- Preface
-
1 Origins and History of Partial Differential Equations
- 1.1 Functions of Several Variables and the Notion of a PDE
- 1.2 Three Equations, Three Behaviours
- 1.3 Cauchy, Riemann, and the Classification
-
2 General Second-Order Equations and Their Classification
- 2.1 The General Second-Order Linear Equation
- 2.2 Hypersurfaces, Hyperplanes, and the Cauchy Problem
- 2.3 The Characteristic Condition and Its Geometric Meaning
- 2.4 The Discriminant and the Classification
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3 Canonical Forms and How to Reach Them
- 3.1 Why Canonical Forms Are the Goal
- 3.2 Reduction in the Hyperbolic, Parabolic, and Elliptic Cases
- 3.3 Worked Examples of Reduction to Canonical Form
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4 Hyperbolic Equations and the Method of Characteristics
- 4.1 Two Characteristic Families and the Cauchy Problem
- 4.2 Worked Examples of Solving Hyperbolic Equations
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5 Parabolic Equations and the Structure of Their Solutions
- 5.1 One Characteristic Family, One Free Coordinate
- 5.2 Worked Examples of Solving Parabolic Equations
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6 Elliptic Equations and the Structure of Their Solutions
- 6.1 Complex Characteristics and the Laplacian Canonical Form
- 6.2 Worked Examples of Solving Elliptic Equations
- 7 Closing Remarks on the Method of Characteristics
An Introduction to Second-Order Partial Differential Equations and the Method of Characteristics
A Foundation Course in Classical PDE Theory
This book is 100% completeLast updated on 2026-08-16
How second-order PDEs are classified via characteristics into hyperbolic, parabolic, and elliptic types, reduced to canonical forms, and solved explicitly.
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An educational module focused exclusively on finding solutions to linear second-order partial differential equations using the method of characteristics.
The course considers the three classes into which linear second-order PDEs are classified—hyperbolic, parabolic, and elliptic—and shows how the method of characteristics transforms each equation into a canonical form, making its underlying solution theory more transparent.
Worked examples are presented throughout, leading to either general or particular solutions depending on the prescribed initial data, and drawing on integration techniques and ordinary differential equation methods where appropriate to solve the resulting canonical forms.
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Abdurrahman Tukur
Abdurrahman Tukur is a master's student in applied mathematics specializing in disease epidemiology modeling; a zbMATH Open reviewer writing post-publication reviews to help visitors assess a paper's relevance to their research needs; a writer committed to science communication, primarily through his course books tied to mathematical biology and relevant mathematical tools; and an essayist, sharing his personal writing through his website, own2share.
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