偏微分方程partialdiffierentialequation

1、Orthogonal Function Expansion 正交函數(shù)展開(kāi),Introduction of the Eigenfunction ExpansionAbstract SpaceFunction SapceLinear Operator and Orthogonal Function,,Introduction -The Eigenfunction Expansion,Consider the equation :,W

2、ith the b.c’s :,The g.s. is where u1,u2 are linearly index. Fucs. And C1,C2 are arb consts.,For b.c’s :,,The condition of nontrivial sol. of c1,c2 to be existed if ? :,,The Euler Col

3、umn,The g.s.,For the b.c.’s : ?,And for nontrivial solution,?,i.e.,So that we obtain the Eigen values,And the corresponding eigenfucs (nontrivial sols) are,According the

4、 analysis, we will have unless the end force P such that:,? Function Space,,,,,,,,,,Rn,Q,Z,N,Inner Product Space,Hilbert Space,Normed Space,Metric Space,由R→Rn(有序性喪失),,,Abstract Space,完備性:每一Cauchy系列均收斂,Topolo

5、gical Space,Banach Space,,Topological Space,Definition,A topological space is a non-empty set E together with a family of subsets of E satisfying the following axioms:,(3) The intersection

6、 of any finite number of sets in X belongs to X i.e.,(1),(2) The union of any number of sets in X belongs to X i.e.,,Metric Space,Definition,A metric space is a 2-tuple (X,d) where X is a set and d is a metric on X, that

7、 is, a function d : X × X → R, such that,d(x, y) ≥ 0     (non-negativity) d(x, y) = 0   if and only if   x = y     (identity) d(x, y) = d(y, x)     (sy

8、mmetry) d(x, z) ≤ d(x, y) + d(y, z)     (triangle inequality).,,Cauchy Sequence,Definition: Complete space,A sequence (Xn):in a metric space X=(X,d) is said to be Cauchy if for every there is

9、an N=N(e) such that,x is called the limit of (Xn) and we write,or, simply,,for m,n>N ?,Any Cauchy Sequence in X is convergence,; d ?0,Definition: Completeness,,Ball and Sphere,Definition:,Given a point

10、 and a real number r>0, we define three of sets:,(a) (Open ball),(b) (Closed ball),(c)

11、 (Sphere),In all three case, x0 is called the center and r the radius.,Furthermore, the definition immediately implies that,,A mapping from a normed space X into a normed s

12、pace Y is called an operator. A mapping from X into the scalar filed R or C is called a functional.,The set of all biunded linear operator from a given normed space X into a given normed space Y can be made into a normed

13、 space, which is denoted by B(x,y). Similarly, the set of all bounded linear functionals on X becomes a normed space, which is called the dual space X’ of X.,Definition (Open set and closed set):,A subset M of a metric s

14、pace X is said to be open if is contains a ball about each of its points. A subset K of X is said to be closed if its complement (in X ) is open, that is, Kc=X-K is open.,Normed Space,Definition of Normed Space,,Definiti

15、on of Banach Space,Here a norm on a vector space X is a real-value function on X whose value at an is denoted by,Here x and y are arbitrary vector in X and is any scalar,A Banach space is a complete n

16、ormed space.,,A metric d induced by a norm on a normed space X satisfies,Lemma (Translation invariance),(a) d(x+a,y+a)=d(x,y),(b),For all x, y and every scalar,Proof.,,Let X be the vector space of all order

17、ed pairs of real numbers. Show norms on X are defined by,The sphere,In a normed space X is called the unit sphere.,,,,,Unit Sphere in LP,,If a normed space X contains a

18、sequence (en) with the property that every there is a uniquie sequence of scalars (an) such that,Then (en) is called basis for X. series,Which has the sum x is then called the expansion of x,,A i

19、nner product space is a vector space X with an inner product define on X.,Here, the inner product is the mapping of into the scale filed, such that,Inner Product Space,,Hilbert Space,Examples of finite-dimens

20、ional Hilbert spaces include,1. The real numbers    with     the vector dot product of and,2. The complex numbers    with     the vector dot prod

21、uct of and the complex conjugate of .,Hence inner product spaces are normed spaces, and Hilber spaces are Banach spaces.,A Hilbert space is a complete inner product space.,(Norm),(Metric),,Euclidean space Rn,The s

22、pace Rn is a Hilbert space with inner product define by,Where,,Space L2[a,b],Hilbert sequence space l2,With the inner product,The norm,Space lp,The space lp with is not inner product space, hence not a Hi

23、lbert space,,Orthonormal Sets and Sequences,Orthogonality of elements plays a basis role in inner product and Hilbert spaces.The vectors form a basis for R3, so that every has a unique representation.,,Contin

