Advertisement
| solving partial differential equations using laplace transforms: Transform Methods for Solving Partial Differential Equations, Second Edition Dean G. Duffy, 2004-07-15 Transform methods provide a bridge between the commonly used method of separation of variables and numerical techniques for solving linear partial differential equations. While in some ways similar to separation of variables, transform methods can be effective for a wider class of problems. Even when the inverse of the transform cannot be found analytically, numeric and asymptotic techniques now exist for their inversion, and because the problem retains some of its analytic aspect, one can gain greater physical insight than typically obtained from a purely numerical approach. Transform Methods for Solving Partial Differential Equations, Second Edition illustrates the use of Laplace, Fourier, and Hankel transforms to solve partial differential equations encountered in science and engineering. The author has expanded the second edition to provide a broader perspective on the applicability and use of transform methods and incorporated a number of significant refinements: New in the Second Edition: · Expanded scope that includes numerical methods and asymptotic techniques for inverting particularly complicated transforms · Discussions throughout the book that compare and contrast transform methods with separation of variables, asymptotic methods, and numerical techniques · Many added examples and exercises taken from a wide variety of scientific and engineering sources · Nearly 300 illustrations--many added to the problem sections to help readers visualize the physical problems · A revised format that makes the book easier to use as a reference: problems are classified according to type of region, type of coordinate system, and type of partial differential equation · Updated references, now arranged by subject instead of listed all together As reflected by the book's organization, content, and many examples, the author's focus remains firmly on applications. While the subject matter is classical, this book gives it a fresh, modern treatment that is exceptionally practical, eminently readable, and especially valuable to anyone solving problems in engineering and the applied sciences. |
| solving partial differential equations using laplace transforms: The Laplace Transform Richard Bellman, Robert S. Roth, 1984 The classical theory of the Laplace Transform can open many new avenues when viewed from a modern, semi-classical point of view. In this book, the author re-examines the Laplace Transform and presents a study of many of the applications to differential equations, differential-difference equations and the renewal equation. |
| solving partial differential equations using laplace transforms: Mathematical Physics with Partial Differential Equations James Kirkwood, 2011-12-01 Mathematical Physics with Partial Differential Equations is for advanced undergraduate and beginning graduate students taking a course on mathematical physics taught out of math departments. The text presents some of the most important topics and methods of mathematical physics. The premise is to study in detail the three most important partial differential equations in the field – the heat equation, the wave equation, and Laplace's equation. The most common techniques of solving such equations are developed in this book, including Green's functions, the Fourier transform, and the Laplace transform, which all have applications in mathematics and physics far beyond solving the above equations. The book's focus is on both the equations and their methods of solution. Ordinary differential equations and PDEs are solved including Bessel Functions, making the book useful as a graduate level textbook. The book's rigor supports the vital sophistication for someone wanting to continue further in areas of mathematical physics. - Examines in depth both the equations and their methods of solution - Presents physical concepts in a mathematical framework - Contains detailed mathematical derivations and solutions— reinforcing the material through repetition of both the equations and the techniques - Includes several examples solved by multiple methods—highlighting the strengths and weaknesses of various techniques and providing additional practice |
| solving partial differential equations using laplace transforms: Laplace Transforms and Their Applications to Differential Equations N.W. McLachlan, 2014-11-19 This introduction to modern operational calculus offers a classic exposition of Laplace transform theory and its application to the solution of ordinary and partial differential equations. The treatment is addressed to graduate students in engineering, physics, and applied mathematics and may be used as a primary text or supplementary reading. Chief topics include the theorems or rules of the operational calculus, evaluation of integrals and establishment of mathematical relationships, derivation of Laplace transforms of various functions, the Laplace transform for a finite interval, and other subjects. Many problems and illustrative examples appear throughout the book, which is further augmented by helpful Appendixes. Dover (2014) republication of the 1962 (Dover) revised edition of Modern Operational Calculus with Applications in Technical Mathematics, Macmillan, London, 1948. See every Dover book in print at www.doverpublications.com |
