# Download Elements of Partial Differential Equations by Ian Sneddon PDF for Free

## Elements of Partial Differential Equations by Ian Sneddon PDF Free Download

If you are looking for a book that introduces the elements of partial differential equations in a clear and concise way, you might want to check out Elements of Partial Differential Equations by Ian Sneddon. This book is a classic in the field of applied mathematics, and it has been widely used by students and researchers for over 50 years. In this article, we will give you an overview of the book, its highlights, and its pros and cons. We will also provide you with a link to download the PDF version of the book for free.

## elements of partial differential equations by ian sneddon pdf free download

## Introduction

Before we dive into the details of the book, let us first review some basic concepts and definitions related to partial differential equations.

### What are partial differential equations?

A partial differential equation (PDE) is an equation that involves partial derivatives of an unknown function with respect to two or more independent variables. For example, the heat equation

$$\frac\partial u\partial t = k \frac\partial^2 u\partial x^2$$ is a PDE that describes how the temperature $u$ changes over time $t$ and space $x$ in a one-dimensional rod with thermal conductivity $k$. PDEs can also have higher dimensions, such as the wave equation

$$\frac\partial^2 u\partial t^2 = c^2 \left( \frac\partial^2 u\partial x^2 + \frac\partial^2 u\partial y^2 + \frac\partial^2 u\partial z^2 \right)$$ that models the propagation of waves with speed $c$ in a three-dimensional medium.

### Why are they important?

PDEs are important because they arise naturally in many fields of science and engineering, such as physics, chemistry, biology, economics, and more. They can be used to describe phenomena such as heat transfer, fluid flow, sound waves, electromagnetism, quantum mechanics, population dynamics, stock prices, and so on. Therefore, understanding PDEs is essential for solving real-world problems and advancing scientific knowledge.

### How to solve them?

Solving PDEs is not an easy task, as they often involve complex mathematical techniques and numerical methods. Depending on the type and form of the PDE, different approaches can be used, such as:

Analytical methods: These methods aim to find exact or approximate solutions of PDEs using mathematical tools such as calculus, algebra, complex analysis, Fourier analysis, etc. Some examples of analytical methods are separation of variables, method of characteristics, Laplace transform, Green's function, etc.

Numerical methods: These methods use computers to approximate solutions of PDEs by discretizing the domain and applying iterative or direct algorithms. Some examples of numerical methods are finite difference, finite element, finite volume, spectral, etc.

Qualitative methods: These methods focus on the properties and behavior of solutions of PDEs without finding explicit formulas. Some examples of qualitative methods are existence and uniqueness, stability and convergence, maximum principle, energy method, etc.

In this article, we will mainly focus on the analytical methods, as they are the main topic of the book by Ian Sneddon.

## Overview of the book

Now that we have a general idea of what PDEs are and why they matter, let us take a closer look at the book Elements of Partial Differential Equations by Ian Sneddon.

### Author and background

The author of the book is Ian Naismith Sneddon (1919-2000), a Scottish mathematician and professor at the University of Glasgow. He was an expert in applied mathematics, especially in the fields of PDEs, integral equations, and special functions. He wrote several books and papers on these topics, and he was also involved in various scientific committees and societies. He received many honors and awards for his contributions to mathematics and education.

The book was first published in 1957 by McGraw-Hill as part of the International Series in Pure and Applied Mathematics. It was based on the courses given by Sneddon to undergraduate and graduate students at the University of Glasgow and the University College of North Staffordshire. The book has been reprinted several times by different publishers, such as Dover Publications in 2006.

### Aim and scope

The aim of the book is to present the elements of the theory of PDEs in a form suitable for the use of students and research workers whose main interest in the subject lies in finding solutions of particular equations rather than in the general theory. The book covers both classical and modern methods of solving PDEs, with an emphasis on physical applications and examples. The book assumes that the reader has a basic knowledge of calculus, complex analysis, and ordinary differential equations.

The book consists of six chapters, each divided into several sections. The chapters are:

First-order equations: This chapter introduces the concept of a characteristic curve and shows how to solve some important types of first-order PDEs, such as linear equations, quasilinear equations, nonlinear equations, etc.

