Preface xix
Acknowledgements xxiii
1 A Comprehensive Guide to Partial Differential Equations: Theory, Numerical Methods, and Applications 1
Jaya Gupta and Shubham Mishra
1.1 Introduction 1
1.2 Ordinary vs. Partial 2
1.3 Origins and Historical Development 4
1.4 Numerical Approaches Used to Solve Partial Differential Equations 5
1.5 Classical Basis Function Methods Used for Solving Partial Differential Equations 6
1.6 Conclusion 11
2 Analytical Solutions of Various Types of Differential Equations via Variants of the Differential Transform Method 15
Adarsh Bhardwaj and Geeta Arora
2.1 Introduction 16
2.2 Methodology 18
2.3 Numerical Examples 25
2.4 Conclusion 46
3 Numerical Solution of Burgers' Equation Using Orthogonal Collocation Method 51
Shelly Arora, Muskan, Abdul-Majeed Ayiebre and Happy Kumar
3.1 Introduction 51
3.2 Orthogonal Collocation Method 52
3.3 Burgers' Equation 54
3.4 Methodology 54
3.5 Results and Discussion 56
3.6 Conclusions 63
4 Soliton Solutions and Chaotic Wave Dynamics in the Combined KdV–mKdV Equation 67
Hamood Ur Rehman, Muhammad Shoaib Saleem and Ifrah Iqbal
List of Abbreviations 68
4.1 Introduction 68
4.2 Description of MSSEM 70
4.3 Simplest Equation Method 79
4.4 Chaotic Analysis 82
4.5 Multistability and Sensitivity Analysis 85
4.6 Results and Discussion 87
4.7 Novelty and Comparative Analysis 89
4.8 Conclusion 90
5 A Comparative Analysis on the Numerical Evaluation of Multiple Integrals Appearing in Mathematical Physics via Different Orthogonal Polynomial Wavelets 95
Ashish Rayal, Mohammad Izadi and Sabri T.M. Thabet
5.1 Introduction 96
5.2 Orthogonal Wavelets 98
5.3 Function Approximation via Orthogonal Wavelets 100
5.4 Wavelets Scheme for Numerous Integrals 101
5.5 Error Estimation 104
5.6 Method Implementation 104
5.7 Conclusion 114
6 Analytical and Numerical Solutions of Different Types of Schrödinger Equations 119
Keyur Modha and Mamta Kapoor
6.1 Introduction 120
6.2 Analytical Solutions of the Time-Independent Schrödinger Equation 123
6.3 Analytical Solutions of the Time-Dependent Schrödinger Equation 127
6.4 Numerical Approaches—Finite Difference Methods 131
6.5 Advanced Numerical Methods: Spectral and Finite Element Approaches 134
6.6 Nonlinear Schrödinger Equation 136
6.7 Multidimensional Schrödinger Equation 142
6.8 Applications in Physics and Optics 147
6.9 Conclusions and Future Directions 153
7 A Computational Study of Reaction–Convection–Diffusion Processes through Quintic Hermite Collocation Method 157
Inderpreet Kaur
7.1 Introduction 157
7.2 Numerical Approach 159
7.3 Error Analysis 162
7.4 Stability Analysis 162
7.5 Numerical Examples 165
7.6 Conclusions 176
7.7 Future Scope 177
8 Numerical Solution of Generalized Burger–Huxley Equation Using Cubic Hermite Splines 181
Shelly Arora, Ishika Mittal, Abdul-Majeed Ayiebre and Happy Kumar
8.1 Introduction 181
8.2 Generalized Burgers–Huxley Equation 182
8.3 Hermite Interpolation 183
8.4 Implementation of Hermite Collocation Technique 185
8.5 Results and Discussion 188
8.6 Conclusions 195
9 Evaluation of Runge–Kutta Methods for Time Integration of the Burgers' Equation Utilizing the Compact Finite Difference Scheme 197
Geeta Arora
9.1 Introduction 198
9.2 Sixth-Order Compact Finite Difference Scheme for the First-Order Derivative 199
9.3 Compact Finite Difference Approximation for the Second-Order Derivative 203
9.4 Numerical Implementation of the Compact Scheme for Burgers' Equation 205
9.5 Runge–Kutta Schemes 206
9.6 Numerical Results 213
9.7 Conclusion 221
10 Analytical Approximate Solutions for 2D Fractional PDEs: A Conformable Fractional Reduced Differential Transform Method 225
Maher Jneid and Abir Chaouk
10.1 Introduction 225
10.2 Analysis of CFRDTM 227
10.3 Applications and Discussion 229
10.4 Conclusion 249
11 Semianalytical Solution of Nonlinear Fractional Partial Differential Equations Using Accelerated HPTM 253
Shabnam Jasrotia and Prince Singh
11.1 Introduction 253
11.2 Preliminaries 257
11.3 AHPTM 259
11.4 Condition of Convergence of AHPTM 262
11.5 Application 262
11.6 Conclusion 283