| solving partial differential equations using laplace transforms: Transform Methods for Solving Partial Differential Equations Dean G. Duffy, 2004-07-15 Transform methods provide a bridge between the commonly used method of separation of variables and numerical techniques for solving linear partial differential equations. While in some ways similar to separation of variables, transform methods can be effective for a wider class of problems. Even when the inverse of the transform cannot be found ana |
| solving partial differential equations using laplace transforms: Notes on Diffy Qs Jiri Lebl, 2019-11-13 Version 6.0. An introductory course on differential equations aimed at engineers. The book covers first order ODEs, higher order linear ODEs, systems of ODEs, Fourier series and PDEs, eigenvalue problems, the Laplace transform, and power series methods. It has a detailed appendix on linear algebra. The book was developed and used to teach Math 286/285 at the University of Illinois at Urbana-Champaign, and in the decade since, it has been used in many classrooms, ranging from small community colleges to large public research universities. See https: //www.jirka.org/diffyqs/ for more information, updates, errata, and a list of classroom adoptions. |
| solving partial differential equations using laplace transforms: An Introduction to Laplace Transforms and Fourier Series Phil Dyke, 2000-10-27 This introduction to Laplace transforms and Fourier series is aimed at second year students in applied mathematics. It is unusual in treating Laplace transforms at a relatively simple level with many examples. Mathematics students do not usually meet this material until later in their degree course but applied mathematicians and engineers need an early introduction. Suitable as a course text, it will also be of interest to physicists and engineers as supplementary material. |
| solving partial differential equations using laplace transforms: A Third Order Differential Equation W. R. Utz, 1955 |
| solving partial differential equations using laplace transforms: The Laplace Transform Joel L. Schiff, 2014-01-15 |
| solving partial differential equations using laplace transforms: Mathematical Foundations for Linear Circuits and Systems in Engineering John J. Shynk, 2016-01-26 Extensive coverage of mathematical techniques used in engineering with an emphasis on applications in linear circuits and systems Mathematical Foundations for Linear Circuits and Systems in Engineering provides an integrated approach to learning the necessary mathematics specifically used to describe and analyze linear circuits and systems. The chapters develop and examine several mathematical models consisting of one or more equations used in engineering to represent various physical systems. The techniques are discussed in-depth so that the reader has a better understanding of how and why these methods work. Specific topics covered include complex variables, linear equations and matrices, various types of signals, solutions of differential equations, convolution, filter designs, and the widely used Laplace and Fourier transforms. The book also presents a discussion of some mechanical systems that mathematically exhibit the same dynamic properties as electrical circuits. Extensive summaries of important functions and their transforms, set theory, series expansions, various identities, and the Lambert W-function are provided in the appendices. The book has the following features: Compares linear circuits and mechanical systems that are modeled by similar ordinary differential equations, in order to provide an intuitive understanding of different types of linear time-invariant systems. Introduces the theory of generalized functions, which are defined by their behavior under an integral, and describes several properties including derivatives and their Laplace and Fourier transforms. Contains numerous tables and figures that summarize useful mathematical expressions and example results for specific circuits and systems, which reinforce the material and illustrate subtle points. Provides access to a companion website that includes a solutions manual with MATLAB code for the end-of-chapter problems. Mathematical Foundations for Linear Circuits and Systems in Engineering is written for upper undergraduate and first-year graduate students in the fields of electrical and mechanical engineering. This book is also a reference for electrical, mechanical, and computer engineers as well as applied mathematicians. John J. Shynk, PhD, is Professor of Electrical and Computer Engineering at the University of California, Santa Barbara. He was a Member of Technical Staff at Bell Laboratories, and received degrees in systems engineering, electrical engineering, and statistics from Boston University and Stanford University. |
| solving partial differential equations using laplace transforms: Introduction to Partial Differential Equations with Applications E. C. Zachmanoglou, Dale W. Thoe, 2012-04-20 This text explores the essentials of partial differential equations as applied to engineering and the physical sciences. Discusses ordinary differential equations, integral curves and surfaces of vector fields, the Cauchy-Kovalevsky theory, more. Problems and answers. |
