The method of least squares can also be applied for the approximate construction of char-acteristic values and eigenfunctions of the kernel Kx, s, similarly to the way in which it can be
Trang 1This requirement leads to the algebraic system of equations
∂I
∂A j =0, j=1, , n, (16.4.9.5) and hence, on the basis of (16.4.9.4), by differentiating with respect to the parameters
A1, , A nunder the integral sign, we obtain
1
2
∂I
∂A j =
a ψ j (x, λ)
ψ0(x, λ) +
n
i=1
A i ψ i (x, λ)
dx=0, j =1, , n (16.4.9.6)
Using the notation
c ij (λ) =
a ψ i (x, λ)ψ j (x, λ) dx, (16.4.9.7)
we can rewrite system (16.4.9.6) in the form of the normal system of the method of least
squares:
c11(λ)A1+ c12(λ)A2+· · · + c1n (λ)A n = –c10(λ),
c21(λ)A1+ c22(λ)A2+· · · + c2n (λ)A n = –c20(λ),
⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅
c n1(λ)A1+ c n2(λ)A2+· · · + c nn (λ)A n = –c n0(λ).
(16.4.9.8)
Note that if ϕ0(x)≡ 0, then ψ0(x) = –f (x) Moreover, since c ij (λ) = c ji (λ), the matrix of
system (16.4.9.8) is symmetric
16.4.9-2 Construction of eigenfunctions
The method of least squares can also be applied for the approximate construction of
char-acteristic values and eigenfunctions of the kernel K(x, s), similarly to the way in which it can be done in the collocation method Namely, by setting f (x)≡ 0and ϕ0(x)≡ 0, which
implies ψ0(x) ≡ 0, we determine approximate values of the characteristic values from the algebraic equation
det[c ij (λ)] =0 After this, approximate eigenfunctions can be found from the homogeneous system of the
form (16.4.9.8), where, instead of λ, the corresponding approximate value is substituted.
16.4.10 Bubnov–Galerkin Method
16.4.10-1 Description of the method
Let
ε [y(x)]≡y (x) – λ
a K (x, t)y(t) dt – f (x) =0 (16.4.10.1) Similarly to the above reasoning, we seek an approximate solution of equation (16.4.10.1)
in the form of a finite sum
Y n (x) = f (x) +
n
i=1
A i ϕ i (x), i=1, , n, (16.4.10.2)
Trang 2where the ϕ i (x) (i =1, , n) are some given linearly independent functions (coordinate
functions) and A1, , A n are indeterminate coefficients On substituting the expres-sion (16.4.10.2) into the left-hand side of equation (16.4.10.1), we obtain the residual
ε [Y n (x)] =
n
j=1
A j
ϕ j (x) – λ
a K (x, t)ϕ j (t) dt
– λ
a K (x, t)f (t) dt. (16.4.10.3)
According to the Bubnov–Galerkin method, the coefficients A i (i = 1, , n) are
defined from the condition that the residual is orthogonal to all coordinate functions
ϕ1(x), , ϕ n (x) This gives the system of equations
a ε [Y n (x)]ϕ i (x) dx =0, i=1, , n,
or, by virtue of (16.4.10.3),
n
j=1
(α ij – λβ ij )A j = λγ i i=1, , n, (16.4.10.4)
where
α ij =
a ϕ i (x)ϕ j (x) dx, β ij =
a
a K (x, t)ϕ i (x)ϕ j (t) dt dx,
γ i =
a
a K (x, t)ϕ i (x)f (t) dt dx, i , j =1, , n.
If the determinant of system (16.4.10.4)
D (λ) = det[α ij – λβ ij]
is nonzero, then this system uniquely determines the coefficients A1, , A n In this case, formula (16.4.10.2) gives an approximate solution of the integral equation (16.4.10.1)
16.4.10-2 Characteristic values
The equation D(λ) =0gives approximate characteristic values 2λ1, , 2 λ nof the integral
equation On finding nonzero solutions of the homogeneous linear system
n
j=1
(α ij – 2λ k β ij)2A(j k) =0, i=1, , n,
we can construct approximate eigenfunctions 2Y(k)
n (x) corresponding to characteristic
val-ues 2λ k:
2
Y(k)
n (x) =
n
i=1
2
A(k)
i ϕ (x).
