Predict
Read the equation before calculating. Identify likely behavior, constraints and exceptional cases.
Integral
Equations
An Interactive Course in Theory,
Methods, and Computation
A first course in integral equations, built around choosing a method, solving the problem, and checking what the answer actually establishes.
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Download the free worked-solutions PDF
Explore the full contents
Choose a method. Check the answer.
A First Course in
Integral Equations
An answer is useful when you can explain why it works.
A small residual can hide a large error. A substitution check can confirm a solution without proving it’s unique. A convergence condition can be sufficient without being necessary. This course keeps those distinctions visible while you learn the methods.
You move from Fredholm and Volterra equations to spectral theory, numerical methods, regularization and mathematical models of memory and potential. Hand calculation, theorems and computation stay connected throughout.
- 24Chapters in 6 parts
- 321Exercises with hints or answers
- 60Figures and diagrams
- 6Recurring benchmark problems
Look Inside
Read a worked example, inspect the method-selection map, and see how the course is organized. Select a page to enlarge it.
Predict. Solve. Check. Vary.
The larger examples use the same four-step pattern. Each step asks you to do something with the mathematics.
Solve
Choose a method, state its hypotheses and carry the calculation through.
Check
Substitute the result, examine uniqueness and test numerical accuracy.
Vary
Change a parameter or assumption. Find out which conclusions survive.
The same 6 benchmark problems return across the course, so you can compare several methods on an equation you already understand.
Who It’s For
Students and independent readers who want to connect mathematical theory with a calculation they can defend.
I recommend working through the early verification examples before jumping to a numerical method. They establish the checks you’ll need later, especially when the data are noisy.
Start With
- Calculus, including differentiation and integration.
- Basic ordinary differential equations.
- Linear algebra: matrices, linear systems and eigenvalues.
Learn Along the Way
Norms, integral operators, compactness and the functional-analysis ideas used in the course are introduced inside the book. A prerequisite refresher and diagnostic answers help you fill gaps.
This is a first course in integral equations, not a replacement for introductory calculus or a complete treatment of singular and boundary integral theory.
Table of Contents
Follow the course from reading an equation to defending a solution from noisy data. Open a part to see its chapters.
Part 1Learn to Read the Equation
- 1
Why Integral Equations?
Accumulation, interactions, memory and the first worked model.
- 2
Types of Equations and Kernels
Fredholm and Volterra forms; equation kinds, kernel structure and eigenvalues.
- 3
The Mathematical Toolkit
Calculus, norms, orthogonality, linear algebra and Laplace transforms.
- 4
Verification and Method Selection
Substitution checks, uniqueness, residuals and a decision map for choosing methods.
Part 2Solve the Classical Problems
- 5
Separable Kernels and Finite-Dimensional Reduction
Reduce a separable kernel to a finite system and examine exceptional parameters.
- 6
Successive Approximations and the Neumann Series
Iterated kernels, convergence conditions, resolvents and computable error bounds.
- 7
Volterra Equations and Resolvent Kernels
Triangular domains, factorial estimates and the construction of Volterra resolvents.
- 8
Convolution and Transform Methods
Convolution kernels, Laplace transforms and inversion.
- 9
Initial-Value Problems as Integral Equations
Convert differential equations to integral form while preserving initial conditions.
- 10
Boundary-Value Problems and Green's Functions
Construct Green's functions and connect boundary conditions to integral kernels.
Part 3Understand Existence, Uniqueness and Resonance
- 11
Integral Operators and Function Spaces
Function spaces, bounded operators and compactness with concrete examples.
- 12
The Fredholm Alternative
Solvability, compatibility and the role of a nontrivial null space.
- 13
Symmetric Kernels and Spectral Methods
Orthogonal eigenfunctions, eigenfunction expansions and symmetric kernels.
- 14
Resolvents and Fredholm Determinants
Resolvent structure, Fredholm determinants and resonance.
Part 4Extend the Methods Carefully
- 15
Abel Equations and Weakly Singular Kernels
Abel inversion, fractional integration and integrable singularities.
- 16
Nonlinear Integral Equations
Hammerstein and Urysohn forms, contraction arguments and Picard iteration.
- 17
Cauchy Principal Values and Singular Equations
Principal values, endpoint behavior and the limits of ordinary quadrature.
Part 5Compute with Error under Control
- 18
Quadrature and the Nyström Method
Quadrature rules, linear systems and the Nyström interpolation formula.
- 19
Numerical Volterra Methods
Time marching, stability, product integration and the cost of memory.
- 20
Collocation and Galerkin Methods
Trial spaces, projection methods, basis choice and residual checks.
- 21
First-Kind Equations and Regularization
Ill-posedness, truncated SVD, Tikhonov regularization and noisy data.
Part 6Build and Judge Complete Models
- 22
Memory and Hereditary Models
Relaxation, creep and integro-differential models with memory.
- 23
Boundary Integral Formulations
Layer potentials, jump relations and potential problems on a circle and ellipse.
- 24
From Data to a Defensible Solution
Choose a model and method, quantify uncertainty and defend a reconstruction.
Appendices and Reference Material
The course also includes the tools you need to revisit a prerequisite, choose a method or check your work.
- Prerequisite Refresher and Diagnostic Answers
- Integral and Transform Identities
- Method-Selection Guide and Kernel Index
- Computation, Reproducibility and the Companion Labs
- Hints and Answers; Selected Solutions
- Onward Reading, Bibliography and Index
Free Worked Solutions
A complete solution for every exercise.
Use this companion after you’ve tried the problems in the main book. It follows the same chapter and exercise numbering, with the method, intermediate steps and checks worked through.
Complete Worked Solutions
- All 321 exercises solved in full.
- 126 pages in a 0.9 MB PDF.
- Free access. No signup or purchase required.
Before You Begin
A few practical details about the course and the editions.
Are solutions included in the textbook?
All 321 exercises have a hint or answer at the back of the textbook. It also includes 66 selected worked solutions. The separate Complete Worked Solutions PDF is free to download and covers every exercise. You don’t need an Amazon purchase or an account to get it.
Do I need to write code?
No. The computational work is optional, and the book prints the figures and tables needed to follow the discussion. You can work through the core course with pencil and paper.
What does “interactive” mean?
You’ll predict results, check answers, vary assumptions and work through self-checks and exercises. The book also describes optional companion labs. It isn’t an online course or a subscription.
Can I buy the PDF and EPUB here?
The main book’s PDF and EPUB aren’t sold here at present. Choose the Kindle edition or paperback on Amazon. The separate worked-solutions companion is available here as a free PDF.
Get Integral Equations
Choose the edition you’ll use.
Keep the Kindle edition in your reading library, or work through the paperback alongside your notes.
Current prices and availability are shown on Amazon. Use the Kindle link for your regional Amazon store. The paperback link opens Amazon.com.