Handbook of mathematics for engineers and scienteists part 130 pps

E., Numerical Solution of Integral Equations of the Second Kind, Cambridge University Press, Cambridge, 1997.. Brunner, H., Collocation Methods for Volterra Integral and Related Function

R EFERENCES FOR C HAPTER 16 871

On applying the quadrature formula from Subsection 16.4.11 and neglecting the approxi-mation error, we transform relations (16.5.3.42) into the nonlinear system of algebraic (or transcendental) equations

y i–

n



j=1

A j K ij (y j ) = f i i=1, , n, (16.5.3.43)

for the approximate values y i of the solution y(x) at the nodes x1, , x n , where f i = f (x i)

and K ij (y j ) = K(x i , t j , y j ), and A j are the coefficients of the quadrature formula

The solution of the nonlinear system (16.5.3.43) gives values y1, , y nfor which by interpolation we find an approximate solution of the integral equation (16.5.3.41) on the

entire interval [a, b] For the analytic expression of an approximate solution, we can take

the function

2y(x) = f(x) +n

j=1

A j K (x, x j , y j).

References for Chapter 16

Atkinson, K E., Numerical Solution of Integral Equations of the Second Kind, Cambridge University Press,

Cambridge, 1997.

Bakhvalov, N S., Numerical Methods [in Russian], Nauka Publishers, Moscow, 1973.

Bateman, H and Erd´elyi, A., Tables of Integral Transforms Vols 1 and 2, McGraw-Hill, New York, 1954 Bitsadze, A.V., Integral Equation of the First Kind, World Scientific Publishing Co., Singapore, 1995 Brunner, H., Collocation Methods for Volterra Integral and Related Functional Differential Equations,

Cam-bridge University Press, CamCam-bridge, 2004.

Cochran, J A., The Analysis of Linear Integral Equations, McGraw-Hill, New York, 1972.

Corduneanu, C., Integral Equations and Applications, Cambridge University Press, Cambridge, 1991 Courant, R and Hilbert, D., Methods of Mathematical Physics Vol 1, Interscience, New York, 1953 Delves, L M and Mohamed, J L., Computational Methods for Integral Equations, Cambridge University

Press, Cambridge, 1985.

Demidovich, B P., Maron, I A., and Shuvalova, E Z., Numerical Methods Approximation of Functions and

Differential and Integral Equations [in Russian], Fizmatgiz, Moscow, 1963.

Ditkin, V A and Prudnikov, A P., Integral Transforms and Operational Calculus, Pergamon Press, New

York, 1965.

Dzhuraev, A., Methods of Singular Integral Equations, Wiley, New York, 1992.

Gakhov, F D and Cherskii, Yu I., Equations of Convolution Type [in Russian], Nauka Publishers, Moscow,

1978.

Gohberg, I C and Krein, M G., The Theory of Volterra Operators in a Hilbert Space and Its Applications

[in Russian], Nauka Publishers, Moscow, 1967.

Golberg, A (Editor), Numerical Solution of Integral Equations, Plenum Press, New York, 1990.

Gorenflo, R and Vessella, S., Abel Integral Equations: Analysis and Applications, Springer-Verlag, Berlin,

1991.

Goursat, E., Cours d’Analyse Math´ematique, III,3 me

´ed., Gauthier–Villars, Paris, 1923.

Gripenberg, G., Londen, S.-O., and Staffans, O., Volterra Integral and Functional Equations, Cambridge

University Press, Cambridge, 1990.

Hackbusch, W., Integral Equations: Theory and Numerical Treatment, Birkh¨auser Verlag, Boston, 1995 Jerry, A J., Introduction to Integral Equations with Applications, Marcel Dekker, New York, 1985.

Kantorovich, L V and Akilov, G P., Functional Analysis in Normed Spaces, Macmillan, New York, 1964 Kantorovich, L V and Krylov, V I., Approximate Methods of Higher Analysis, Interscience, New York, 1958 Kanwal, R P., Linear Integral Equations, Birkh¨auser Verlag, Boston, 1997.

Kolmogorov, A N and Fomin, S V., Introductory Real Analysis, Prentice-Hall, Englewood Cliffs, New

Jersey, 1970.

Kondo, J., Integral Equations, Clarendon Press, Oxford, 1991.

Korn, G A and Korn, T M., Mathematical Handbook for Scientists and Engineers, Dover Publications, New

York, 2000.

872 INTEGRALEQUATIONS

Krasnosel’skii, M A., Topological Methods in the Theory of Nonlinear Integral Equations, Macmillan, New

York, 1964.