It can be shown that the Bubnov–Galerkin method is equivalent to the replacement of
the kernel K(x, t) by some degenerate kernel K(n) (x, t) Therefore, for the approximate solution Y n (x) we have an error estimate similar to that presented in Subsection 16.4.7-2.
Trang 3Example Let us find the first two characteristic values of the integral equation
ε[y(x)]≡y(x) – λ
1
0 K(x, t)y(t) dt =0 , where
K(x, t) =
t for t≤x,
On the basis of (16.4.10.5), we have
ε[y(x)] = y(x) – λ
x
0 ty(t) dt +
1
x xy(t) dt
.
We set
Y2(x) = A1x + A2x2.
In this case
ε[Y2(x)] = A1x + A2x2– λ1
3A1x3+14A2x4+ x 12A1 +13A2
– 12A1x3+13A2x4
=
= A1 1 –12λ
x+16λx3
+ A2 –13λx + x2+121 λx4
.
On orthogonalizing the residual ε[Y2(x)], we obtain the system
1
0
ε[Y2(x)]x dx =0 ,
1
0 ε[Y2(x)]x2dx= 0 ,
or the following homogeneous system of two algebraic equations with two unknowns:
A1 ( 120 – 48λ) + A2 ( 90 – 35λ) =0 ,
A1( 630 – 245λ) + A2( 504 – 180λ) =0 (16.4 10 6 )
On equating the determinant of system (16.4.10.6) with zero, we obtain the following equation for the characteristic values:
D(λ)≡ 120 – 48λ 90 – 35λ
630 – 245λ 504 – 180λ
= 0 Hence,
λ2– 26 03λ+ 58 15 = 0 (16 4 10 7 ) Equations (16.4.10.7) imply
2λ1 = 2 462 . and 2λ2 = 23 568 .
For comparison we present the exact characteristic values:
λ1=14π2= 2 467 . and λ2= 94π2= 22 206 ., which can be obtained from the solution of the following boundary value problem equivalent to the original equation:
y xx (x) + λy(x) =0 ; y(0 ) = 0 , y x ( 1 ) = 0 Thus, the error of 2λ1is approximately equal to 0 2 % and that of 2λ2, to 6 %.
16.4.11 Quadrature Method
16.4.11-1 General scheme for Fredholm equations of the second kind
In the solution of an integral equation, the reduction to the solution of systems of algebraic equations obtained by replacing the integrals with finite sums is one of the most effective tools The method of quadratures is related to the approximation methods It is widespread
in practice because it is rather universal with respect to the principle of constructing algo-rithms for solving both linear and nonlinear equations
Trang 4Just as in the case of Volterra equations, the method is based on a quadrature formula (see Subsection 16.1.5):
a ϕ (x) dx =
n
j=1
A j ϕ (x j ) + ε n [ϕ], (16.4.11.1)
where the x j are the nodes of the quadrature formula, A j are given coefficients that do not
depend on the function ϕ(x), and ε n [ϕ] is the error of replacement of the integral by the
sum (the truncation error)
If in the Fredholm integral equation of the second kind
y (x) – λ
a K (x, t)y(t) dt = f (x), a≤x≤b, (16.4.11.2)
we set x = x i (i =1, , n), then we obtain the following relation that is the basic formula
for the method under consideration:
y (x i ) – λ
a K (x i , t)y(t) dt = f (x i), i=1, , n. (16.4.11.3) Applying the quadrature formula (16.4.11.1) to the integral in (16.4.11.3), we arrive at the following system of equations:
y (x i ) – λ
n
j=1
A j K (x i , x j )y(x j ) = f (x i ) + λε n [y]. (16.4.11.4)
By neglecting the small term λε n [y] in this formula, we obtain the system of linear algebraic equations for approximate values y i of the solution y(x) at the nodes x1, , x n:
y i – λ
n
j=1
A j K ij y j = f i i=1, , n, (16.4.11.5)
where K ij = K(x i , x j ), f i = f (x i)
The solution of system (16.4.11.5) gives the values y1, , y n, which determine an
approximate solution of the integral equation (16.4.11.2) on the entire interval [a, b] by
interpolation Here for the approximate solution we can take the function obtained by linear
interpolation, i.e., the function that coincides with y i at the points x iand is linear on each
of the intervals [x i , x i+1] Moreover, for an analytic expression of the approximate solution
to the equation, a function
j=1
A j K (x, x j )y j (16.4.11.6)
can be chosen, which also takes the values y1, , y n at the points x1, , x n
Example Consider the equation
y(x) –12
1
0 xty(t) dt = 56x.
