Mathematical Methods for Physics: Problems and Solutions

Description

This book presents mathematical tools to solve partial differential equations, typical of physical problems. It explains in a detailed manner the process of solving the problems that typically arise in the context of physics. Although there are a large number of textbooks on this topic, few go so deep into the topic. One of the original and unique features of this book is emphasis on the mathematical formulation of the problems, as well as the analysis of several alternative ways to solve them. Importantly, the book provides a graphical analysis of the results when appropriate. It describes a wide scope of the problems, with detailed solutions and the methods involved, ranging from cases in one to three dimensions, from Cartesian to polar, cylindrical, and spherical coordinates and includes properties and applications of the Fourier transform to solve partial differential equations. Cover Half Title Title Page Copyright Page Table of Contents Preface Chapter 1: Harmonic Oscillator and Green’s Function 1.1: Damped Harmonic Oscillator 1.2: Properties of the Ordinary Linear Differential Equation for a Forced Oscillator 1.3: General Definition of Green’s Functions 1.4: Expansion of Green’s Function in a Series of Orthogonal Eigenfunctions 1.5: Green’s Function of an Oscillator with Friction 1.6: Movement of an Oscillator under the Influence of a Constant Force, Solved by Two Methods 1.7: Oscillator Forced by a Rectangular Hit, Solved with Green’s Functions 1.8: Movement of a Mass after an Instantaneous Exponential Hit 1.9: Shape of a String in Mechanical Equilibrium, Solved by the Green’s Function Method 1.10: Case Study: Transversal Displacement of a Tense String Glued to an Elastic Plane 1.11: Forced Harmonic Oscillator, Solved with Green’s Functions Chapter 2: Problems in One Dimension 2.1: Closed String 2.2: Sturm–Liouville Problem with Boundary Conditions of the Second and Third Kind 2.3: Stationary String in a Gravitational Field 2.4: Static String with Boundary Conditions of the Third Kind at Both Ends 2.5: String with a Point Mass Hanging from One of Its Ends 2.6: String with a Point Mass in Its Center and Second and Third Type Boundary Conditions 2.7: Static Form of a String with a Mass 2.8: Heat Conduction through a Semi-Insulated Bar 2.9: Variation of the Temperature of a Thin Rod as a Function of Time 2.10: Thermal Conduction in a Bar with Insulated Ends 2.11: Variation of the Temperature of a Bar as a Function of Time 2.12: Relaxation of Temperature in a Rod with a Local Heat Source 2.13: Heat Transfer in an Insulated Bar According to Newton’s Law 2.14: Case Study: Heat Transfer in a Semi-Infinite 1D Bar: Periodically Varying Temperature 2.15: Case Study: Vibrations of Two United Bars 2.16: Distribution of Temperature in a Non-Homogeneous Bar 2.17: Case Study: Variation in the Ion Concentration in a Rod with Flux across Its Ends 2.18: Oscillations of a Non-Homogeneous String 2.19: Forced Oscillations of a String 2.20: Case Study: Oscillations of a String Subject to an External Force 2.21: Case Study: Oscillations of the Gas in a Semi-Open Tube 2.22: Variation of the Temperature in a Thin Rod Exchanging Heat through Its Surface 2.23: Distribution of Temperature in a Thin Wire with Losses on Its Surface 2.24: Oscillations of a Finite String with Friction 2.25: Propagation of a Thermal Pulse in a Thin Bar with Insulated Ends 2.26: Forced Oscillations of a Hanging String in a Gravitational Field 2.27: Temperature Equilibrium in a Bar with Heat Sources 2.28: Case Study: String under a Gravitational Field 2.29: String with Oscillations Forced in One of Its Ends 2.30: Oscillations of a String with a Force That Increases Linearly in Time 2.31: Case Study: Lateral Photoeffect 2.32: Oscillations of a String under the Influence of a Gravitational Field 2.33: Dynamic String with Free Ends and a Point Mass at x = x0 2.34: Oscillations in a String Interrupted by a Spring 2.35: Point Like Heat Exchange Chapter 3: Bidimensional Problems 3.1: Forced Oscillations of a Membrane 3.2: Oscillations of a Membrane Fixed at Two Boundaries 3.3: Electrostatic Field inside a Semi-Infinite Region 3.4: Distribution of Electrostatic Potential in a Rectangle 3.5: Distribution of Temperature in a Semi-Insulated and Semi-Infinite Slab 3.6: Oscillations of a Semi-Fixed Membrane 3.7: Stationary Temperature in a Rectangle with Heat Losses through Its Boundaries 3.8: Case Study: Heat Leak from a Rectangle 3.9: Rectangular Hit on a Square Membrane 3.10: Case Study: Distribution of Temperature in a Peltier Element 3.11: Case Study: Charged Filament inside a Prism 3.12: Case Study: Capacity in a Rectangular Tube 3.13: Temperature Distribution inside a Box Heated by Two Transistors Chapter 4: Three-Dimensional Problems 4.1: Stationary Temperature Distribution inside a Prism with a Thin Heater in One of Its Faces 4.2: Case Study: Forced Gas Oscillations in a Prism: Case of a Homogeneous Force 4.3: Case Study: Forced Gas Oscillations in a Prism: Case of an Inhomogeneous Force 4.4: Case Study: Optimization of the Size of an Atomic Bomb: Diffusion Equation in Cartesian Coordinates 4.5: Oscillations of a Gas in a Cube 4.6: Stationary Temperature Distribution inside a Prism 4.7: Variation of the Temperature inside a Cube: From Poisson