Partial Differential Equations: Foundations and Applications offers a clear, precise, and conceptually rich introduction to one of the most fundamental areas of mathematics. Far beyond abstract formalism, partial differential equations form the language through which the laws of nature are expressed, governing phenomena in physics, engineering, and the life sciences - from heat diffusion and wave propagation to fluid motion, electromagnetism, and quantum systems.

This volume is designed for undergraduate and postgraduate students in mathematics, physics, and engineering, as well as for motivated self-learners and researchers seeking a unified and reliable reference. It balances mathematical depth with conceptual clarity, integrating theory and application so that abstract methods emerge from physical motivations.

This book includes:

• Fundamental definitions, classification, and formation of partial differential equations
• First-order partial differential equations: method of characteristics, Lagrange's method, Charpit's method, and applications such as transport equations and Burgers' equation
• Second-order partial differential equations: classification into hyperbolic, parabolic, and elliptic types, canonical forms, and associated physical models
• Separation of variables and Sturm-Liouville theory, with orthogonal functions and eigenfunction expansions
• Fourier series and Fourier transforms, convergence theorems, Parseval's identity, and applications to the heat and wave equations
• The heat, wave, and Laplace equations in one or more dimensions, steady-state and time-dependent solutions, and coordinate-based techniques
• Laplace transform methods for problems on semi-infinite domains, impulsive sources, and delta function initial conditions

Each chapter is structured to develop both analytical techniques and physical insight, supported by solved examples and graded exercises with answers. Special attention is given to connecting mathematical derivations with their physical interpretations, ensuring the reader gains not only procedural skill but also a comprehensive insight of the underlying principles of the subject.