Trang 5Let us choose the nodes x1 = 0, x2= 12, x3= 1and calculate the values of the right-hand side f (x) = 56x and of the kernel K(x, t) = xt at these nodes:
f(0 ) = 0 , f 12
= 125, f( 1 ) = 56,
K(0 , 0 ) = 0 , K 0 , 12
= 0 , K(0 , 1 ) = 0 , K 12, 0 = 0 , K 12,12
= 14,
K 12, 1 = 12, K(1 , 0 ) = 0 , K 1 ,12
= 12, K(1 , 1 ) = 1
On applying Simpson’s rule (see Subsection 16.1.5)
1
0
F (x) dx≈ 1
6
F( 0 ) + 4F 12
+ F (1 )
to determine the approximate values y i (i =1 , 2 , 3) of the solution y(x) at the nodes x iwe obtain the system
y1= 0 ,
11
12y2–241y3= 125, –122y2+1112y3= 56,
whose solution is y1 = 0, y2 = 12, y3 = 1 In accordance with the expression (16.4.11.6), the approximate solution can be presented in the form
2y(x) = 5
6x+12 × 1
6 0 + 4 × 1
2 × 1
2x+ 1 × 1 ×x
= x.
We can readily verify that it coincides with the exact solution.
16.4.11-2 Construction of the eigenfunctions
The method of quadratures can also be applied for solutions of homogeneous Fredholm equations of the second kind In this case, system (16.4.11.5) becomes homogeneous
(f i = 0) and has a nontrivial solution only if its determinant D(λ) is equal to zero The algebraic equation D(λ) =0of degree n for λ makes it possible to find the roots 2 λ1, , 2 λ n,
which are approximate values of n characteristic values of the equation The substitution
of each value 2λ k (k =1, , n) into (16.4.11.5) for f i ≡ 0leads to the system of equations
y(k)
i – 2λ k
n
j=1
A j K ij y(k)
j =0, i=1, , n,
whose nonzero solutions y(i k) make it possible to obtain approximate expressions for the eigenfunctions of the integral equation:
2y k (x) = 2 λ k
n
j=1
A j K (x, x j )y j(k)
If λ differs from each of the roots 2 λ k, then the nonhomogeneous system of linear
algebraic equations (16.4.11.5) has a unique solution In the same case, the homogeneous system of equations (16.4.11.5) has only the trivial solution
16.4.12 Systems of Fredholm Integral Equations of the Second
Kind
16.4.12-1 Some remarks
A system of Fredholm integral equations of the second kind has the form
y i (x) – λ
n
j=1
a K ij (x, t)y j (t) dt = f i (x), a≤x≤b, i=1, , n. (16.4.12.1)
Trang 6Assume that the kernels K ij (x, t) are continuous or square integrable on the square S =
{a≤x≤b , a ≤t≤ b} and the right-hand sides f i (x) are continuous or square integrable
on [a, b] We also assume that the functions y i (x) to be defined are continuous or square integrable on [a, b] as well The theory developed above for Fredholm equations of the
second kind can be completely extended to such systems In particular, it can be shown that for systems (16.4.12.1), the successive approximations converge in mean-square to the
solution of the system if λ satisfies the inequality
|λ|< 1
where
n
i=1
n
j=1
a
a |Kij (x, t)|2dx dt = B ∗2<∞. (16.4.12.3)
If the kernel K ij (x, t) satisfies the additional condition
a K
2
ij (x, t) dt≤A ij, a≤x≤b, (16.4.12.4)
where A ijare some constants, then the successive approximations converge absolutely and uniformly
If all kernels K ij (x, t) are degenerate, then system (16.4.12.1) can be reduced to a linear
algebraic system It can be established that for a system of Fredholm integral equations, all Fredholm theorems are satisfied
16.4.12-2 Method of reducing a system of equations to a single equation
System (16.4.12.1) can be transformed into a single Fredholm integral equation of the
second kind Indeed, let us introduce the functions Y (x) and F (x) on [a, nb – (n –1)a] by
setting
Y (x) = y i x – (i –1)(b – a)
, F (x) = f i x – (i –1)(b – a)
, for
(i –1)b – (i –2)a≤x≤ib – (i –1)a.