| solving partial differential equations using laplace transforms: Partial Differential Equations and Boundary-Value Problems with Applications Mark A. Pinsky, 2011 Building on the basic techniques of separation of variables and Fourier series, the book presents the solution of boundary-value problems for basic partial differential equations: the heat equation, wave equation, and Laplace equation, considered in various standard coordinate systems--rectangular, cylindrical, and spherical. Each of the equations is derived in the three-dimensional context; the solutions are organized according to the geometry of the coordinate system, which makes the mathematics especially transparent. Bessel and Legendre functions are studied and used whenever appropriate throughout the text. The notions of steady-state solution of closely related stationary solutions are developed for the heat equation; applications to the study of heat flow in the earth are presented. The problem of the vibrating string is studied in detail both in the Fourier transform setting and from the viewpoint of the explicit representation (d'Alembert formula). Additional chapters include the numerical analysis of solutions and the method of Green's functions for solutions of partial differential equations. The exposition also includes asymptotic methods (Laplace transform and stationary phase). With more than 200 working examples and 700 exercises (more than 450 with answers), the book is suitable for an undergraduate course in partial differential equations. |
| solving partial differential equations using laplace transforms: Student Solutions Manual, Partial Differential Equations & Boundary Value Problems with Maple George A. Articolo, 2009-07-22 Student Solutions Manual, Partial Differential Equations & Boundary Value Problems with Maple |
| solving partial differential equations using laplace transforms: Integral Transforms and Their Applications B. Davies, 2013-11-27 This book is intended to serve as introductory and reference material for the application of integral transforms to a range of common mathematical problems. It has its im mediate origin in lecture notes prepared for senior level courses at the Australian National University, although I owe a great deal to my colleague Barry Ninham, a matter to which I refer below. In preparing the notes for publication as a book, I have added a considerable amount of material ad- tional to the lecture notes, with the intention of making the book more useful, particularly to the graduate student - volved in the solution of mathematical problems in the physi cal, chemical, engineering and related sciences. Any book is necessarily a statement of the author's viewpoint, and involves a number of compromises. My prime consideration has been to produce a work whose scope is selective rather than encyclopedic; consequently there are many facets of the subject which have been omitted--in not a few cases after a preliminary draft was written--because I v believe that their inclusion would make the book too long. |
| solving partial differential equations using laplace transforms: Fourier and Laplace Transforms , 2003-08-07 This textbook presents in a unified manner the fundamentals of both continuous and discrete versions of the Fourier and Laplace transforms. These transforms play an important role in the analysis of all kinds of physical phenomena. As a link between the various applications of these transforms the authors use the theory of signals and systems, as well as the theory of ordinary and partial differential equations. The book is divided into four major parts: periodic functions and Fourier series, non-periodic functions and the Fourier integral, switched-on signals and the Laplace transform, and finally the discrete versions of these transforms, in particular the Discrete Fourier Transform together with its fast implementation, and the z-transform. This textbook is designed for self-study. It includes many worked examples, together with more than 120 exercises, and will be of great value to undergraduates and graduate students in applied mathematics, electrical engineering, physics and computer science. |
| solving partial differential equations using laplace transforms: Beginning MATLAB and Simulink Sulaymon Eshkabilov, 2019-11-28 Employ essential and hands-on tools and functions of the MATLAB and Simulink packages, which are explained and demonstrated via interactive examples and case studies. This book contains dozens of simulation models and solved problems via m-files/scripts and Simulink models which help you to learn programming and modeling essentials. You’ll become efficient with many of the built-in tools and functions of MATLAB/Simulink while solving engineering and scientific computing problems. Beginning MATLAB and Simulink explains various practical issues of programming and modelling in parallel by comparing MATLAB and Simulink. After reading and using this book, you'll be proficient at using MATLAB and applying the source code from the book's examples as templates for your own projects in data science or engineering. What You Will Learn Get started using MATLAB and Simulink Carry out data visualization with MATLAB Gain the programming and modeling essentials of MATLAB Build a GUI with MATLAB Work with integration and numerical root finding methods Apply MATLAB to differential equations-based models and simulations Use MATLAB for data science projects Who This Book Is For Engineers, programmers, data scientists, and students majoring in engineering and scientific computing. |
| solving partial differential equations using laplace transforms: Introduction To Partial Differential Equations (With Maple), An: A Concise Course Zhilin Li, Larry Norris, 2021-09-23 The book is designed for undergraduate or beginning level graduate students, and students from interdisciplinary areas including engineers, and others who need to use partial differential equations, Fourier series, Fourier and Laplace transforms. The prerequisite is a basic knowledge of calculus, linear algebra, and ordinary differential equations.The textbook aims to be practical, elementary, and reasonably rigorous; the book is concise in that it describes fundamental solution techniques for first order, second order, linear partial differential equations for general solutions, fundamental solutions, solution to Cauchy (initial value) problems, and boundary value problems for different PDEs in one and two dimensions, and different coordinates systems. Analytic solutions to boundary value problems are based on Sturm-Liouville eigenvalue problems and series solutions.The book is accompanied with enough well tested Maple files and some Matlab codes that are available online. The use of Maple makes the complicated series solution simple, interactive, and visible. These features distinguish the book from other textbooks available in the related area. |
