Handbook of mathematics for engineers and scienteists part 130 pps

REFERENCES FOR CHAPTER 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 x 1 , , x n ,wheref 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 y 1 , , y n for 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 y(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, Cambridge, 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.andCherskii,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.andKorn,T.M.,Mathematical Handbook for Scientists and Engineers, Dover Publications, New York, 2000. 872 INTEGRAL EQUATIONS 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. Ky the, 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.andPr ¨ ossdorf, S., Singular Integral Operators, Springer-Verlag, Berlin, 1986. Mikhlin,S. G. andSmolitskiy,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 Publications, 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.andManzhirov,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.,etal.,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 a 0 a 1 a n–1 , n = 1, 2, ,(17.1.1.2) where C = y 0 is an arbitrary constant and u n is 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 solution y n of the nonhomogeneous equation (17.1.1.3): y n = Cu n + y n , n = 1, 2, ,(17.1.1.4) where C = y 0 is an arbitrary constant, u n is defined by (17.1.1.2), and y 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 a 1 a 2 a n–1 f 0 .(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 DIFFERENCE EQUATIONS AND OTHER FUNCTIONAL EQUATIONS 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 n that, 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 y n = ϕ(n, C)(17.1.2.2) or in implicit form Φ(n, y n , C)=0. SpecificvaluesofC 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+1 has the form y n+1 = f(n, y n ). (17.1.2.3) The Cauchy problem consists of finding a solution of this equation with a given initial value of y 0 . The next value y 1 is calculated by substituting the initial value into the right-hand side of equation (17.1.2.3) for n = 0: y 1 = f (0, y 0 ). (17.1.2.4) Then, taking n = 1 in (17.1.2.3), we get y 2 = f (1, y 1 ). (17.1.2.5) Substituting the previous value (17.1.2.4) into this relation, we find y 2 = f  1, f (0, y 0 )  . Taking n = 2 in (17.1.2.3) and using the calculated value y 2 ,wefind y 3 , etc. In a similar way, one finds subsequent values y 4 , y 5 , Example. Consider the Cauchy problem for the nonlinear difference equation y n+1 = ay β n ; y 0 = 1. Consecutive calculations yield y 1 = a, y 2 = a β+1 , y 3 = a β 2 +β+1 , , y n = 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. DIFFERENCE EQUATIONS OF INTEGER ARGUMENT 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 n+1 u n + a n , u 0 = 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 u 0 = 1, u 1 = y 0 – a 0 . 2 ◦ .Lety ∗ n be 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 ◦ .Lety (1) n and y (2) n be two particular solutions of equation (17.1.2.6) with y (1) n ≠ y (2) n .Then the substitution w n = 1 y n – y (1) n + 1 y (1) n – y (2) n , 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, , y 0 = λ, (17.1.2.7) where 0 < a ≤ 4, b > 0,and0 ≤ λ ≤ b. 876 DIFFERENCE EQUATIONS AND OTHER FUNCTIONAL EQUATIONS 1 ◦ .Let a = b = 4, λ = 4 sin 2 θ (0 ≤ θ ≤ π 2 ). Then problem (17.1.2.7) has the closed-form solution y n = 4 sin 2 (2 n θ), n = 0, 1, 2 ◦ .Let a = 4, b = 1, λ =sin 2 θ (0 ≤ θ ≤ π 2 ). Then problem (17.1.2.7) has the closed-form solution y n =sin 2 (2 n θ), n = 0, 1, 3 ◦ .Let0 ≤ a ≤ 4 and b = 1. In this case, the solutions of the logistic equation have the following properties: (a) There are equilibrium solutions y n = 0 and y n =(a – 1)/a. (b) If 0 ≤ y 0 ≤ 1,then0 ≤ y n ≤ 1. (c) If a = 0,theny n = 0. (d) If 0 < a ≤ 1,theny n → 0 as n →∞. (e) If 1 < a ≤ 3,theny n → (a – 1)/a as n →∞. (f) If 3 < a < 3.449 ,theny n oscillates between the two points: y = 1 2a (a + 1 √ a 2 – 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 y 0 , y 1 , y 2 , 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 P 0 , P 1 , P 2 , on the graph of the function z = f(y) with the abscissas y 0 , y 1 , y 2 , determined by equation (17.1.2.8). O y ξ z zy= zfy= () y 2 y 1 y 2 y 0 y 1 Q 1 Q 0 P * P 2 P 1 P 0 Figure 17.1. Construction, using the graph of the function z = f(y), of the points with abscissas y 0 , y 1 , y 2 , that satisfy the difference equations (17.1.2.8). 17.1. DIFFERENCE EQUATIONS OF INTEGER ARGUMENT 877 This construction consists of the following steps: 1. Through the point P 0 =(y 0 , y 1 ) with y 1 = f(y 0 ), we draw a horizontal line. This line crosses the iteration axis at the point Q 0 =(y 1 , y 1 ). 2. Through the point Q 0 , we draw a vertical line. This line crosses the graph of the function f(y) at the point P 1 =(y 1 , y 2 ) with y 2 = f (y 1 ). 3. Repeating the operations of steps 1 and 2, we obtain the following sequence on the graph of f(y): P 0 =(y 0 , f (y 0 )), P 1 =(y 1 , f (y 1 )), P 2 =(y 2 , f (y 2 )), 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 ξ I such that f(ξ)=ξ. B RAUER 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 . AsetE is called the domain of attraction of a fixed point ξ of a function f(y)ifthe sequence y n+1 = f(y n )convergestoξ for any y 0 E. If ξ = f(ξ)and|f  (ξ)| < 1,thenξ is an attracting fixed point: there is a neighborhood of ξ belonging 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) ≠ 0 and |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” (for 0 < 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 aλ 2 + bλ + c = 0.(17.1.3.2) 1 ◦ .Letb 2 – 4ac > 0. Then the quadratic equation (17.1.3.2) has two different real roots λ 1 = –b + √ b 2 – 4ac 2a , λ 2 = –b – √ b 2 – 4ac 2a , and the general solution of the difference equation (17.1.3.1) is given by the formula y n = C 1 λ 1 λ n 2 – λ n 1 λ 2 λ 1 – λ 2 + C 2 λ n 1 – λ n 2 λ 1 – λ 2 ,(17.1.3.3) where C 1 and C 2 are arbitrary constants. Solution (17.1.3.3) satisfies the initial conditions y 0 = C 1 , y 1 = C 2 . . n,(16.5.3.43) for the approximate values y i of the solution y(x) at the nodes x 1 , , x n ,wheref 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. 1991. Korn,G.A.andKorn,T.M.,Mathematical Handbook for Scientists and Engineers, Dover Publications, New York, 2000. 872 INTEGRAL EQUATIONS Krasnosel’skii, M. A., Topological Methods in the Theory of Nonlinear Integral. explicit form y n = ϕ(n, C)(17.1.2.2) or in implicit form Φ(n, y n , C)=0. SpecificvaluesofC define specific solutions of the equation (particular solutions). Any constant solution y n = ξ of equation