The book Partial Differential Equations through Examples and Exercises has evolved from the lectures and exercises that the authors have given for more than fifteen years, mostly for mathematics, computer science, physics and chemistry students.

1 Introduction.- 1.1 Basic Notions.- 1.2 The Cauchy-Kowalevskaya Theorem.- 1.3 Equations of Mathematical Physics.- 2 First Order PDEs.- 2.1 Quasi-linear PDEs.- 2.2 Pfaff’s Equations.- 2.3 Nonlinear First Order PDEs.- 3 Classification of the Second Order PDEs.- 3.1 Two Independent Variables.- 3.2 n Independent Variables.- 3.3 Wave, Potential and Heat Equation.- 4 Hyperbolic Equations.- 4.1 Cauchy Problem for the One-dimensional Wave Equation.- 4.2 Cauchy Problem for the n-dimensional Wave Equation.- 4.3 The Fourier Method of Separation Variables.- 4.4 The Sturm—Liouville Problem.- 4.5 Miscellaneous Problems.- 4.6 The Vibrating String.- 5 Elliptic Equations.- 5.1 Dirichlet Problem.- 5.2 The Maximum Principle.- 5.3 The Green Function.- 5.4 The Harmonic Functions.- 5.5 Gravitational Potential.- 6 Parabolic Equations.- 6.1 Cauchy Problem.- 6.2 Mixed Type Problem.- 6.3 Heat conduction.- 7 Numerical Methods.- 7.0.1 Preliminaries.- 7.0.2 Examples and Exercises.- 8 Lebesgue’s Integral, Fourier Transform.- 8.1 Lebesgue’s Integral and the L2(Q) Space.- 8.2 Delta Nets.- 8.3 The Surface Integrals.- 8.4 The Fourier Transform.- 9 Generalized Derivative and Sobolev Spaces.- 9.1 Generalized Derivative.- 9.2 Sobolev Spaces.- 10 Some Elements from Functional Analysis.- 10.1 Hilbert Space.- 10.2 The Fredholm Alternatives.- 10.3 Normed Vector Spaces.- 11 Functional Analysis Methods in PDEs.- 11.1 Generalized Dirichlet Problem.- 11.2 The Generalized Mixed Problems.- 11.3 Numerical Solutions.- 11.4 Miscellaneous.- 12 Distributions in the theory of PDEs.- 12.1 Basic Properties.- 12.2 Fundamental Solutions.

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