T11.4. LINEAR EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRATION 1403 2 ◦ . The solution bounded at the endpoint x = 1 and unbounded at the endpoint x =–1: y(x)=Af (x)– B π  1 –1 g(x) g(t) f(t) dt t – x , g(x)=(1 + x) α (1 – x) –α ,(3) where α is the solution of the trigonometric equation (2) on the interval –1 < α < 0. 3 ◦ . The solution unbounded at the endpoints: y(x)=Af(x)– B π  1 –1 g(x) g(t) f(t) dt t – x + Cg(x), g(x)=(1 + x) α (1 – x) –1–α , where C is an arbitrary constant and α is the solution of the trigonometric equation (2) on the interval –1 < α < 0. 4. y(x) – λ  1 0  1 t – x – 1 x + t –2xt  y(t) dt = f(x), 0<x <1. Tricomi’s equation. Solution: y(x)= 1 1 + λ 2 π 2  f(x)+  1 0 t α (1 – x) α x α (1 – t) α  1 t – x – 1 x + t – 2xt  f(t) dt  + C(1 – x) β x 1+β , α = 2 π arctan(λπ)(–1 < α < 1), tan βπ 2 = λπ (–2 < β < 0), where C is an arbitrary constant. 5. y(x) + λ  ∞ 0 e –|x–t| y(t) dt = f(x). Solution for λ >– 1 2 : y(x)=f (x)– λ √ 1 + 2λ  ∞ 0 exp  – √ 1 + 2λ |x – t|  f(t) dt +  1 – λ + 1 √ 1 + 2λ   ∞ 0 exp  – √ 1 + 2λ (x + t)  f(t) dt. 6. y(x) – λ  ∞ –∞ e –|x–t| y(t) dt =0, λ >0. Lalesco–Picard equation. Solution: y(x)= ⎧ ⎪ ⎨ ⎪ ⎩ C 1 exp  x √ 1 – 2λ  + C 2 exp  –x √ 1 – 2λ  for 0 < λ < 1 2 , C 1 + C 2 x for λ = 1 2 , C 1 cos  x √ 2λ – 1  + C 2 sin  x √ 2λ – 1  for λ > 1 2 , where C 1 and C 2 are arbitrary constants. 1404 INTEGRAL EQUATIONS 7. y(x) + λ  ∞ –∞ e –|x–t| y(t) dt = f(x). 1 ◦ . Solution for λ >– 1 2 : y(x)=f (x)– λ √ 1 + 2λ  ∞ –∞ exp  – √ 1 + 2λ |x – t|  f(t) dt. 2 ◦ .Ifλ ≤ – 1 2 , for the equation to be solvable the conditions  ∞ –∞ f(x)cos(ax) dx = 0,  ∞ –∞ f(x)sin(ax) dx = 0, where a = √ –1 – 2λ, must be satisfied. In this case, the solution has the form y(x)=f(x)– a 2 + 1 2a  ∞ 0 sin(at)f(x + t) dt (–∞ < x < ∞). In the class of solutions not belonging to L 2 (–∞, ∞), the homogeneous equation (with f(x) ≡ 0) has a nontrivial solution. In this case, the general solution of the corresponding nonhomogeneous equation with λ ≤ – 1 2 has the form y(x)=C 1 sin(ax)+C 2 cos(ax)+f(x)– a 2 + 1 4a  ∞ –∞ sin(a|x – t|)f(t) dt. 8. y(x) + A  b a e λ|x–t| y(t) dt = f (x). 1 ◦ . The function y = y(x) obeys the following second-order linear nonhomogeneous ordi- nary differential equation with constant coefficients: y  xx + λ(2A – λ)y = f  xx (x)–λ 2 f(x). (1) The boundary conditions for (1) have the form y  x (a)+λy(a)=f  x (a)+λf(a), y  x (b)–λy(b)=f  x (b)–λf(b). (2) Equation (1) under the boundary conditions (2) determines the solution of the original integral equation. 2 ◦ .Forλ(2A – λ)<0, the general solution of equation (1) is given by y(x)=C 1 cosh(kx)+C 2 sinh(kx)+f(x)– 2Aλ k  x a sinh[k(x – t)] f(t) dt, k =  λ(λ – 2A), (3) where C 1 and C 2 are arbitrary constants. For λ(2A – λ)>0, the general solution of equation (1) is given by y(x)=C 1 cos(kx)+C 2 sin(kx)+f(x)– 2Aλ k  x a sin[k(x – t)] f (t) dt, k =  λ(2A – λ). (4) For λ = 2A, the general solution of equation (1) is given by y(x)=C 1 + C 2 x + f(x)–4A 2  x a (x – t)f(t) dt.