Krasnov, M L., Kiselev, A I., and Makarenko, G I., Problems and Exercises in Integral Equations, Mir

Publishers, Moscow, 1971.

Krein, M G., Integral equations on a half-line with kernels depending upon the difference of the arguments

[in Russian], Uspekhi Mat Nauk, Vol 13, No 5 (83), pp 3–120, 1958.

Krylov, V I., Bobkov, V V., and Monastyrnyi, P I., Introduction to the Theory of Numerical Methods Integral

Equations, Problems, and Improvement of Convergence [in Russian], Nauka i Tekhnika, Minsk, 1984.

Kythe, P K and Puri, P., Computational Methods for Linear Integral Equations, Birkh¨auser Verlag, Boston,

2002.

Ladopoulos, E G., Singular Integral Equations: Linear and Non-Linear Theory and Its Applications in Science

and Engineering, Springer-Verlag, Berlin, 2000.

Lavrentiev, M M., Some Improperly Posed Problems of Mathematical Physics, Springer-Verlag, New York,

1967.

Lovitt, W V., Linear Integral Equations, Dover Publications, New York, 1950.

Mikhlin, S G., Linear Integral Equations, Hindustan Publishing, Delhi, 1960.

Mikhlin, S G and Pr¨ossdorf, S., Singular Integral Operators, Springer-Verlag, Berlin, 1986.

Mikhlin, S G and Smolitskiy, K L., Approximate Methods for Solution of Differential and Integral Equations,

American Elsevier, New York, 1967.

Muskhelishvili N I., Singular Integral Equations: Boundary Problems of Function Theory and Their

Appli-cations to Mathematical Physics, Dover PubliAppli-cations, New York, 1992.

Petrovskii, I G., Lectures on the Theory of Integral Equations, Graylock Press, Rochester, 1957.

Pipkin, A C., A Course on Integral Equations, Springer-Verlag, New York, 1991.

Polyanin, A D and Manzhirov, A V., A Handbook of Integral Equations, CRC Press, Boca Raton, 1998 Porter, D and Stirling, D S G., Integral Equations: A Practical Treatment, from Spectral Theory to

Applica-tions, Cambridge University Press, Cambridge, 1990.

Precup, R., Methods in Nonlinear Integral Equations, Kluwer Academic, Dordrecht, 2002.

Pr¨ossdorf, S and Silbermann, B., Numerical Analysis for Integral and Related Operator Equations,

Birk-h¨auser Verlag, Basel, 1991.

Sakhnovich, L A., Integral Equations with Difference Kernels on Finite Intervals, Birkh¨auser Verlag, Basel,

1996.

Samko, S G., Kilbas, A A., and Marichev, O I., Fractional Integrals and Derivatives Theory and

Applica-tions, Gordon & Breach, New York, 1993.

Tricomi, F G., Integral Equations, Dover Publications, New York, 1985.

Tslaf, L Ya., Variational Calculus and Integral Equations [in Russian], Nauka Publishers, Moscow, 1970 Verlan’, A F and Sizikov, V S., Integral Equations: Methods, Algorithms, and Programs [in Russian],

Naukova Dumka, Kiev, 1986.

Zabreyko, P P., Koshelev, A I., et al., Integral Equations: A Reference Text, Noordhoff International

Publish-ing, Leyden, 1975.

Chapter 17

Difference Equations and

Other Functional Equations

17.1 Difference Equations of Integer Argument

17.1.1 First-Order Linear Difference Equations of Integer Argument

17.1.1-1 First-order homogeneous linear difference equations General solution

Let y n = y(n) be a function of integer argument n =0,1, 2, A first-order homogeneous

linear difference equation has the form

y n+1 + a n y n=0 (17.1.1.1) Its general solution can be written in the form

y n = Cu n, u n= (–1)n a0a1 a n–1, n=1, 2, , (17.1.1.2)

where C = y0is an arbitrary constant and u nis a particular solution

17.1.1-2 First-order nonhomogeneous linear difference equations General solution

A first-order nonhomogeneous linear difference equation has the form

y n+1 + a n y n = f n. (17.1.1.3) The general solution of the nonhomogeneous linear equation (17.1.1.3) can be rep-resented as the sum of the general solution (17.1.1.2) of the corresponding homogeneous equation (17.1.1.1) and a particular solution2y nof the nonhomogeneous equation (17.1.1.3):

y n = Cu n+2y n, n=1, 2, , (17.1.1.4)

where C = y0is an arbitrary constant, u nis defined by (17.1.1.2), and

2y n=

n–1



j=0

u n

u j+1 f j = f n–1 –a n–1 f n–2 +a n–2 a n–1 f n–3–· · ·+(–1)n–1 a1a2 a n–1 f0 (17.1.1.5)

17.1.1-3 First-order linear difference equations with constant coefficients

A first-order linear difference equation with constant coefficients has the form

y n+1 – ay n = f n.

