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1997 | Buch

Introduction Kapitel lesen Erstes Kapitel lesen

Partial Differential Equations through Examples and Exercises

verfasst von: Endre Pap, Arpad Takači, Djurdjica Takači

Verlag: Springer Netherlands

Buchreihe : Kluwer Texts in the Mathematical Sciences

Enthalten in: Professional Book Archive

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Über dieses Buch

The book Partial Differential Equations through Examples and Exercises has evolved from the lectures and exercises that the authors have given for more than fifteen years, mostly for mathematics, computer science, physics and chemistry students. By our best knowledge, the book is a first attempt to present the rather complex subject of partial differential equations (PDEs for short) through active reader-participation. Thus this book is a combination of theory and examples. In the theory of PDEs, on one hand, one has an interplay of several mathematical disciplines, including the theories of analytical functions, harmonic analysis, ODEs, topology and last, but not least, functional analysis, while on the other hand there are various methods, tools and approaches. In view of that, the exposition of new notions and methods in our book is "step by step". A minimal amount of expository theory is included at the beginning of each section Preliminaries with maximum emphasis placed on well selected examples and exercises capturing the essence of the material. Actually, we have divided the problems into two classes termed Examples and Exercises (often containing proofs of the statements from Preliminaries). The examples contain complete solutions, and also serve as a model for solving similar problems, given in the exercises. The readers are left to find the solution in the exercises; the answers, and occasionally, some hints, are still given. The book is implicitly divided in two parts, classical and abstract.

