Integral Equations

More documents

Recommendations

Info

Contents1 Preface 42 Notion and examples of integral equations (IEs). Fredholm IEs of the firstand second kind 52.1 Primitive function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 Examples of solution to integral equations and ordinary differential equations 64 Reduction of ODEs to the Volterra IE and proof of the unique solvabilityusing the contraction mapping 84.1 Reduction of ODEs to the Volterra IE . . . . . . . . . . . . . . . . . . . . . . . 84.2 Principle of contraction mappings . . . . . . . . . . . . . . . . . . . . . . . . . . 105 Unique solvability of the Fredholm IE of the 2nd kind using the contractionmapping. Neumann series 125.1 Linear Fredholm IE of the 2nd kind . . . . . . . . . . . . . . . . . . . . . . . . . 125.2 Nonlinear Fredholm IE of the 2nd kind . . . . . . . . . . . . . . . . . . . . . . . 135.3 Linear Volterra IE of the 2nd kind . . . . . . . . . . . . . . . . . . . . . . . . . . 145.4 Neumann series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145.5 Resolvent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 IEs as linear operator equations. Fundamental properties of completely continuousoperators 176.1 Banach spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176.2 IEs and completely continuous linear operators . . . . . . . . . . . . . . . . . . . 186.3 Properties of completely continuous operators . . . . . . . . . . . . . . . . . . . 207 Elements of the spectral theory of linear operators. Spectrum of a completelycontinuous operator 218 Linear operator equations: the Fredholm theory 238.1 The Fredholm theory in Banach space . . . . . . . . . . . . . . . . . . . . . . . 238.2 Fredholm theorems for linear integral operators and the Fredholm resolvent . . . 269 IEs with degenerate and separable kernels 279.1 Solution of IEs with degenerate and separable kernels . . . . . . . . . . . . . . . 279.2 IEs with degenerate kernels and approximate solution of IEs . . . . . . . . . . . 319.3 Fredholm’s resolvent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3210 Hilbert spaces. Self-adjoint operators. Linear operator equations with completelycontinuous operators in Hilbert spaces 3410.1 Hilbert space. Selfadjoint operators . . . . . . . . . . . . . . . . . . . . . . . . . 3410.2 Completely continuous integral operators in Hilbert space . . . . . . . . . . . . . 3510.3 Selfadjoint operators in Hilbert space . . . . . . . . . . . . . . . . . . . . . . . . 3710.4 The Fredholm theory in Hilbert space . . . . . . . . . . . . . . . . . . . . . . . . 372
10.5 The Hilbert–Schmidt theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4111 IEs with symmetric kernels. Hilbert–Schmidt theory 4211.1 Selfadjoint integral operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4211.2 Hilbert–Schmidt theorem for integral operators . . . . . . . . . . . . . . . . . . 4512 Harmonic functions and Green’s formulas 4712.1 Green’s formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4712.2 Properties of harmonic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 4913 Boundary value problems 5014 Potentials with logarithmic kernels 5114.1 Properties of potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5214.2 Generalized potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5615 Reduction of boundary value problems to integral equations 5616 Functional spaces and Chebyshev polynomials 5916.1 Chebyshev polynomials of the first and second kind . . . . . . . . . . . . . . . . 5916.2 Fourier–Chebyshev series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6017 Solution to integral equations with a logarithmic singulatity of the kernel 6317.1 <strong>Integral</strong> equations with a logarithmic singulatity of the kernel . . . . . . . . . . 6317.2 Solution via Fourier–Chebyshev series . . . . . . . . . . . . . . . . . . . . . . . . 6418 Solution to singular integral equations 6618.1 Singular integral equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6618.2 Solution via Fourier–Chebyshev series . . . . . . . . . . . . . . . . . . . . . . . . 6619 Matrix representation 7119.1 Matrix representation of an operator in the Hilbert space . . . . . . . . . . . . . 7119.2 Summation operators in the spaces of sequences . . . . . . . . . . . . . . . . . . 7220 Matrix representation of logarithmic integral operators 7420.1 Solution to integral equations with a logarithmic singularity of the kernel . . . . 7621 Galerkin methods and basis of Chebyshev polynomials 7921.1 Numerical solution of logarithmic integral equation by the Galerkin method . . . 813
Page 1: Karlstad UniversityDivision for Eng
Page 5 and 6: 2 Notion and examples of integral e
Page 7 and 8: Subtracting termwise we obtain an o
Page 9 and 10: with the initial conditiony(x 0 ) =
Page 11 and 12: Example 7 The space C[a, b] of cont
Page 13 and 14: where the kernel K(x, y) is a conti
Page 15 and 16: and so on, obtaining for the (n + 1
Page 17 and 18: Thus the common term of the series
Page 19 and 20: Proof. To prove the theorem, it is
Page 21 and 22: Theorem 9 (Superposition principle)
Page 23 and 24: A, being completely continuous in t
Page 25 and 26: These conditions are equivalent to
Page 27 and 28: In particular, if λ is a regular v
Page 29 and 30: so thatanda 11 =a 12 =a 21 =a 22 =f
Page 31 and 32: Therefore the solution isφ(x) = f(
Page 33 and 34: ∫ bc n = . . .a∫ ba∣K(y 1 , y
Page 35 and 36: is called the Euclidian space.In a
Page 37 and 38: Now note that K N (x, y) in (189) i
Page 39 and 40: Proof. Assume that y n ∈ Im T and
Page 41 and 42: We see that, on the one hand, vecto
Page 43 and 44: Multiply equality (213) by ¯φ 0 a
Page 45 and 46: Replacing x by y abd vice versa, we
Page 47 and 48: 12 Harmonic functions and Green’s
Page 49 and 50: Since on Ω(x, r) we havegrad y Φ(
Page 51 and 52: From Green’s first formula applie
Page 53 and 54:
which proves the continuity of the
Page 55 and 56:
where B 1 is a constant taken accor
Page 57 and 58:
Theorem 36 The operators I − K 0
Page 59 and 60:
Theorem 41 Let D ∈ R 2 be a domai
Page 61 and 62:
n = 2 π∫1−1f(x)U n (x) √ 1
Page 63 and 64:
A similar statement is valid for th
Page 65 and 66:
In some cases one can determine the
Page 67 and 68:
Coefficient a 0 vanishes because (s
Page 69 and 70:
Expand the right-hand side in the C
Page 71 and 72:
We see that the equation has the un
Page 73 and 74:
or, in other notation,andorL = {l n
Page 75 and 76:
where T n (x) = cos(n arccos x) are
Page 77 and 78:
and the integral equationwhereΦ N
Page 79 and 80:
Let us expand the functiong(s) = 1l
Page 81 and 82:
Assume that X n and Y n are the spa
show all

Integral Equations

Create successful ePaper yourself

Delete template?

Save as template?