12 Solution of Fuzzy Riccati Differential Equation Using Fuzzy Sumudu Decomposition Method 287
Asha N., Deepak Kumar Sah, Sreenivasulu Ballem, K. Venkateshwarlu, Kottakkaran Sooppy Nisar, Ramalingaiah Kadari and Devika Dabke
12.1 Introduction 288
12.2 Novelty of the Work 290
12.3 Preliminaries 290
12.4 Fuzzy Riccati Differential Equation 292
12.5 Numerical Results 297
12.6 Conclusion 309
13 Chaotic Behavior of Fractional Delay Differential Equations 311
S. Priyadharsini, E. Kungumaraj, S. Saravanakumar and T. Sathiyaraj
13.1 Introduction 312
13.2 Preliminaries 317
13.3 Main Result: Method of Steps 318
13.4 Limitations 323
13.5 Computational Method 324
13.6 Example 327
13.7 Significance of the Proposed Model 330
13.8 Conclusion 332
14 Physics-Informed Neural Network for Solving Coupled Ordinary Differential Equations 335
Priyanka Chandra and Raja Das
14.1 Introduction 336
14.2 Differential Equations 338
14.3 Physics-Informed Neural Networks 340
14.4 Implementation of PINNs in ODEs 341
14.5 Results and Discussion 346
14.6 Conclusion 355
15 Artificial Neural Network–Based Approximation of Nonlinear ODEs with Comparative Analysis of Activation Functions 361
Hetvi Patel, Dhvani Patel and Yogeshwari F. Patel
15.1 Introduction 362
15.2 Theoretical Background of Multilayer Perceptron Neural Network 365
15.3 Illustration 372
15.4 Conclusion 396
16 AI-Driven Numerical and Machine Learning Approaches for Solving Differential Equations 399
S. Padmanabhan, Pallavi G., Kavya K. N. and Vijayalakshmi S.
16.1 Introduction and Motivation 400
16.2 Traditional Numerical Approaches 405
16.3 Machine Learning Approaches for Differential Equations 408
16.4 Hybrid AI–Numerical Techniques 413
16.5 Data Pipeline and High-Dimensionality Handling 416
16.6 Applications 420
16.7 Case Studies and Experimental Results 422
16.8 Comparison of Methods 426
16.9 Challenges and Research Gaps 430
16.10 Tools and Implementation Details 430
16.11 Recent Research (2020–2025) and Literature Gaps 431
16.12 Future Directions and Emerging Trends 432
16.13 Conclusion 434
17 PDE-Constrained Neural Fusion for Infrared and Visible Images 439
Gargi Trivedi
17.1 Introduction 439
17.2 Mathematical Foundations 440
17.3 Methodology 444
17.4 Training Algorithm and Experimental Setup 447
17.5 Results and Analysis 450
17.6 Conclusion 454
18 Numerical Methods in Biological and Medical Modeling—Diverse Applications 457
Ravneet Kaur and Mamta Kapoor
18.1 Introduction 458
18.2 Simulation of Population Dynamics Using Differential Equation Solvers 461
18.3 Modeling Diffusion–Reaction Processes in Biology and Medicine 463
18.4 Numerical Methods for Epidemiology 466
18.5 High-Performance and Data-Driven Methods 472
18.6 Comparative Analysis of Numerical Approaches 473
18.7 Conclusion and Future Perspectives 476
19 Time-Delayed Modeling of Diabetic Neuropathy Using Delay Differential Equations 479
Dipesh, Ankita Tiwari, Gunal Malhotra and Pankaj Kumar
19.1 Introduction 480
19.2 Mathematical Model 482
19.3 Numerical Example 490
19.4 Conclusion 491
19.5 Future Scope 492
20 Computational Modeling and Numerical Simulation of Hepatitis B Dynamics with Nonlinear Incidence 495
Mustaq Ahmad, Archana Singh Bhadauria and Mahmood Khalid Jasim
20.1 Introduction 496
20.2 Model Formulation 497
20.3 Parameter Values 498
20.4 Analytical Analysis 499
20.5 Numerical Methods 506
20.6 Summary 517
20.7 Conclusion 517
21 Mathematical Analysis of a Plankton–Fish Interaction System in the Presence of Environmental Toxins and Additional Food 521
Amit Sharma and R.P. Kaur
21.1 Introduction 522
21.2 Model System 523
21.3 Dynamical Properties of Model System 525
21.4 Stability Analysis 526
21.5 Existence of Various Equilibria 527
21.6 Stability of the System Around Different Equilibrium States 527
21.7 Global Stability 531
21.8 Hopf-Bifurcation Analysis 533
21.9 Direction of Bifurcating Solutions 534
21.10 Numerical Simulation 535
21.11 Role of Available Additional Food (b1 and c1) to Zooplankton and Fish 538
21.12 Impact of e, b1, and c1 in Termination of Planktonic Blooms 540
21.13 Discussion and Conclusion 541
References 542
Appendix 546
Index 549