Second-order equations: This chapter discusses the classification of second-order PDEs into three types: elliptic, parabolic, and hyperbolic. It also explains how to use the method of separation of variables to solve some canonical equations, such as Laplace's equation, heat equation, wave equation, etc.

Fourier series and integrals: This chapter reviews the theory of Fourier series and integrals and shows how they can be used to represent solutions of PDEs with periodic or non-periodic boundary conditions. It also introduces some special functions that arise from Fourier expansions, such as Bessel functions, Legendre polynomials, etc.

Boundary value problems and eigenfunctions: This chapter deals with the problem of finding solutions of PDEs that satisfy certain conditions on the boundary of the domain. It explains how to use eigenfunction expansions to construct solutions in terms of orthogonal functions that satisfy homogeneous boundary conditions. It also discusses some properties and applications of eigenvalues and eigenfunctions.

Green's functions and integral equations: This chapter introduces the concept of a Green's function as a solution of a PDE with a delta function source term. It shows how to use Green's functions to solve boundary value problems with non-homogeneous boundary conditions or source terms. It also relates Green's functions to integral equations and shows how to solve them using various techniques.

Some special topics: This chapter covers some advanced topics that are not usually included in introductory courses on PDEs. These topics are: conformal mapping, potential theory, variational methods, asymptotic methods, etc.

## Highlights of the book

In this section, we will highlight some key features and examples from each chapter of the book that illustrate its strengths and usefulness.

### First-order equations and characteristics

One of the main advantages of this chapter is that it provides a geometric interpretation of first-order PDEs using characteristic curves. A characteristic curve is a curve along which a solution or its derivative is constant or satisfies a simple relation. For example,

```html

```html u}\partial x + \frac\partial u\partial y = 0$$ has characteristic curves given by $$y = x + c$$ where $c$ is a constant. Along these curves, the solution $u$ is constant. Therefore, the general solution of this equation is $$u = f(x - y)$$ where $f$ is an arbitrary function of one variable. The characteristic curves and the solution surface are shown in Figure 1.

Figure 1: Characteristic curves and solution surface of the equation $$\frac\partial u\partial x + \frac\partial u\partial y = 0$$

The method of characteristics can also be used to solve more complicated types of first-order PDEs, such as quasilinear equations, nonlinear equations, and systems of equations. The book by Sneddon provides many examples and exercises to illustrate these methods.

### Second-order equations and separation of variables

This chapter introduces one of the most powerful and widely used techniques for solving second-order PDEs: the method of separation of variables. This method assumes that the solution can be written as a product of functions, each depending on one independent variable only. For example,

$$u(x,y) = X(x)Y(y)$$ By substituting this form into the PDE and dividing by $u$, we obtain an equation that involves only one independent variable on each side. For example, for the heat equation

$$\frac\partial u\partial t = k \frac\partial^2 u\partial x^2$$ we get

$$\frac1k\frac1Y\fracdYdt = \frac1X\fracd^2Xdx^2$$ Since the left-hand side depends only on $t$ and the right-hand side depends only on $x$, they must both be equal to a constant, say $-\lambda$. This gives us two ordinary differential equations to solve:

$$\fracdYdt + k \lambda Y = 0 \quad \textand \quad \fracd^2Xdx^2 + \lambda X = 0$$ The solutions of these equations depend on the value and sign of $\lambda$. For example, if $\lambda > 0$, we have

$$Y(t) = A e^-k \lambda t \quad \textand \quad X(x) = B \cos(\sqrt\lambda x) + C \sin(\sqrt\lambda x)$$ where $A$, $B$, and $C$ are arbitrary constants. Therefore, a possible solution of the heat equation is

$$u(x,t) = (B \cos(\sqrt\lambda x) + C \sin(\sqrt\lambda x)) e^-k \lambda t$$ However, this is not the only solution. In fact, there are infinitely many values of $\lambda$ that satisfy the boundary conditions of the problem. For example, if we impose that $u(0,t) = u(L,t) = 0$ for some positive constant $L$, we get that $\lambda$ must be of the form $\lambda_n = (n \pi / L)^2$ for some positive integer $n$. Therefore, we have a family of solutions:

$$u_n(x,t) = (B_n \cos(n \pi x / L) + C_n \sin(n \pi x / L)) e^-k (n \pi / L)^2 t$$ The general solution is then obtained by adding all these solutions together, using the principle of superposition. This gives us

$$u(x,t) = \sum_n=1^\infty (B_n \cos(n \pi x / L) + C_n \sin(n \pi x / L)) e^-k (n \pi / L)^2 t$$ The coefficients $B_n$ and $C_n$ can be determined by using the initial condition of the problem, such as $u(x,0) = f(x)$. This involves using Fourier series and integrals, which are discussed in the next chapter.

### Fourier series and integrals

This chapter reviews the theory of Fourier series and integrals, which are essential tools for solving PDEs with periodic or non-periodic boundary conditions. The main idea is to represent a function as a linear combination of trigonometric functions, such as sines and cosines, or exponential functions, such as complex exponentials. For example, a Fourier series of a function $f(x)$ defined on the interval $[-L,L]$ is given by

$$f(x) = \fraca_02 + \sum_n=1^\infty (a_n \cos(n \pi x / L) + b_n \sin(n \pi x / L))$$ where the coefficients $a_n$ and $b_n$ are given by

$$a_n = \frac1L \int_-L^L f(x) \cos(n \pi x / L) dx \quad \textand \quad b_n = \frac1L \int_-L^L f(x) \sin(n \pi x / L) dx$$ A Fourier integral of a function $f(x)$ defined on the whole real line is given by

$$f(x) = \frac1\sqrt2\pi\int_-\infty^\infty F(k) e^ikx dk$$ where the function $F(k)$ is called the Fourier transform of $f(x)$ and is given by

$$F(k) = \frac1\sqrt2\pi\int_-\infty^\infty f(x) e^-ikx dx$$ The inverse Fourier transform is given by

$$f(x) = \frac1\sqrt2\pi\int_-\infty^\infty F(k) e^ikx dk$$ Fourier series and integrals have many properties and applications that are useful for solving PDEs. For example, they can be used to find solutions of PDEs with non-homogeneous boundary conditions or source terms, as we saw in the previous chapter. They can also be used to simplify the calculations of eigenvalues and eigenfunctions, as we will see in the next chapter. Moreover, they can be used to study the convergence and stability of solutions, as well as their frequency and energy spectra.

### Boundary value problems and eigenfunctions

This chapter deals with the problem of finding solutions of PDEs that satisfy certain conditions on the boundary of the domain. These conditions can be either homogeneous or non-homogeneous, depending on whether they involve the value of the solution or its derivative. For example, for a function $u(x,y)$ defined on a rectangular domain $[0,L] \times [0,H]$, some possible boundary conditions are:

Dirichlet boundary conditions: These specify the value of the solution on the boundary. For example,