| solving partial differential equations using laplace transforms: Introduction to Partial Differential Equations Peter J. Olver, 2013-11-08 This textbook is designed for a one year course covering the fundamentals of partial differential equations, geared towards advanced undergraduates and beginning graduate students in mathematics, science, engineering, and elsewhere. The exposition carefully balances solution techniques, mathematical rigor, and significant applications, all illustrated by numerous examples. Extensive exercise sets appear at the end of almost every subsection, and include straightforward computational problems to develop and reinforce new techniques and results, details on theoretical developments and proofs, challenging projects both computational and conceptual, and supplementary material that motivates the student to delve further into the subject. No previous experience with the subject of partial differential equations or Fourier theory is assumed, the main prerequisites being undergraduate calculus, both one- and multi-variable, ordinary differential equations, and basic linear algebra. While the classical topics of separation of variables, Fourier analysis, boundary value problems, Green's functions, and special functions continue to form the core of an introductory course, the inclusion of nonlinear equations, shock wave dynamics, symmetry and similarity, the Maximum Principle, financial models, dispersion and solutions, Huygens' Principle, quantum mechanical systems, and more make this text well attuned to recent developments and trends in this active field of contemporary research. Numerical approximation schemes are an important component of any introductory course, and the text covers the two most basic approaches: finite differences and finite elements. |
| solving partial differential equations using laplace transforms: Partial Differential Equations Victor Henner, Tatyana Belozerova, Alexander Nepomnyashchy, 2019-11-20 Partial Differential Equations: Analytical Methods and Applications covers all the basic topics of a Partial Differential Equations (PDE) course for undergraduate students or a beginners’ course for graduate students. It provides qualitative physical explanation of mathematical results while maintaining the expected level of it rigor. This text introduces and promotes practice of necessary problem-solving skills. The presentation is concise and friendly to the reader. The teaching-by-examples approach provides numerous carefully chosen examples that guide step-by-step learning of concepts and techniques. Fourier series, Sturm-Liouville problem, Fourier transform, and Laplace transform are included. The book’s level of presentation and structure is well suited for use in engineering, physics and applied mathematics courses. Highlights: Offers a complete first course on PDEs The text’s flexible structure promotes varied syllabi for courses Written with a teach-by-example approach which offers numerous examples and applications Includes additional topics such as the Sturm-Liouville problem, Fourier and Laplace transforms, and special functions The text’s graphical material makes excellent use of modern software packages Features numerous examples and applications which are suitable for readers studying the subject remotely or independently |
| solving partial differential equations using laplace transforms: Differential Equations with Discontinuous Coefficients Ward Conrad Sangren, 1953 |
| solving partial differential equations using laplace transforms: Laplace Transforms Mohamed F. El-Hewie, 2013-04-15 This is a revised edition of the chapter on Laplace Transforms, which was published few years ago in Part II of My Personal Study Notes in advanced mathematics. In this edition, I typed the cursive scripts of the personal notes, edited the typographic errors, but most of all reproduced all the calculations and graphics in a modern style of representation. The book is organized into six chapters equally distributed to address: (1) The theory of Laplace transformations and inverse transformations of elementary functions, supported by solved examples and exercises with given answers; (2) Transformation of more complex functions from elementary transformation; (3) Practical applications of Laplace transformation to equations of motion of material bodies and deflection, stress, and strain of elastic beams; (4) Solving equations of state of motion of bodies under inertial and gravitational forces. (5) Solving heat flow equations through various geometrical bodies; and (6) Solving partial differential equations by the operational algebraic properties of transforming and inverse transforming of partial differential equations. During the editing process, I added plenty of comments of the underlying meaning of the arcane equations such that the reader could discern the practical weight of each mathematical formula. In a way, I attempted to convey a personal sense and feeling on the significance and philosophy of devising a mathematical equation