24、uous functions,Let X be the inner product space of all real-valued continuous functions on [0,2π] with inner product defined by,An orthogonal sequence in X is (un), where,Another orthogonal sequence in X is (vn), where,

25、,,Hence an orthonormal sequence sequence is (en),From (vn) we obtain the orthonormal sequence ( ) where,Homework,1. Does d(x,y)=(x-y)2 define a metric on the set of all real numbers?,2. Show that

26、 defines a metric on the set of all real numbers.,3. Let Show that the open interval (a,b) is an incomplete subspace of R, whereas the c

27、losed interval [a,b] is complete.,4. Prove that the eigenfunction and eigenvalue are orthogonalization and real for the the Sturm-Lioville System.,6. For the very special case and , the self

28、-adjoint eigenvalue equation becomes,5. Show the following when linear second-order difference equation is expressed in self-adjoint form:,(a) The Wronskian is equal to constant divided by the initial coeffic

29、ient p,(b) A second solution is given by,Use this obtain a “second” solution of the following,(a ) Legendre’s equation,(b ) Laguerre’s equation,(c ) Hermite’s equation,,Function Space,(A) L2 [a,b] space:,Space of real fu

30、cs. f(x) which is define on [a,b] and square integrable i.e .,In the language of vector space, we say that,“any n linearly indep vectors form a basis in E ”space”. Similarly, in function spaceIt is possible to choose a

31、set of basis function such that any function, satisfying Appropriate condition can be expressed as a linear combination to a basis in L2[a,b]Certainly, any such set of fucs. Must have infinitely many numbers; that is,

32、such a L2[a,b] comprises infinitely many dimensions.,(B) Schwarz Inequality:,Given f(x), g(x) in L2[a,b], Define,Proof:,(C) Linear Dependence, Independence:,Criterion: A set of fucs.,In L2 [a,b] is linear dep.(indep.) i

33、f its,If its Gramian (G) vanishes (does not vanish), where,The proof is the same as in linear vector space.,(D) The orthogonal System,A set of real fucs.,…….is called an orthogonal set of fucs.,In L2[a,b] if these fucs.

34、are define in L2[a,b] if all the integral,exist and are zero for all pairs of distinct,,Properties of Complete System,Theorem: Let f(x), F(x) be defined on L2 [a,b] for which,Then we have,Proof:,Since f+F, and f-F are sq

35、uare integer able, from the completeness relation,?,Theorem:,Every square integer able fnc. f(x) is uniquely determined (except for its value at a finite number of points) by its Fourier series.,Proof:,Suppose there are

36、 two fucs. f(x),g(x) having the identical Fourier series representation,i.e.,Then using we find,?g(x)=f(x) at the pts of continuity of the integrand,?g(x) and f(x) coincide ev

37、erywhere, except possibly at a finite number of pts. of discontinuity,Proof: Since…,And let,We can prove f(x)=g(x) at every point.,?,Theorem:,An continuous fuc. f(x) which is orthogonal to all the fucs. of the comp

38、lete system must be identically zero.,Proof: Since…,Assume x2>x1,Take,Theorem:,The fourier series of every square integer able fuc. f(x) can be integrated term by term. In other words, if,Then,Where x1,x2 are any

39、 points on the inteval [a,b],,The Sturm-Liouville Problem,Self-adjoint Operator,For a linear operator L the analog of a quadratic form for a matrix is the integral,Because of the analogy with the transposed matrix, it is

40、 convenient to define the linear operator,Comparing the integrands,,The operator L is said to be self-adjoint.,As the adjoint operator L. The necessary and sufficient condition that,,The Sturm-Liouville Boundary Value Pr

41、oblem,A differential equation defined on the interval          having the form of,and the boundary conditions,is called as Sturm-Liouville boundary value problem or Sturm-L

42、iouville system, where ,              ; the weighting function r(x)>0 are given functions; a1 , a2 , b1 , b2   are given constants;

43、and the eigenvalue is an unspecified parameter.,,The Regular Sturm-Liouville Equation,It is a special kind of boundary value problem which consists of a second-order homogeneous linear differential equation and linea

44、r homogeneous boundary conditions of the form,where the p, q and r are real and continuous functions such that p has a continuous derivative, and p(x) > 0, r(x) > 0 for all x on a real interval a ? x ? b; and

45、 ? is a parameter independent of x. L is the linear homogeneous differential operator defined by L(y) = [p(x)y´]´+q(x)y.And two supplementary boundary conditions,where A1 , A2 , B1 and B2 are real constant

