results and Discussion
19.4 Conclusion
19.5 Future Scope
References
20. Computational Modeling and Numerical Simulation of Hepatitis B Dynamics with Nonlinear Incidence
Mustaq Ahmad, Archana Singh Bhadauria and Mahmood Khalid Jasim
20.1 Introduction
20.2 Model Formulation
20.2.1 Model Assumptions
20.3 Parameter Values
20.4 Analytical Analysis
20.4.1 Positivity and Boundedness of the Model
20.4.2 Equilibrium Points
20.4.3 Basic Reproduction Number 0
20.4.4 Stability Analysis
20.5 Numerical Methods
20.5.1 Runge–Kutta Fourth-Order Method
20.5.2 Nonstandard Finite Difference Scheme
20.5.3 Numerical Simulation and Discussion
20.5.4 Comparison of Methods
20.6 Summary
20.7 Conclusion
References
21. Mathematical Analysis of a Plankton–Fish Interaction System in the Presence of Environmental Toxins and Additional Food
Amit Sharma and R.P. Kaur
21.1 Introduction
21.2 Model System
21.3 Dynamical Properties of Model System
21.4 Stability Analysis
21.5 Existence of Various Equilibria
21.6 Stability of the System Around Different Equilibrium States
21.7 Global Stability
21.8 Hopf-Bifurcation Analysis
21.9 Direction of Bifurcating Solutions
21.10 Numerical Simulation
21.11 Role of Available Additional Food (b1 and c1) to Zooplankton and Fish
21.12 Impact of e, b1, and c1 in Termination of Planktonic Blooms
21.13 Discussion and Conclusion
References
Appendix
Index
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