Using (17.1.1.2), (17.1.1.4), and (17.1.1.5) for a n = –a, we obtain its general solution

y n = Ca n+

n–1



j=0

a n–j–1 f

j.

873

874 DIFFERENCEEQUATIONS ANDOTHERFUNCTIONALEQUATIONS

17.1.2 First-Order Nonlinear Difference Equations of Integer

Argument

17.1.2-1 First-order nonlinear equations General and particular solutions

Let y n = y(n) be a function of integer argument n =0, 1, 2, A first-order nonlinear

difference equation, in the general case, has the form

F (n, y n , y n+1) =0 (17.1.2.1)

A solution of the difference equation (17.1.2.1) is defined as a discrete function y nthat,

being substituted into the equation, turns it into identity The general solution of a difference

equation is the set of all its solutions The general solution of equation (17.1.2.1) depends

on an arbitrary constant C The general solution can be written either in explicit form

or in implicit formΦ(n, y n , C) =0 Specific values of C define specific solutions of the equation (particular solutions).

Any constant solution y n = ξ of equation (17.1.2.1), with ξ independent of n, is called

an equilibrium solution.

17.1.2-2 Cauchy’s problem and its solution

A difference equation resolved with respect to the leading term y n+1has the form

The Cauchy problem consists of finding a solution of this equation with a given initial value

of y0

The next value y1is calculated by substituting the initial value into the right-hand side

of equation (17.1.2.3) for n =0:

Then, taking n =1in (17.1.2.3), we get

Substituting the previous value (17.1.2.4) into this relation, we find y2 = f 1, f (0, y0)

Taking n =2in (17.1.2.3) and using the calculated value y2, we find y3, etc In a similar

way, one finds subsequent values y4, y5,

Example Consider the Cauchy problem for the nonlinear difference equation

y n+1= ay n β; y0= 1.

Consecutive calculations yield

y1= a, y2= a β+1, y3= a β2+β+1, ., y = a β n–1+β n–2+···+β+1= a β

n–1

β–1

Remark As a rule, solutions of nonlinear difference equations cannot be found in closed form (i.e., in terms of a single, not a recurrent, formula).

17.1 D IFFERENCE E QUATIONS OF I NTEGER A RGUMENT 875

17.1.2-3 Riccati difference equation

The Riccati difference equation has the general form

y n y n+1 = a n y n+1 + b n y n + c n, n=0, 1, , (17.1.2.6)

with the constants a n , b n , c n satisfying the condition a n b n + c n ≠ 0

1◦ The substitution

y n= u u n+1

n + a n, u0=1, leads us to the linear second-order difference equation

u n+2 + (a n+1 – b n )u n+1 – (a n b n + c n )u n=0 with the initial conditions

u0=1, u1= y0– a0.

2◦ Let y ∗

nbe a particular solution of equation (17.1.2.6) Then the substitution

z n= 1

y n – y ∗ n, n=0,1, ,

reduces equation (17.1.2.6) to the first-order linear nonhomogeneous difference equation

z n+1+ (y ∗ n – a n)

2

a n b n + c n z n+

y ∗

n – a n

a n b n + c n =0 With regard to the solution of this equation see Paragraph 17.1.1-2

3◦ Let y( 1 )

n and y n(2)be two particular solutions of equation (17.1.2.6) with y(n1)≠y(2 )

n Then

the substitution

y n – y n(1)

y(1 )

n – y n(2)

, n=0, 1, ,

reduces equation (17.1.2.6) to the first-order linear homogeneous difference equation

w n+1+ (y

( 1 )

n – a n)2

a n b n + c n w n=0, n=0, 1,

With regard to the solution of this equation see Paragraph 17.1.1-1

17.1.2-4 Logistic difference equation

Consider the initial-value problem for the logistic difference equation

y n+1 = ay n



1– y n

b

 , n=0,1, ,

y0= λ,

(17.1.2.7)

where0< a≤ 4, b >0, and0 ≤λ≤b

876 DIFFERENCEEQUATIONS ANDOTHERFUNCTIONALEQUATIONS

1◦ Let

a = b =4, λ=4sin2θ (0 ≤θ≤ π

2).