Inhaltsverzeichnis

Frontmatter
Chapter 1. Introduction
Abstract
A partial differential equation (briefly PDE) for a function u = u(x1,… ,xn) is a relation of the form
$$F\left( {{x_1}, \ldots ,{x_n},{{\partial u} \over {\partial {x_1}}}, \ldots ,{{\partial u} \over {\partial {x_n}}},{{{\partial ^2}u} \over {\partial x_1^2}}, \ldots ,{{{\partial ^m}u} \over {\partial x_n^m}}} \right) = 0,$$
where F is a given function of the independent variables (x1,… ,xn) n > 1, and of the (unknown) function u of a finite number of its partial derivatives. The order of the PDE (1.1) is the order of the highest derivative that occurs.
Endre Pap, Arpad Takači, Djurdjica Takači
Chapter 2. First Order PDEs
Abstract
A first order PDE is an equation of the form {Eq 17_2.1} where F is a given and u = u(x1,x2,…,xn) is the unknown function of n independent variables x 1 ,x 2 ,…,x n . A solution of (2.1) is called integral surface.
Endre Pap, Arpad Takači, Djurdjica Takači
Chapter 3. Classification of the Second Order PDEs
Abstract
The quasi-linear second order PDE on some region Q ⊂ R2 is given by
$$A\left( {x,y} \right){{{\partial ^2}u} \over {\partial {x^2}}} + 2B\left( {x,y} \right){{{\partial ^2}u} \over {\partial {y^2}}} = F\left( {x,y,u,{{\partial u} \over {\partial x}},{{\partial u} \over {\partial y}}} \right).$$
.
Endre Pap, Arpad Takači, Djurdjica Takači
Chapter 4. Hyperbolic Equations
Abstract
The one-dimensional wave equation given on the set Ω2 = {(x, t) x ∈ R, t > 0} {Eq 71_4.1} for a > 0, with conditions {Eq 71_4.2} where f ∈ C2(R), g ∈ Cl(R) and F ∈ C2(Ω2) are given functions, is called the Cauchy problem for one dimensional wave equation. If F = 0, then we are dealing with a homogeneous wave equation, otherwise it is a nonhomogeneous wave equation.
Endre Pap, Arpad Takači, Djurdjica Takači
Chapter 5. Elliptic Equations
Abstract
We consider the equation given by {Eq 143_1} where aij = aji i,j = 1,2,… n, aij, bij,c ∈ C(Q) and L is an elliptic operator, i.e., {Eq 143_2} for all (p1,P2,…,Pn) ∈ Rn} \ {(0,0,…,0)} and for every x ∈ Q, where Q is a bounded region of Rn with a boundary ∂Q, which has piecewise continuous normal at each point.
Endre Pap, Arpad Takači, Djurdjica Takači
Chapter 6. Parabolic Equations
Abstract
The problem for heat equation given as {Eq 183_6.1} with condition {Eq 183_6.2} where a > 0, f is bounded and f ∈ C(Rn), F ∈ C2(Rn x [0,∞)), and all its second order partial derivatives are bounded on every set of the form Rn x [0,T] is called the Cauchy problem.
Endre Pap, Arpad Takači, Djurdjica Takači
Chapter 7. Numerical Methods
Abstract
The most commonly used method for obtaining the approximate solutions of certain partial differential equations is the finite differences method. In order to employ this method we replace the continuous independent variables x,y, z, t,…, by a finite numbers of discrete variablesxi, yj, zm, tn,}…, namely we determine suitable mesh points
$$\begin{array}{*{20}{c}} {{x_i} = {x_0} + ih,}&{i = 0, \pm 1, \pm 2, \ldots ,} \\ {{y_j} = {y_0} + jk,}&{j = 0, \pm 1, \pm 2, \ldots ,} \end{array}$$
and so on. Replacing each of derivatives by a suitable difference quotient, a difference equation for i,j = 0, ±1,±2,…, is obtained. It represents a system of algebraic equations, whose solutions can be treated as the approximate solutions of the considered problem at the mesh points.
Endre Pap, Arpad Takači, Djurdjica Takači
Chapter 8. Lebesgue’s Integral and the Fourier Transform
Abstract
Let A ⊂ R n be a (Lebesgue) measurable set. For a measurable function f : A → [0, ∞] its Lebesgue integral on a measurable subset E of A is given by
$$\int\limits_E {f\left( x \right)dx = \sup } \int\limits_E {s\left( x \right)dx,} $$
where the supremum is taken over all simple functions s, \(s = \sum\nolimits_{i = 1}^n {{a_i}\chi {E_i},} \) which satisfy the inequality 0 ≤ s ≤ f (we are using the convention 0 · ∞ = 0) and Χ E i . are the characteristic functions of sets E i ∈ ∑ (i = 1,…,n) and ∪E i = E, where ∑ is the σ-algebra of measurable sets.
Endre Pap, Arpad Takači, Djurdjica Takači
Chapter 9. Generalized Derivative and Sobolev Spaces
Abstract
Let α = (α1, ⋧, α n ) ∈ Z n + and Q ⊂ R n is a region.
Endre Pap, Arpad Takači, Djurdjica Takači
Chapter 10. Some Elements from Functional Analysis
Abstract
A Hilbert space is a vector space (real or complex) H endowed with a scalar product (· | ·), i.e., (real or complex) valued bilinear functional defined on H × H with the properties for all x,y,z ∈ H :
$$\matrix{ {\left( {h1} \right)\left( {ax\left| y \right.} \right) = a\left( {x\left| y \right.} \right)for\,every\,scalar\,a;} \hfill \cr {\left( {h2} \right)\left( {x + y\left| z \right.} \right) = \left( {x\left| z \right.} \right) + \left( {y\left| z \right.} \right);} \hfill \cr {\left( {h3} \right)\left( {x\left| y \right.} \right) = \overline {\left( {y\left| x \right.} \right)} ;} \hfill \cr {\left( {h4} \right)\left( {x\left| x \right.} \right) > 0\,for\,x \ne 0;} \hfill \cr {\left( {h5} \right)\left( {x\left| x \right.} \right) = 0\,for\,x = 0;} \hfill \cr }$$
and H is a complete metric space with respect to the metric ∥ x − y∥ induced by the norm \(\left\| x \right\| = \sqrt {\left( {x\left| x \right.} \right)} \).
Endre Pap, Arpad Takači, Djurdjica Takači
Chapter 11. Functional Analysis Methods in PDEs
Abstract
The equation {Eq 329_1}is uniformly elliptic in the region Q ⊂ R n if there exist constants C 1 > 0 and C 2 > 0 such that {Eq 329_2} where A(x)=[a ij ] n×n
Endre Pap, Arpad Takači, Djurdjica Takači
Chapter 12. Distributions in the theory of PDEs
Abstract
In this chapter 0 denotes an open set in R n .
Endre Pap, Arpad Takači, Djurdjica Takači
Backmatter
Metadaten
Titel
Partial Differential Equations through Examples and Exercises
verfasst von
Endre Pap
Arpad Takači
Djurdjica Takači
Copyright-Jahr
1997
Verlag
Springer Netherlands
Electronic ISBN
978-94-011-5574-8
Print ISBN
978-94-010-6349-4
DOI
https://doi.org/10.1007/978-94-011-5574-8