Melkonian S. Mathematical Methods and Boundary Value Problems

Infinity Books, 2023. — 494 p.Preface.
The Laplace Transform.
Introduction.
Further Properties and Initial-Value Problems.
Convolutions and Generalized Functions.
Table of Laplace Transforms.
Series Solutions of Ordinary Differential Eqs.
Basic Concepts.
Solutions About Ordinary Points.
Solutions About Regular Singular Points.
Cauchy-Euler Equations.
The General Equation y′′ + p(x)y′ + q(x)y = 0.
Bessel’s Equation.
Fourier Series.
Periodic Functions.
Functions Defined on Finite Intervals.
Summary of Fourier Series.
Partial Differential Equations.
The Heat Equation.
The Bar with Zero Boundary Conditions.
The Bar with Nonzero Boundary Conditions.
The Bar with Insulated Ends.
The Wave Equation.
Laplace’s Equation.
Solutions Within Rectangular Regions.
Polynomial Solutions.
Regions with Circular Boundaries.
Solutions Inside a Circle.
Solutions Outside a Circle.
Solutions Within an Annulus.
Summary of Solutions.
Sturm-Liouville Problems.
Regular and Periodic Problems.
General Theory.
Orthonormal Families.
Singular Problems.
Bessel’s Equation.
The Vibrating Membrane.
The Fourier Transform.
Fundamental Properties.
Applications.
Partial Differential Equations.
The Heat Equation on (−∞, ∞).
Laplace’s Equation in the Upper Half-Plane.
The Wave Equation on (−∞, ∞).
Summary of Solutions.
Table of Fourier Transforms.
A Notes.
B Solutions.
Bibliography.
Index.