Then problem (17.1.2.7) has the closed-form solution

y n=4sin2(2n θ), n=0, 1,

2◦ Let

a=4, b=1, λ= sin2θ (0 ≤θ≤ π

2).

Then problem (17.1.2.7) has the closed-form solution

y n= sin2(2n θ), n=0,1,

3◦ Let0 ≤a ≤ 4and b = 1 In this case, the solutions of the logistic equation have the following properties:

(a) There are equilibrium solutions y n=0and y n = (a –1)/a.

(b) If0 ≤y0≤ 1, then0 ≤y n≤ 1

(c) If a =0, then y n=0

(d) If0< a≤ 1, then y n →0as n → ∞.

(e) If1< a≤ 3, then y n → (a –1)/a as n → ∞.

(f) If3< a <3.449 , then y noscillates between the two points:

y= 1

2a (a +1 √ a2–2a–3)

17.1.2-5 Graphical construction of solutions to nonlinear difference equations

Consider nonlinear difference equations of special form

y n+1 = f (y n), n=0,1, (17.1.2.8)

The points y0, y1, y2, are constructed on the plane (y, z) on the basis of the graph

z = f (y) and the straight line z = y, called the iteration axis.

Figure 17.1 shows the result of constructing the points P0, P1, P2, on the graph of the function z = f (y) with the abscissas y0, y1, y2, determined by equation (17.1.2.8).

O

y ξ

z=f y( )

y2

y1

y2

y0 y1

Q1

Q0

P*

P2

P1

P0

Figure 17.1 Construction, using the graph of the function z = f (y), of the points with abscissas y0, y1, y2,

that satisfy the difference equations (17.1.2.8).

17.1 D IFFERENCE E QUATIONS OF I NTEGER A RGUMENT 877

This construction consists of the following steps:

1 Through the point P0= (y0, y1) with y1= f (y0), we draw a horizontal line This line

crosses the iteration axis at the point Q0= (y1, y1)

2 Through the point Q0, we draw a vertical line This line crosses the graph of the

function f (y) at the point P1= (y1, y2) with y2= f (y1)

3 Repeating the operations of steps 1 and 2, we obtain the following sequence on the

graph of f (y):

P0= (y0, f (y0)), P1 = (y1, f (y1)), P2= (y2, f (y2)), .

In the case under consideration, for n → ∞, the points y n converge to a fixed ξ,

which determines an equilibrium solution satisfying the algebraic (transcendental) equation

ξ = f (ξ).

17.1.2-6 Convergence to a fixed point Qualitative behavior of solutions

A fixed point of a mapping f of a set I is a point ξI such that f (ξ) = ξ.

BRAUER FIXED POINT THEOREM If f (y) is a continuous function on the interval I =

{a≤y ≤b}and f (I) ⊂ I, then f(y) has a fixed point in I.

A set E is called the domain of attraction of a fixed point ξ of a function f (y) if the sequence y n+1 = f (y n ) converges to ξ for any y0 E

If ξ = f (ξ) and|f  (ξ)|<1, then ξ is an attracting fixed point: there is a neighborhood of

ξbelonging to its domain of attraction

Figure 17.2 illustrates the qualitative behavior of sequences (17.1.2.8) starting from

points sufficiently close to a fixed point ξ such that f  (ξ)≠ 0and|f  (ξ)| ≠ 1

According to the behavior of the iteration process in a neighborhood of the fixed point, the cases represented in Fig 17.2 may be called one-dimensional analogues of a “stable node” (for0< f  (ξ) <1; see Fig 17.2 a), “stable focus” (for –1< f  (ξ) <0; see Fig 17.2 b),

“unstable node” (for 1 < f  (ξ); see Fig 17.2 c), or “unstable focus” (for f  (ξ) < –1; see

Fig 17.2 d).

17.1.3 Second-Order Linear Difference Equations with Constant

Coefficients

17.1.3-1 Homogeneous linear equations

A second-order homogeneous linear difference equation with constant coefficients has the form

ay n+2 + by n+1 + cy n=0 (17.1.3.1) The general solution of this equation is determined by the roots of the quadratic equation

1◦ Let b2–4ac>0 Then the quadratic equation (17.1.3.2) has two different real roots

λ1= –b +

√

b2–4ac

2a , λ2 = –b –

√

b2–4ac

and the general solution of the difference equation (17.1.3.1) is given by the formula

y n = C1λ1λ

n

2 – λ n1λ2

λ1– λ2 + C2

λ n

1 – λ n2

λ1– λ2, (17.1.3.3)

where C1and C2are arbitrary constants Solution (17.1.3.3) satisfies the initial conditions

y0= C1, y1= C2.