46、s such that A1 and A2 not both zero and B1 and B2 are not both zero.,A1y(a)+A2y´(a) = 0 B1y(b)+B2y´(b) = 0 .,Definition 1.1 : Consider the Sturm-Liouville problem consisting of the differentail equati

47、on and supplementary conditions. The value of the parameter in for which there exists nontrivial solution of the problem is called the eigenvalue of the problem. The corresponding nontrivial solution,is called th

48、e eigenfunction of the problem. The Sturm-Liouville problem is also called an eigenvalue problem.,,The Nonhomogeneous Sturm-Liouville Problems,And as in regular Sturm-Liouville problems we assume that p, p?, q, and r

49、are continuous on a ? x ? b and p(x) > 0, r(x) > 0 there.We solve the problem by making use of the eigenfunctions of the corresponding homogeneous problem consisting of the differential equation,Consider boun

50、dary value problem consisting of the nonhomogeneous differential equation,L[y] = - [p(x)y´]´+q(x)y = ?w(x)y+f(x),,where ? is a given constant and f is a given function on a a ? x ? b and the boundary cond

51、itions,A1y(a)+A2y´(a)=0 B1y(b)+B2y´(b)=0 .,,The Bessel's Differential Equation,In the Sturm-Liouville Boundary Value Problem, there is an important special case called Bessel's Differential Eq

52、uation which arises in numerous problems, especially in polar and cylindrical coordinates. Bessel's Differential Equation is defined as:             

53、;                                    

54、0;                                    .

55、,where   is a non-negative real number. The solutions of this equation are called Bessel Functions of order n . Although the order n can be any real number, the scope of this section is limited to non-neg

56、ative integers, i.e.,             , unless specified otherwise. Since Bessel's differential equation is a second order ordinary differential equation, t

57、wo sets of functions, the Bessel function of the first kind Jn(x) and the Bessel function of the second kind (also known as the Weber Function) Yn(x) , are needed to form the general solution:,,Five Approaches,The Bess

58、el functions are introduced here by means of a generating function. Other approaches are possible. Listing the various possibilities, we have     .,Gram-Schmidt Orthogonalization,We now demand th

59、at each solution be multiplied by,We star with n=0, letting,Then normalize,The presence of the new un(x) will guarantee linear independence.,Fro n=1, let,This demand of orthogonality leads to,As is normalized

60、 to unity, we have,Fixing the value of a10. Normalizing, we have,,We demand that be orthogonal to,Where,,The equation can be replaced by,And aij becomes,The coefficients aij are given by,If some order normaliza

61、tion is selected,Orthogonal polynomial Generated by Gram-Schmidt Orthogonalization of,,2. Series solution of Bessel’s differential equation,Using y’ for dy/dx and for d2y/dx2 . Again, assuming a solution of the for

62、m,Inserting these coefficients in our assumed series solution, we have,Inserting these coefficients in our assumed series solution, we have,With the result that…..,3. Generating  function,,Expanding this function

63、in a Laurent series, we obtain,It is instructive to compare.,The coefficient of tn, Jn(x), is defined to be Bessel function of the first kind of integral order n. Expanding the exponential, we have a product of Maclaurin

64、 series in xt/2 and –x/2t, respectively.,,,The coefficient tn is then,For a given s we get tn(n>=0) from r=n+s;,Bessel function J0(x), J1(x) and J2(x),,4. Contour integral: Some writers prefer to start with contour i

65、ntegral definitions of the Hankel function, and develop the Bessel function Jv(x) from the Hankel functions.,The integral representation,,(Schlsefli integral),may easily be established as a Cauchy integral for

66、 v=n, that is , an integer. [Recognizing that the numerator is the generating function and integrating around the origin],,Cut line,,,5. Direct solution of physical problems, Fraunhofer diffraction with a circular

67、 aperture illusterates this. Incidentally, can be treated by series expansion if desired. Feynman develop Bessel function from a consideration of cavity resonators.,The parameter B us given by,In the theory of di

68、ffraction through a circular aperture we encounter the integral,Feynman develop Bessel function from a consideration of cavity resonators. (Homework 1),,The intensity of the light in the diffraction pattern is proportio

69、nal to Ф2 and,Fro green light Hence, if a=0.5 cm,(2) Using only the generating function,Explicit series form Jn(x), shoe that Jn(x) has odd or even parity according to whether n is odd or even,

70、this,(3) Show by direct differentiation that,Satisfies the two recurrence relations,And bessel‘s differential equation,Homework,(4) Show that,Thus generating modified Bessel function In(x),(5) The chebyshev polynomials (