that transcends into real-life emulation. The reader will find this edition dense with graphic illustrations that should spare the reader the trouble of searching other references in order to infer any missing steps. In my view, detailed graphic illustrations could soothe the harshness of arcane mathematical jargon, as well as expose the merits of the assumption contemplated in the formulation. In lieu of offering a dense textbook on Laplace Transforms, I opted to stick to my personal notes that give the memorable zest of a subject that could easily remembered when not frequently used. Brief Outline of Contents: CHAPTER 1. THE LAPLACE TRANSFORMATION AND INVERSE TRANSFORMATION 1.1. Integral transforms 1.2. Some elementary Laplace transforms 1.3. The Laplace transformation of the sum of two functions 1.4. Sectionally or piecewise continuous functions 1.5. Functions of exponential order 1.7. Null functions 1.8. Inverse Laplace transforms 1.10. Laplace transforms of derivatives 1.11. Laplace transforms of integrals 1.12. The first shift theorem of multiplying the object function by eat 1.15. Determination of the inverse Laplace transforms by the aid of partial fractions 1.16. Laplace's solution of linear differential equations with constant coefficients CHAPTER 2. GENERAL THEOREMS ON THE LAPLACE TRANSFORMATION 2.1. The unit step function 2.2. The second translation or shifting property 2.4. The unit impulse function 2.5. The unit doublet 2.7. Initial value theorem 2.8. Final value theorem 2.9. Differentiation of transform 2.11. Integration of transforms 2.12. Transforms of periodic functions 2.13. The product theorem-Convolution 2.15. Power series method for the determination of transforms and inverse transforms 2.16. The error function or probability integral 2.22. The inversion integral CHAPTER 3. ELECTRICAL APPLICATIONS OF THE LAPLACE TRANSFORMATION CHAPTER 4. DYNAMICAL APPLICATIONS OF LAPLACE TRANSFORMS CHAPTER 5. STRUCTURAL APPLICATIONS 5.1. Deflection of beams CHAPTER 6. USING LAPLACE TRANSFORMATION IN SOLVING LINEAR PARTIAL DIFFERENTIAL EQUATIONS 6.1. Transverse vibrations of a stretched string under gravity 6.2. Longitudinal vibrations of bars 6.3. Partial differential equations of transmission lines 6.4. Conduction of heat 6.5. Exercise on using Laplace Transformation in solving Linear Partial Differential Equations |
| solving partial differential equations using laplace transforms: Information Processing and Management of Uncertainty in Knowledge-Based Systems Eyke Hüllermeier, Rudolf Kruse, Frank Hoffmann, 2010-06-17 This book constitutes the proceedings of the 13th conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems, held in Dortmund, Germany, in June 2010. |
| solving partial differential equations using laplace transforms: Transform Methods for Solving Partial Differential Equations Dean G. Duffy, 1994-02-16 For most scientists and engineers, the only analytic technique for solving linear partial differential equations is separation of variables. In Transform Methods for Solving Partial Differential Equations, the author uses the power of complex variables to demonstrate how Laplace and Fourier transforms can be harnessed to solve many practical, everyday problems experienced by scientists and engineers. Unlike many mathematics texts, this book provides a step-by-step analysis of problems taken from scientific and engineering literature. Detailed solutions are given in the back of the book. This essential text/reference draws from the latest literature on transform methods to provide in-depth discussions on the joint transform problem, the Cagniard-de Hoop method, and the Wiener-Hopf technique. Some 1,500 references are included as well. |
| solving partial differential equations using laplace transforms: Introductory Applications of Partial Differential Equations G. L. Lamb, Jr., 1995-04-17 INTRODUCTORY APPLICATIONS OF PARTIAL DIFFERENTIAL EQUATIONS With Emphasis on Wave Propagation and Diffusion This is the ideal text for students and professionals who have somefamiliarity with partial differential equations, and who now wishto consolidate and expand their knowledge. Unlike most other textson this topic, it interweaves prior knowledge of mathematics andphysics, especially heat conduction and wave motion, into apresentation that demonstrates their interdependence. The result isa superb teaching text that reinforces the reader's understandingof both mathematics and physics. Rather than presenting themathematics in isolation and out of context, problems in this textare framed to show how partial differential equations can be usedto obtain specific information about the physical system beinganalyzed. Designed for upper-level students, professionals and researchers inengineering, applied mathematics, physics, and optics, ProfessorLamb's text is lucid in its presentation and comprehensive in itscoverage of all the important topic areas, including: * One-Dimensional Problems * The Laplace Transform Method * Two and Three Dimensions * Green's Functions * Spherical Geometry * Fourier Transform Methods * Perturbation Methods * Generalizations and First Order Equations In addition, this text includes a supplementary chapter of selectedtopics and handy appendices that review Fourier Series, LaplaceTransform, Sturm-Liouville Equations, Bessel Functions, andLegendre Polynomials. |
