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Differential equations, Partial
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Video
This video playlist covers the topic of: 1. PDE 1  Introduction 2.PDE 2  Three fundamental examples 3.PDE 3  Transport equation: derivation 4.PDE 4  Transport equation: general solution 5. PDE 5  Method of characteristics 6. PDE 6  Transport with decay and nonlinear transport 7.PDE 7  Wave equation: intuition 8.PDE 8  Wave equation: derivation 9.PDE 9  Wave equation: general solution 10.PDE 10  Wave equation: d'Alembert's formula 11.PDE 11  Wave equation: d'Alembert examples 12.PDE 12  Wave equation: characteristics 13.PDE 13  Wave equation: separation of variables 14.FA 1  Fourier series introduction 15.FA 2  Computing Fourier series 16.PDE  Heat equation: intuition 17.PDE  Finite differences: introduction
 Course related:
 AMA3723 Further Mathematical Methods for Finance
 Subjects:
 Mathematics and Statistics
 Keywords:
 Differential equations Partial
 Resource Type:
 Video

ebook
This book consists of ten weeks of material given as a course on ordinary differential equations (ODEs) for second year mathematics majors at the University of Bristol. It is the first course devoted solely to differential equations that these students will take. This book consists of 10 chapters, and the course is 12 weeks long. Each chapter is covered in a week, and in the remaining two weeks I summarize the entire course, answer lots of questions, and prepare the students for the exam. I do not cover the material in the appendices in the lectures. Some of it is basic material that the students have already seen that I include for completeness and other topics are "tasters" for more advanced material that students will encounter in later courses or in their project work. Students are very curious about the notion of chaos, and I have included some material in an appendix on that concept. The focus in that appendix is only to connect it with ideas that have been developed in this course related to ODEs and to prepare them for more advanced courses in dynamical systems and ergodic theory that are available in their third and fourth years.
 Subjects:
 Mathematics and Statistics
 Keywords:
 Differential equations Textbooks Differential equations Partial
 Resource Type:
 ebook

ebook
This is a complete college textbook, including a detailed Table of Contents, seventeen Chapters (each with a set of relevant homework problems), a list of References, two Appendices, and a detailed Index. The book is intended to enable students to: Solve first, second, and higherorder, linear, timeinvariant (LTI) ordinary differential equations (ODEs) with initial conditions and excitation, using both timedomain and Laplacetransform methods; Solve for the frequency response of an LTI system to periodic sinusoidal excitation and plot this response in standard form; Explain the role of the time constant in the response of a firstorder LTI system, and the roles of natural frequency, damping ratio, and resonance in the response of a secondorder LTI system; Derive and analyze mathematical models (ODEs) of loworder mechanical systems, both translational and rotational, that are composed of inertial elements, spring elements, and damping devices; Derive and analyze mathematical models (ODEs) of loworder electrical circuits composed of resistors, capacitors, inductors, and operational amplifiers; Derive (from ODEs) and manipulate Laplace transfer functions and block diagrams representing outputtoinput relationships of discrete elements and of systems; Define and evaluate stability for an LTI system; Explain proportional, integral, and derivative types of feedback control for singleinput, singleoutput (SISO), LTI systems; Sketch the locus of characteristic values, as a control parameter varies, for a feedbackcontrolled SISO, LTI system; Use MATLAB as a tool to study the time and frequency responses of LTI systems. The book's general organization is: Chapters 110 deal primarily with the ODEs and behaviors of firstorder and secondorder dynamic systems; Chapters 11 and 12 discuss the ODEs and behaviors of mechanical systems having two degrees of freedom, i.e., fourthorder systems; Chapters 13 and 14 introduce classical feedback control; Chapter 15 presents the basic features of proportional, integral, and derivative types of classical control; Chapters 16 and 17 discuss methods for analyzing the stability of classical control systems. The general minimum prerequisite for understanding this book is the intellectual maturity of a juniorlevel (thirdyear) college student in an accredited fouryear engineering curriculum. A mathematical secondorder system is represented in this book primarily by a single secondorder ODE, not in the statespace form by a pair of coupled firstorder ODEs. Similarly, a twodegreesoffreedom (fourthorder) system is represented by two coupled secondorder ODEs, not in the statespace form by four coupled firstorder ODEs. The book does not use bond graph modeling, the general and powerful, but complicated, modern tool for analysis of complex, multidisciplinary dynamic systems. The homework problems at the ends of chapters are very important to the learning objectives, so the author attempted to compose problems of practical interest and to make the problem statements as clear, correct, and unambiguous as possible. A major focus of the book is computer calculation of system characteristics and responses and graphical display of results, with use of basic (not advanced) MATLAB commands and programs. The book includes many examples and homework problems relevant to aerospace engineering, among which are rolling dynamics of flight vehicles, spacecraft actuators, aerospace motion sensors, and aeroelasticity. There are also several examples and homework problems illustrating and validating theory by using measured data to identify first and secondorder system dynamic characteristics based on mathematical models (e.g., time constants and natural frequencies), and system basic properties (e.g., mass, stiffness, and damping). Applications of real and simulated experimental data appear in many homework problems. The book contains somewhat more material than can be covered during a single standard college semester, so an instructor who wishes to use this as a onesemester course textbook should not attempt to cover the entire book, but instead should cover only those parts that are most relevant to the course objectives.
 Keywords:
 Differential equations Engineering mathematics Differential equations Partial Textbooks
 Resource Type:
 ebook

ebook
Elementary Differential Equations with Boundary Value Problems is written for students in science, engineering, and mathematics who have completed calculus through partial differentiation. An elementary text should be written so the student can read it with comprehension without too much pain. I have tried to put myself in the student's place, and have chosen to err on the side of too much detail rather than not enough. An elementary text can't be better than its exercises. This text includes 1695 numbered exercises, many with several parts. They range in difficulty from routine to very challenging. An elementary text should be written in an informal but mathematically accurate way, illustrated by appropriate graphics. I have tried to formulate mathematical concepts succinctly in language that students can understand. I have minimized the number of explicitly stated theorems and definitions, preferring to deal with concepts in a more conversational way, copiously illustrated by 250 completely worked out examples. Where appropriate, concepts and results are depicted in 144 figures. Although I believe that the computer is an immensely valuable tool for learning, doing, and writing mathematics, the selection and treatment of topics in this text reflects my pedagogical orientation along traditional lines. However, I have incorporated what I believe to be the best use of modern technology, so you can select the level of technology that you want to include in your course. The text includes 336 exercises – identified by the symbols C and C/G – that call for graphics or computation and graphics. There are also 73 laboratory exercises – identified by L – that require extensive use of technology. In addition, several sections include informal advice on the use of technology. If you prefer not to emphasize technology, simply ignore these exercises and the advice.
 Subjects:
 Mathematics and Statistics
 Keywords:
 Textbooks Differential equations Partial Boundary value problems
 Resource Type:
 ebook

Courseware
In this course, we study elliptic Partial Differential Equations (PDEs) with variable coefficients building up to the minimal surface equation. Then we study Fourier and harmonic analysis, emphasizing applications of Fourier analysis. We will see some applications in combinatorics / number theory, like the Gauss circle problem, but mostly focus on applications in PDE, like the CalderonZygmund inequality for the Laplacian, and the Strichartz inequality for the Schrodinger equation. In the last part of the course, we study solutions to the linear and the nonlinear Schrodinger equation. All through the course, we work on the craft of proving estimates.
 Subjects:
 Mathematics and Statistics
 Keywords:
 Fourier analysis Differential equations Partial
 Resource Type:
 Courseware