Let us define a kernel K(x, t) on the square{a≤x≤nb – (n –1)a, a≤t≤nb – (n –1)a}
as follows:
K (x, t) = K ij x – (i –1)(b – a), t – (j –1)(b – a)
for
(i –1)b – (i –2)a≤x≤ib – (i –1)a, (j –1)b – (j –2)a≤t≤jb – (j –1)a.
Now system (16.4.12.1) can be rewritten as the single Fredholm equation
Y (x) – λ
nb–(n–1)a
a K (x, t)Y (t) dt = F (x), a≤x≤nb – (n –1)a.
If the kernels K ij (x, t) are square integrable on the square S ={a≤x≤b , a≤t≤b}and
the right-hand sides f i (x) are square integrable on [a, b], then the kernel K(x, t) is square
integrable on the new square
S n={a< x < nb – (n –1)a, a < t < nb – (n –1)a},
and the right-hand side F (x) is square integrable on [a, nb – (n –1)a].
Trang 7If condition (16.4.12.4) is satisfied, then the kernel K(x, t) satisfies the inequality
a K
2(x, t) dt≤A ∗, a < x < nb – (n –1)a, where A ∗is a constant
16.5 Nonlinear Integral Equations
16.5.1 Nonlinear Volterra and Urysohn Integral Equations
16.5.1-1 Nonlinear integral equations with variable integration limit
Nonlinear Volterra integral equations have the form
a K x , t, y(t)
dt = F x , y(x)
where K x , t, y(t)
is the kernel of the integral equation and y(x) is the unknown function (a≤x≤b) All functions in (16.5.1.1) are usually assumed to be continuous
16.5.1-2 Nonlinear integral equations with constant integration limits
Nonlinear Urysohn integral equations have the form
a K x , t, y(t)
dt = F (x, y(x)), α≤x≤β, (16.5.1.2)
where K x , t, y(t)
is the kernel of the integral equation and y(x) is the unknown function Usually, all functions in (16.5.1.2) are assumed to be continuous and the case of α = a and
β = b is considered.
Conditions for existence and uniqueness of the solution of an Urysohn equation are discussed below in Paragraphs 16.5.3-4 and 16.5.3-5
Remark A feature of nonlinear equations is that it frequently has several solutions.
16.5.2 Nonlinear Volterra Integral Equations
16.5.2-1 Method of integral transforms
Consider a Volterra integral equation with quadratic nonlinearity
μy (x) – λ
0 y (x – t)y(t) dt = f (x). (16.5.2.1)
To solve this equation, the Laplace transform can be applied, which, with regard to the convolution theorem (see Section 11.2), leads to a quadratic equation for the transform
2y(p) = L{y(x)}:
μ 2y(p) – λ2y2(p) = 2 f (p).
This implies
μ2–4λ2f (p)
2λ . (16.5.2.2)
The inverse Laplace transform y(x) = L– 1{2y(p)} (if it exists) is a solution to equa-tion (16.5.2.1) Note that for the two different signs in formula (16.5.2.2), there are two corresponding solutions of the original equation