| solving partial differential equations using laplace transforms: Introduction to Partial Differential Equations Aslak Tveito, Ragnar Winther, 2008-01-21 Combining both the classical theory and numerical techniques for partial differential equations, this thoroughly modern approach shows the significance of computations in PDEs and illustrates the strong interaction between mathematical theory and the development of numerical methods. Great care has been taken throughout the book to seek a sound balance between these techniques. The authors present the material at an easy pace and exercises ranging from the straightforward to the challenging have been included. In addition there are some projects suggested, either to refresh the students memory of results needed in this course, or to extend the theories developed in the text. Suitable for undergraduate and graduate students in mathematics and engineering. |
| solving partial differential equations using laplace transforms: Integral Equations and Their Applications Matiur Rahman, 2007 The book deals with linear integral equations, that is, equations involving an unknown function which appears under the integral sign and contains topics such as Abel's integral equation, Volterra integral equations, Fredholm integral integral equations, singular and nonlinear integral equations, orthogonal systems of functions, Green's function as a symmetric kernel of the integral equations. |
| solving partial differential equations using laplace transforms: Applied Stochastic Differential Equations Simo Särkkä, Arno Solin, 2019-05-02 With this hands-on introduction readers will learn what SDEs are all about and how they should use them in practice. |
| solving partial differential equations using laplace transforms: Partial Differential Equations: Graduate Level Problems and Solutions Igor Yanovsky, 2014-10-21 Partial Differential Equations: Graduate Level Problems and SolutionsBy Igor Yanovsky |
| solving partial differential equations using laplace transforms: Differential Equations and Linear Algebra Gilbert Strang, 2015-02-12 Differential equations and linear algebra are two central topics in the undergraduate mathematics curriculum. This innovative textbook allows the two subjects to be developed either separately or together, illuminating the connections between two fundamental topics, and giving increased flexibility to instructors. It can be used either as a semester-long course in differential equations, or as a one-year course in differential equations, linear algebra, and applications. Beginning with the basics of differential equations, it covers first and second order equations, graphical and numerical methods, and matrix equations. The book goes on to present the fundamentals of vector spaces, followed by eigenvalues and eigenvectors, positive definiteness, integral transform methods and applications to PDEs. The exposition illuminates the natural correspondence between solution methods for systems of equations in discrete and continuous settings. The topics draw on the physical sciences, engineering and economics, reflecting the author's distinguished career as an applied mathematician and expositor. |
| solving partial differential equations using laplace transforms: Mathematics for Machine Learning Marc Peter Deisenroth, A. Aldo Faisal, Cheng Soon Ong, 2020-04-23 The fundamental mathematical tools needed to understand machine learning include linear algebra, analytic geometry, matrix decompositions, vector calculus, optimization, probability and statistics. These topics are traditionally taught in disparate courses, making it hard for data science or computer science students, or professionals, to efficiently learn the mathematics. This self-contained textbook bridges the gap between mathematical and machine learning texts, introducing the mathematical concepts with a minimum of prerequisites. It uses these concepts to derive four central machine learning methods: linear regression, principal component analysis, Gaussian mixture models and support vector machines. For students and others with a mathematical background, these derivations provide a starting point to machine learning texts. For those learning the mathematics for the first time, the methods help build intuition and practical experience with applying mathematical concepts. Every chapter includes worked examples and exercises to test understanding. Programming tutorials are offered on the book's web site. |
| solving partial differential equations using laplace transforms: Fractional Calculus Dumitru Baleanu, 2012 This title will give readers the possibility of finding very important mathematical tools for working with fractional models and solving fractional differential equations, such as a generalization of Stirling numbers in the framework of fractional calculus and a set of efficient numerical methods. |
| solving partial differential equations using laplace transforms: An Introduction to Partial Differential Equations with MATLAB Matthew P. Coleman, Vladislav Bukshtynov, 2024-08-01 The first two editions of An Introduction to Partial Differential Equations with MATLAB® gained popularity among instructors and students at various universities throughout the world. Plain mathematical language is used in a friendly manner to provide a basic introduction to partial differential equations (PDEs). Suitable for a one- or two-semester introduction to PDEs and Fourier series, the book strives to provide physical, mathematical, and historical motivation for each topic. Equations are studied based on method of solution, rather than on type of equation. This third edition of this popular textbook updates the structure of the book by increasing the role of the computational portion, compared to previous editions. The redesigned content will be extremely useful for students of mathematics, physics, and engineering who would like to focus on the practical aspects of the study of PDEs, without sacrificing mathematical rigor. The authors have maintained flexibility in the order of topics. In addition, students will be able to use what they have learned in some later courses (for example, courses in numerical analysis, optimization, and PDE-based programming). Included in this new edition is a substantial amount of material on reviewing computational methods for solving ODEs (symbolically and numerically), visualizing solutions of PDEs, using MATLAB®'s symbolic programming toolbox, and applying various schemes from numerical analysis, along with suggestions for topics of course projects. Students will use sample MATLAB® or Python codes available online for their practical experiments and for completing computational lab assignments and course projects. |