$$u(0,y) = u(L,y) = u(x,0) = u(x,H) = 0$$

Neumann boundary conditions: These specify the normal derivative of the solution on the boundary. For example,

```html \frac\partial u\partial y(x,0) = \frac\partial u\partial y(x,H) = 0$$

Mixed boundary conditions: These specify a combination of the value and the normal derivative of the solution on the boundary. For example,

$$u(0,y) = u(L,y) = 0, \quad \frac\partial u\partial y(x,0) = g(x), \quad \frac\partial u\partial y(x,H) = h(x)$$ To solve a boundary value problem, we need to find a function that satisfies both the PDE and the boundary conditions. This can be done by using the method of separation of variables, as we saw in the previous chapter. However, this method requires that we find the eigenvalues and eigenfunctions of a certain ordinary differential equation that arises from separating the variables. For example, for the Laplace equation

$$\frac\partial^2 u\partial x^2 + \frac\partial^2 u\partial y^2 = 0$$ with Dirichlet boundary conditions, we assumed that $u(x,y) = X(x)Y(y)$ and obtained the equation

$$X'' + \lambda X = 0$$ where $\lambda$ is a constant. To find $\lambda$ and $X$, we need to impose the boundary conditions on $X$, such as $X(0) = X(L) = 0$. This leads to an eigenvalue problem of the form

$$X'' + \lambda X = 0, \quad X(0) = X(L) = 0$$ The eigenvalues and eigenfunctions of this problem are given by

$$\lambda_n = \left(\fracn \piL\right)^2, \quad X_n(x) = \sin\left(\fracn \pi xL\right), \quad n = 1,2,3,\dots$$ The book by Sneddon explains how to find these eigenvalues and eigenfunctions using various methods, such as characteristic equations, trigonometric identities, orthogonality relations, etc. It also shows how to use these eigenfunctions to construct solutions of PDEs using Fourier series and integrals.

### Green's functions and integral equations

This chapter introduces another powerful technique for solving boundary value problems: the method of Green's functions. A Green's function is a solution of a PDE with a delta function source term. For example, for the Laplace equation with Dirichlet boundary conditions on a domain $\Omega$, a Green's function is a function $G(x,y)$ that satisfies

$$\nabla^2 G(x,y) = -\delta(x-y), \quad G(x,y) = 0 \quad \textfor \quad x \in \partial \Omega$$ where $\nabla^2$ is the Laplacian operator, $\delta$ is the Dirac delta function, and $\partial \Omega$ is the boundary of $\Omega$. The delta function has the property that it is zero everywhere except at the origin, where it is infinite. It also has the property that it integrates to one over any domain that contains the origin. For example,

$$\int_-\infty^\infty \delta(x) dx = 1$$ The idea behind Green's functions is that they can be used to construct solutions of PDEs with non-homogeneous boundary conditions or source terms by using superposition and convolution. For example, if we want to solve the Laplace equation with a source term $f(x)$ and Dirichlet boundary conditions on $\Omega$, we can write

$$u(x) = -\int_\Omega G(x,y) f(y) dy + \int_\partial \Omega g(y) \frac\partial G\partial n(x,y) ds(y)$$ where $g(y)$ is the boundary condition and $\frac\partial G\partial n$ is the normal derivative of $G$. This formula can be derived by using Green's identity and some properties of the delta function. The book by Sneddon provides many examples and exercises to illustrate how to find and use Green's functions for various PDEs and domains.

Green's functions are also related to integral equations, which are equations that involve unknown functions under integrals. For example, a Fredholm integral equation of the second kind is of the form

$$u(x) = f(x) + \lambda \int_a^b K(x,y) u(y) dy$$ where $f(x)$ and $K(x,y)$ are given functions, and $\lambda$ is a constant. This equation can be solved by using various techniques, such as iteration, kernel expansion, eigenfunction expansion, etc. The book by Sneddon explains how to solve integral equations using these techniques and how to relate them to PDEs using Green's functions.

### Some special topics

This chapter covers some advanced topics that are not usually included in introductory courses on PDEs. These topics are:

Conformal mapping: This is a method of transforming a complex function from one domain to another while preserving angles and shapes. It can be used to solve PDEs on complicated domains by mapping them to simpler domains, such as circles or rectangles.

Potential theory: This is a branch of mathematics that studies harmonic functions, which are solutions of the Laplace equation. It can be used to model physical phenomena such as electrostatics, fluid flow, heat conduction, etc.

Variational methods: These are methods of finding solutions of PDEs by minimizing or maximizing a certain functional, which is a function of functions. For example, the principle of least action states that the motion of a particle is determined by minimizing the action functional, which is the integral of the Lagrangian function.

Asymptotic methods: These are methods of finding approximate solutions of PDEs by using expansions in terms of small or large parameters. For example, the method of perturbation theory uses a series expansion in terms of a small parameter that measures the deviation from an exact solution.

The book by Sneddon gives an introduction to these topics and shows how they can be applied to solve some interesting PDEs.

## Conclusion

In this article, we have given you an overview of the book Elements of Partial Differential Equations by Ian Sneddon. We have summarized its main contents, highlights, and pros and cons. We hope that this article has sparked your interest in learning more about PDEs and their applications.

### Summary and main points

The book Elements of Partial Differential Equations by Ian Sneddon is a classic in the field of applied mathematics, and it has been widely used by students and researchers for over 50 years.

The book introduces the elements of the theory of PDEs in a form suitable for th