to a Diffusion Problem 4.8: Variation of the Pressure inside a Rectangular Prism due to the Periodic Action of a Piston 4.9: Case Study: Variation of Temperature inside a Prism: From Laplace to Poisson Problems 4.10: Case Study: Distribution of Temperature inside a Periodically Heated Prism 4.11: Heating Rectangular Resistor with Different Boundary Conditions 4.12: Heating of a Rectangular Resistor with the Same Boundary Conditions 4.13: Case Study: Distribution of Photocarriers Induced by a Laser 4.14: Heater inside a Prism 4.15: Cube with a Heater Chapter 5: Problems in Polar Coordinates 5.1: Separation of Variables in a Circular Membrane 5.2: Electric Potential in a Circular Sector: Case 1 5.3: Electric Potential in a Circular Sector: Case 2 5.4: Stationary Distribution of the Concentration of Particles in a Sector of an Infinite Cylinder 5.5: Instantaneous Hit on a Membrane with Circular Sector Form 5.6: Linear Heating of a Disk 5.7: Case Study: Laplace’s Problem in a Sector with Non-Homogeneous Boundary Conditions 5.8: Case Study: Temperature Distribution in a Disk with Heaters 5.9: Diffusion in an Infinite Cylinder with Heat Sources 5.10: Variation of the Temperature in a Quarter of a Disk 5.11: Oscillations of a Quarter of a Membrane 5.12: Case Study: Variation of the Temperature in a Cylinder with a Thin Heater 5.13: Forced Oscillations in a Circular Membrane 5.14: Case Study: Stationary Distribution of Temperature Inside the Sector of a Disk 5.15: Variation of the Temperature in Two Semi-Cylinders 5.16: Stationary Temperature inside an Infinite Cylindrical Tube 5.17: Case Study: Time Variation of the Density of Viruses Emitted by a Thin Filament Placed in a Sector of a Disk Chapter 6: Problems in Cylindrical Coordinates 6.1: General Solution of the Heat Equation in a Finite Cylinder with a Hole 6.2: Case Study: Heating of a Cylinder 6.3: Case Study: Stationary Distribution of Temperature inside a Semicylinder 6.4: Case Study: Laplace’s Equation in a Cylinder with No Homogeneous Contours 6.5: Heating of 1/16 of a Cylinder 6.6: Distribution of Temperature inside a Finite Semi-Cylinder with a Centered Hole 6.7: Distribution of Temperature inside a Hollow Cylinder with a Heater 6.8: Case Study: Temperature in a Cylinder with Bases Thermally Insulated 6.9: Case Study: Cylinder with a Heater of xy Symmetry 6.10: Case Study: Semi-Cylinder with a Thin Heater 6.11: Half Cylinder with Two Inward Fluxes 6.12: Heat Flux through Half a Cylinder Chapter 7: Problems in Spherical Coordinates 7.1: Electric Potential between Two Spheric Shells 7.2: Distribution of Temperature inside a Sphere 7.3: Laplace Problem in a Sphere with a Difference of Potential 7.4: Electric Potential inside a Spherical Sector 7.5: Electric Potential of a Metallic Sphere inside a Homogeneous Electric Field 7.6: Case Study: Variation of Temperature in a Sphere Quadrant with Non-Homogeneous Boundaries 7.7: Case Study: Stationary Distribution of Temperature in a Sphere with Heat Sources 7.8: Gas in Two Semi-Spheres 7.9: Case Study: Forced Oscillations in a Semi-Sphere 7.10: Heat Transfer in an Eight of a Sphere 7.11: Case Study: Heated Quarter of a Sphere 7.12: Case Study: Two Concentric Semi-Spheres 7.13: Case Study: Variation of Temperature in a Semisphere 7.14: Case Study: Oscillating Sphere FilledWith Gas 7.15: Case Study: Stationary Distribution of Temperature in a Planet Close to a Star 7.16: Pre-Heated Quarter of a Sphere Chapter 8: Fourier Transform and Its Applications 8.1: Reciprocity of the Fourier Transform 8.2: Fourier Transform of a Bidirectional Pulse 8.3: Loss Spectrum of a Relaxator 8.4: Inverse Fourier Transform of a Function 8.5: Fourier Transform of the Product of Two Functions 8.6: Example of the Calculation of the Fourier Transform of a Product of Two Functions from the Convolution Operation 8.7: Parseval Theorem Formulated for Two Different Functions 8.8: Wiener–Khinchin (WK) Theorem 8.9: Fourier Transform of an Oscillation Modulated by a Gaussian Pulse 8.10: Autoconvolution of a Rectangular Pulse 8.11: Fourier Transform of a Bipolar Triangular Pulse 8.12: Fourier Transform of a Rectangular Pulse 8.13: Fourier Transform of the Convolution of a Triangular Pulse with Itself 8.14: Fourier Transform of a Shifted Rectangular Pulse with a Sine Modulation 8.15: Case Study: Solution of a PDE Using the Fourier Transform: Case 1 8.16: Case Study: Solution of a PDE Using the Fourier Transform: Case 2 8.17: Case Study: Solution of a PDE Using the Fourier Transform: Case 3 8.18: Case Study: Solution of a PDE Using the Fourier Transform: Case 4 8.19: Case Study: Solution of a PDE Using the Fourier Transform: Case 5 8.20: Case Study: Solution of a PDE Using the Fourier Transform: Case 6 8.21: Case Study: Solution of the Diffusion Equation in an Infinite String with Convection Using the Fourier Transform 8.22: Application of the Fourier Transform to Find the Displacements of a String Attached to an Elastic Fabric 8.23: Case Study: Oscillations in an Infinite String with Friction 8.24: Case Study: Fourier Transform to Find the Distribution of Temperature in a Semi-Infinite Bar 8.25: Case Study: Application of the Fourier Transform to Find the Distribution of Temperature in a Semi-Plane 8.26: Fourier Transform of the General Solution of Laplace’s Problem in a Disk Appendix Bibliography Index