| solving partial differential equations using laplace transforms: Mathematical Physics with Partial Differential Equations James Kirkwood, 2012-01-20 Suitable for advanced undergraduate and beginning graduate students taking a course on mathematical physics, this title presents some of the most important topics and methods of mathematical physics. It contains mathematical derivations and solutions - reinforcing the material through repetition of both the equations and the techniques. |
| solving partial differential equations using laplace transforms: Partial Differential Equations T. Hillen, I.E. Leonard, H. van Roessel, 2019-05-15 Provides more than 150 fully solved problems for linear partial differential equations and boundary value problems. Partial Differential Equations: Theory and Completely Solved Problems offers a modern introduction into the theory and applications of linear partial differential equations (PDEs). It is the material for a typical third year university course in PDEs. The material of this textbook has been extensively class tested over a period of 20 years in about 60 separate classes. The book is divided into two parts. Part I contains the Theory part and covers topics such as a classification of second order PDEs, physical and biological derivations of the heat, wave and Laplace equations, separation of variables, Fourier series, D’Alembert’s principle, Sturm-Liouville theory, special functions, Fourier transforms and the method of characteristics. Part II contains more than 150 fully solved problems, which are ranked according to their difficulty. The last two chapters include sample Midterm and Final exams for this course with full solutions. |
| solving partial differential equations using laplace transforms: Partial Differential Equations in Classical Mathematical Physics Isaak Rubinstein, Lev Rubinstein, 1998-04-28 The unique feature of this book is that it considers the theory of partial differential equations in mathematical physics as the language of continuous processes, that is, as an interdisciplinary science that treats the hierarchy of mathematical phenomena as reflections of their physical counterparts. Special attention is drawn to tracing the development of these mathematical phenomena in different natural sciences, with examples drawn from continuum mechanics, electrodynamics, transport phenomena, thermodynamics, and chemical kinetics. At the same time, the authors trace the interrelation between the different types of problems - elliptic, parabolic, and hyperbolic - as the mathematical counterparts of stationary and evolutionary processes. This combination of mathematical comprehensiveness and natural scientific motivation represents a step forward in the presentation of the classical theory of PDEs, one that will be appreciated by both students and researchers alike. |
| solving partial differential equations using laplace transforms: Linear Partial Differential Equations and Fourier Theory Marcus Pivato, 2010-01-07 This highly visual introductory textbook provides a rigorous mathematical foundation for all solution methods and reinforces ties to physical motivation. |
| solving partial differential equations using laplace transforms: Fuzzy Fractional Differential Operators and Equations Tofigh Allahviranloo, 2020-06-15 This book contains new and useful materials concerning fuzzy fractional differential and integral operators and their relationship. As the title of the book suggests, the fuzzy subject matter is one of the most important tools discussed. Therefore, it begins by providing a brief but important and new description of fuzzy sets and the computational calculus they require. Fuzzy fractals and fractional operators have a broad range of applications in the engineering, medical and economic sciences. Although these operators have been addressed briefly in previous papers, this book represents the first comprehensive collection of all relevant explanations. Most of the real problems in the biological and engineering sciences involve dynamic models, which are defined by fuzzy fractional operators in the form of fuzzy fractional initial value problems. Another important goal of this book is to solve these systems and analyze their solutions both theoretically and numerically. Given the content covered, the book will benefit all researchers and students in the mathematical and computer sciences, but also the engineering sciences. |
