Boundary Value Problems for Linear Partial Differential Equations

Boundary value problems play a significant role in modeling systems characterized by established conditions at their boundaries. On the other hand, initial value problems hold paramount importance in comprehending dynamic processes and foreseeing future behaviors. The fusion of these two types of problems yields profound insights into the intricacies of the conduct exhibited by many physical and mathematical systems regulated by linear partial differential equations.

Boundary Value Problems for Linear Partial Differential Equations provides students with the opportunity to understand and exercise the benefits of this fusion, equipping them with realistic, practical tools to study solvable linear models of electromagnetism, fluid dynamics, geophysics, optics, thermodynamics and specifically, quantum mechanics. Emphasis is devoted to motivating the use of these methods by means of concrete examples taken from physical models.

Features

• No prerequisites apart from knowledge of differential and integral calculus and ordinary differential equations.
• Provides students with practical tools and applications
• Contains numerous examples and exercises to help readers understand the concepts discussed in the book.

1. Introduction. 1.1. Partial Differential Equations. 1.2. Boundary and Initial Conditions. 1.3. Local Solvability. 1.4. Characteristics. 1.5. General Solutions. 1.6. Remarkable Lives and Achievements. 2. Linear PDEs. 2.1. Linear Differential Operators. 2.2. 1D Eigenvalue Problems. 2.3. Linear BVPs in Physics. 2.4. Exercises. 3. Separation of Variables Method. 3.1. Separable Homogeneous Linear PDEs. 3.2. Separable Homogeneous BVP. 3.3. Equations of Mathematical Physics. 3.4. Helmholtz Equation. 3.5. Remarkable Lives and Achievements. 3.6. Exercises. 4. Symmetric Differential Operators. 4.1. Hilbert Spaces. 4.2. Orthogonal Sets of Functions. 4.3. Green Functions. 4.4. Symmetric Differential Operators. 4.5. Sturm–Liouville Differential Operators. 4.6. Remarkable Lives and Achievements. 4.7. Exercises. 5. Fourier Analysis. 5.1. Fourier Trigonometric Bases. 5.2. Fourier Series. 5.3. Convergence of Fourier Series. 5.4. Fourier Transform. 5.5. Remarkable Lives and Achievements. 5.6. Exercises. 6. Eigenfunction Expansion Method. 6.1. Preliminary Discussion: Restricted Inhomogeneities. 6.2. Application to Evolution Equations. 6.3. General Discussion: Full Inhomogeneties. 6.4. Unbounded Domains and Fourier Transform. 7. Special Functions. 7.1. Frobenius Series. 7.2. Ordinary Points. 7.3. Regular Singular Points. 7.4. Bessel Equation. 7.5. Euler Gamma Function. 7.6. Remarkable Lives and Achievements. 7.7. Exercises. 8. Cylindrical and Spherical BVPs. 8.1. Cylindrical BVPs. 8.2. Spherical BVPs. 8.3. Beyond Cylindrical and Spherical. 8.4. Exercises.

Born in Madrid, Spain, in 1964, Manuel Mañas obtained his PhD in Physics in 1987 from the Complutense University of Madrid (UCM). Currently holding the position of full professor in Theoretical Physics at UCM, he has also served as a faculty member at the Instituto de Ciencias Matemáticas. Mañas has demonstrated his commitment to the academic sphere by being an active member of the Government Board of the Royal Spanish Mathematical Society.
He has served as a visiting professor at several prestigious international institutions, including Johannes Kepler Linz University (Austria), University of Aveiro (Portugal), University of California at Berkeley (USA), China University of Mining and Technology (Beijing, China), Université Catholique de Louvain (Belgium), SISSA (Trieste, Italy), La Sapienza (Italy), and Leeds University (UK).
Mañas’ dedication to teaching is evident in his comprehensive curriculum. His instructional portfolio spans a diverse array of topics in mathematical physics. He specializes in teaching courses related to partial differential equations, with a particular focus on their applications in physics, encompassing boundary value problems. Moreover, his teaching extends to the fundamental principles of complex analysis in one variable and their practical applications. Furthermore, Mañas has shared his expertise through postgraduate courses, imparting knowledge on a wide range of advanced topics. These include functional analysis, Lie group theory, integrable systems, and orthogonal polynomials.
With 127 scientific publications to his name, Manuel Mañas’ research primarily focuses on mathematical physics and applied mathematics. His expertise spans various areas, with notable contributions in the theory of integrable systems, the field of orthogonal polynomials, and the theory of special functions. Of late, he achieved a significant breakthrough by establishing a spectral theorem similar to Favard’s for bounded banded operators,