(5) The constants C 1 and C 2 in solutions (3)–(5) are determined by conditions (2). T11.4. LINEAR EQUATIONS OF THE SECOND KIND WITH CONSTANT LIMITS OF INTEGRATION 1405 3 ◦ . In the special case a = 0 and λ(2A –λ)>0, the solution of the integral equation is given by formula (4) with C 1 = A(kI c – λI s ) (λ – A)sinμ – k cos μ , C 2 =– λ k A(kI c – λI s ) (λ – A)sinμ – k cos μ , k =  λ(2A – λ), μ = bk, I s =  b 0 sin[k(b – t)]f(t) dt, I c =  b 0 cos[k(b – t)]f (t) dt. 9. y(x) + λ  ∞ –∞ y(t) dt cosh[b(x – t)] = f (x). Solution for b > π|λ|: y(x)=f(x)– 2λb √ b 2 – π 2 λ 2  ∞ –∞ sinh[2k(x – t)] sinh[2b(x – t)] f(t) dt, k = b π arccos  πλ b  . 10. y(x) – λ  ∞ 0 cos(xt)y(t) dt = f(x). Solution: y(x)= f(x) 1 – π 2 λ 2 + λ 1 – π 2 λ 2  ∞ 0 cos(xt)f (t) dt, λ ≠  2/π. 11. y(x) – λ  ∞ 0 sin(xt)y(t) dt = f(x). Solution: y(x)= f(x) 1 – π 2 λ 2 + λ 1 – π 2 λ 2  ∞ 0 sin(xt)f(t) dt, λ ≠  2/π. 12. y(x) – λ  ∞ –∞ sin(x – t) x – t y(t) dt = f (x). Solution: y(x)=f (x)+ λ √ 2π – πλ  ∞ –∞ sin(x – t) x – t f(t) dt, λ ≠  2/π. 13. Ay(x) – B 2π  2π 0 cot  t – x 2  y(t) dt = f(x), 0 ≤ x ≤ 2π. Here the integral is understood in the sense of the Cauchy principal value. Without loss of generality we may assume that A 2 + B 2 = 1. Solution: y(x)=Af (x)+ B 2π  2π 0 cot  t – x 2  f(t) dt + B 2 2πA  2π 0 f(t) dt. 1406 INTEGRAL EQUATIONS 14. y(x) – λ  ∞ 0 e μ(x–t) cos(xt)y(t) dt = f (x). Solution: y(x)= f(x) 1 – π 2 λ 2 + λ 1 – π 2 λ 2  ∞ 0 e μ(x–t) cos(xt)f (t) dt, λ ≠  2/π. 15. y(x) – λ  ∞ 0 e μ(x–t) sin(xt)y(t) dt = f(x). Solution: y(x)= f(x) 1 – π 2 λ 2 + λ 1 – π 2 λ 2  ∞ 0 e μ(x–t) sin(xt)f(t) dt, λ ≠  2/π. 16. y(x) –  ∞ –∞ K(x – t)y(t) dt = f(x). The Fourier transform is used to solve this equation. Solution: y(x)=f (x)+  ∞ –∞ R(x – t)f(t) dt, where R(x)= 1 √ 2π  ∞ –∞  R(u)e iux du,  R(u)=  K(u) 1 – √ 2π  K(u) ,  K(u)= 1 √ 2π  ∞ –∞ K(x)e –iux dx. 17. y(x) –  ∞ 0 K(x – t)y(t) dt = f(x). Wiener–Hopf equation of the second kind. This equation is discussed in the books by Noble (1958), Gakhov and Cherskii (1978), and Polyanin and Manzhirov (1998) in detail. References for Chapter T11 Bitsadze, A. V., Integral Equation of the First Kind, World Scientific Publishing Co., Singapore, 1995. Ditkin, V. A. and Prudnikov, A. P., Integral Transforms and Operational Calculus, Pergamon Press, New York, 1965. Gakhov, F. D., Boundary Value Problems, Dover Publications, New York, 1990. Gakhov,F.D.andCherskii,Yu.I.,Equations of Convolution Type [in Russian], Nauka Publishers, Moscow, 1978. Gorenflo, R. and Vessella, S., Abel Integral Equations: Analysis and Applications, Springer-Verlag, Berlin, 1991. Krasnov, M. L., Kiselev, A. I., and Makarenko, G. I., Problems and Exercises in Integral Equations,Mir Publishers, Moscow, 1971. Lifanov, I. K., Poltavskii, L. N., and Vainikko, G., Hypersingular