| solving partial differential equations using laplace transforms: ORDINARY AND PARTIAL DIFFERENTIAL EQUATIONS : THEORY AND APPLICATIONS Nita H. Shah, 2010-06 This book presents the theoretical concepts of methods of solutions of ordinary and partial differential equations as well as equips the students with the various tools and techniques to model different physical problems using such equations. The book discusses the basic concepts of differential equations, different methods of solving ordinary differential equations and the solution procedure for ordinary differential equations of first order and higher degree. It gives the solution methodology for linear differential equations with constant and variable coefficients and linear differential equations of second order. The book elaborates simultaneous linear differential equations, total differential equations, and partial differential equations along with the series solution of second order linear differential equations. It also covers Bessel's and Legendre's equations and functions, and the Laplace transform. Finally, the book revisits partial differential equations to solve the Laplace equation, wave equation and diffusion equation, and discusses the methods to solve partial differential equations using the Fourier transform. A large number of solved examples as well as exercises at the end of chapters help the students comprehend and strengthen the underlying concepts. The book is intended for undergraduate and postgraduate students of Mathematics (B.A./B.Sc., M.A./M.Sc.), and undergraduate students of all branches of engineering (B.E./B.Tech.), as part of their course in Engineering Mathematics. |
| solving partial differential equations using laplace transforms: The Laplace Transform Joel L. Schiff, 2013-06-05 The Laplace transform is a wonderful tool for solving ordinary and partial differential equations and has enjoyed much success in this realm. With its success, however, a certain casualness has been bred concerning its application, without much regard for hypotheses and when they are valid. Even proofs of theorems often lack rigor, and dubious mathematical practices are not uncommon in the literature for students. In the present text, I have tried to bring to the subject a certain amount of mathematical correctness and make it accessible to un dergraduates. Th this end, this text addresses a number of issues that are rarely considered. For instance, when we apply the Laplace trans form method to a linear ordinary differential equation with constant coefficients, any(n) + an-lY(n-l) + · · · + aoy = f(t), why is it justified to take the Laplace transform of both sides of the equation (Theorem A. 6)? Or, in many proofs it is required to take the limit inside an integral. This is always fraught with danger, especially with an improper integral, and not always justified. I have given complete details (sometimes in the Appendix) whenever this procedure is required. IX X Preface Furthermore, it is sometimes desirable to take the Laplace trans form of an infinite series term by term. Again it is shown that this cannot always be done, and specific sufficient conditions are established to justify this operation. |
| solving partial differential equations using laplace transforms: Elements of Partial Differential Equations Pavel Drábek, Gabriela Holubová, 2007 This textbook presents a first introduction to PDEs on an elementary level, enabling the reader to understand what partial differential equations are, where they come from and how they can be solved. The intention is that the reader understands the basic principles which are valid for particular types of PDEs, and to acquire some classical methods to solve them, thus the authors restrict their considerations to fundamental types of equations and basic methods. Only basic facts from calculus and linear ordinary differential equations of first and second order are needed as a prerequisite. An elementary introduction to the basic principles of partial differential equations. With many illustrations. The book is addressed to students who intend to specialize in mathematics as well as to students of physics, engineering, and economics. |
Step-by-Step Math Problem Solver
QuickMath allows students to get instant solutions to all kinds of math problems, from algebra and equation solving right through to calculus and …
What is Problem Solving? Steps, Process & Techniques …
WHAT IS PROBLEM SOLVING? Solving a problem depends on correctly identifying its cause so the best solution can be selected and …
What is Problem Solving? (Steps, Techniques, Examples…
Problem solving is the process of finding solutions to obstacles or challenges you encounter in your life or work. It is a skill that allows you to …
Step-by-Step Calculator - Symbolab
Symbolab is the best step by step calculator for a wide range of math problems, from basic arithmetic to advanced calculus and linear …
SOLVING | English meaning - Cambridge Dictionary
SOLVING definition: 1. present participle of solve 2. to find an answer to a problem: . Learn more.
Step-by-Step Math Problem Solver
QuickMath allows students to get instant solutions to all kinds of math problems, from algebra and equation solving right through to calculus and matrices.
What is Problem Solving? Steps, Process & Techniques | ASQ
WHAT IS PROBLEM SOLVING? Solving a problem depends on correctly identifying its cause so the best solution can be selected and implemented for sustained results. Problems may affect …
What is Problem Solving? (Steps, Techniques, Examples) - Status.net
Problem solving is the process of finding solutions to obstacles or challenges you encounter in your life or work. It is a skill that allows you to tackle complex situations, adapt to changes, and …
Step-by-Step Calculator - Symbolab
Symbolab is the best step by step calculator for a wide range of math problems, from basic arithmetic to advanced calculus and linear algebra. It shows you the solution, graph, detailed …
SOLVING | English meaning - Cambridge Dictionary
SOLVING definition: 1. present participle of solve 2. to find an answer to a problem: . Learn more.