Integral Equations and Their Applications, Chapman & Hall/CRC Press, Boca Raton, 2004. Mikhlin,S.G.andPr ¨ ossdorf, S., Singular Integral Operators, Springer-Verlag, Berlin, 1986. Muskhelishvili, N. I., Singular Integral Equations: Boundary Problems of Function Theory and Their Appli- cations to Mathematical Physics, Dover Publications, New York, 1992. Noble, B., Methods Based on Wiener–Hopf Technique for the Solution of Partial Differential Equations, Pergamon Press, London, 1958. Polyanin,A.D.andManzhirov,A.V.,Handbook of Integral Equations, CRC Press, Boca Raton, 1998. REFERENCES FOR CHAPTER T11 1407 Prudnikov, A. P., Brychkov, Yu. A., and Marichev, O. I., Integrals and Series, Vol. 5, Inverse Laplace Transforms, Gordon & Breach, New York, 1992. 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. Zabreyko,P.P.,Koshelev,A.I.,etal.,Integral Equations: A Reference Text, Noordhoff International Publish- ing, Leyden, 1975. Chapter T12 Functional Equations T12.1. Linear Functional Equations in One Independent Variable T12.1.1. Linear Difference and Functional Equations Involving Unknown Function with Two Different Arguments T12.1.1-1. First-order linear difference equations involving y(x)andy(x + a). 1. y(x +1) – y(x) =0. This functional equation may be treated as a definition of periodic functions with unit period. 1 ◦ . Solution: y(x)=Θ(x), where Θ(x)=Θ(x + 1) is an arbitrary periodic function with unit period. 2 ◦ . A periodic function Θ(x) with period 1 that satisfies the Dirichlet conditions can be expanded into a Fourier series: Θ(x)= a 0 2 + ∞  n=1  a n cos(2πnx)+b n sin(2πnx)  , where a n = 2  1 0 Θ(x)cos(2πnx) dx, b n = 2  1 0 Θ(x)sin(2πnx) dx. 2. y(x +1) – y(x) = f(x). Solution: y(x)=Θ(x)+¯y(x), where Θ(x)=Θ(x + 1) is an arbitrary periodic function with period 1, and ¯y(x)isanypar- ticular solution of the nonhomogeneous equation. Table T12.1 presents particular solutions of the nonhomogeneous equation for some specific f(x). 3. y(x +1) – ay(x) =0. A homogeneous first-order constant-coefficient linear difference equation. 1 ◦ . Solution for a > 0: y(x)=Θ(x)a x , where Θ(x)=Θ(x + 1) is an arbitrary periodic function with period 1. For Θ(x) ≡ const, we have a particular solution y(x)=Ca x ,whereC is an arbitrary constant. 1409 . Technique for the Solution of Partial Differential Equations, Pergamon Press, London, 1958. Polyanin,A.D.andManzhirov,A.V. ,Handbook of Integral Equations, CRC Press, Boca Raton, 1998. REFERENCES FOR. f(x). Wiener–Hopf equation of the second kind. This equation is discussed in the books by Noble (1958), Gakhov and Cherskii (1978), and Polyanin and Manzhirov (1998) in detail. References for Chapter T11 Bitsadze,. periodic function with period 1, and ¯y(x)isanypar- ticular solution of the nonhomogeneous equation. Table T12.1 presents particular solutions of the nonhomogeneous equation for some specific f(x). 3.