T12.3. FUNCTIONAL EQUATIONS IN SEVERAL INDEPENDENT VARIABLES 1445 15. f x 2 + y 2 = f(x)f(y). Gauss’s equation. Solution: f(x)=exp(Cx 2 ), where C is an arbitrary constant. In addition, the function f(x) ≡ 0 is also a solution. 16. f 2 (x) + f 2 (y) 2 1/2 = f x 2 + y 2 2 1/2 . Solution: f(x)=(ax 2 + b) 1/2 , where a and b are arbitrary positive constants. 17. f (x n + y n ) 1/n = af(x)f (y), n is any number. Solution: f(x)= 1 a exp(Cx n ), where C is an arbitrary constant. In addition, the function f (x) ≡ 0 is also a solution. 18. f n (x) + f n (y) 2 1/n = f x n + y n 2 1/n , n is any number. Solution: f(x)=(ax n + b) 1/n , where a and b are arbitrary positive constants. 19. f x + y f(x) + f x – y f(x) =2f(x)f(y). Solutions: f(x) ≡ 0, f(x)=1 + Cx 2 , where C is an arbitrary constant. 20. f g –1 g(x) + g(y) = af(x)f (y). Generalized Gauss equation. Here, g(x) is an arbitrary monotonic function and g –1 (x)is the inverse of g(x). Solution: f(x)= 1 a exp Cg(x) , where C is an arbitrary constant. The function f(x) ≡ 0 is also a solution. 21. M f(x), f(y) = f M(x, y) . Here, M(x, y)=ϕ –1 ϕ(x)+ϕ(y) 2 is a quasiarithmetic mean for a continuous strictly monotonic function ϕ, with ϕ –1 being the inverse of ϕ. Solution: f(x)=ϕ –1 aϕ(x)+b , where a and b are arbitrary constants. 1446 FUNCTIONAL EQUATIONS T12.3.2-2. Equations involving several unknown functions of a single argument. 22. f(x)g(y) = h(x + y). Here, f(x), g(y), and h(z) are unknown functions. Solution: f(x)=C 1 exp(C 3 x), g(y)=C 2 exp(C 3 y), h(z)=C 1 C 2 exp(C 3 z), where C 1 , C 2 ,andC 3 are arbitrary constants. 23. f(x)g(y) + h(y) = f(x + y). Here, f(x), g(y), and h(z) are unknown functions. Solutions: f(x)=C 1 x + C 2 , g(x)=1, h(x)=C 1 x (first solution); f(x)=C 1 e ax + C 2 , g(x)=e ax , h(x)=C 2 (1 – e ax ) (second solution), where a, C 1 ,andC 2 are arbitrary constants. 24. f 1 (x)g 1 (y) + f 2 (x)g 2 (y) + f 3 (x)g 3 (y) =0. Bilinear functional equation. Two solutions: f 1 (x)=C 1 f 3 (x), f 2 (x)=C 2 f 3 (x), g 3 (y)=–C 1 g 1 (y)–C 2 g 2 (y); g 1 (y)=C 1 g 3 (y), g 2 (y)=C 2 g 3 (y), f 3 (x)=–C 1 f 1 (x)–C 2 f 2 (x), where C 1 and C 2 are arbitrary constants, the functions on the right-hand sides of the solutions are prescribed arbitrarily. 25. f 1 (x)g 1 (y) + f 2 (x)g 2 (y) + f 3 (x)g 3 (y) + f 4 (x)g 4 (y) =0. Bilinear functional equation. Equations of this type often arise in the generalized separation of variables in partial differential equations. 1 ◦ . Solution: f 1 (x)=C 1 f 3 (x)+C 2 f 4 (x), f 2 (x)=C 3 f 3 (x)+C 4 f 4 (x), g 3 (y)=–C 1 g 1 (y)–C 3 g 2 (y), g 4 (y)=–C 2 g 1 (y)–C 4 g 2 (y). It depends on four arbitrary constants C 1 , , C 4 . The functions on the right-hand sides of the solution are prescribed arbitrarily. 2 ◦ . The equation also has two other solutions, f 1 (x)=C 1 f 4 (x), f 2 (x)=C 2 f 4 (x), f 3 (x)=C 3 f 4 (x), g 4 (y)=–C 1 g 1 (y)–C 2 g 2 (y)–C 3 g 3 (y); g 1 (y)=C 1 g 4 (y), g 2 (y)=C 2 g 4 (y), g 3 (y)=C 3 g 4 (y), f 4 (x)=–C 1 f 1 (x)–C 2 f 2 (x)–C 3 f 3 (x), involving three arbitrary constants. T12.3. FUNCTIONAL EQUATIONS IN SEVERAL INDEPENDENT VARIABLES 1447 26. f(x) + g(y) = Q(z), where z = ϕ(x) + ψ(y). Here, one of the two functions f(x)andϕ(x) is prescribed and the other is assumed unknown, also one of the functions g(y)andψ(y) is prescribed and the other is unknown, and the function Q(z) is assumed unknown. (In similar equations with a composite argument, it is assumed that ϕ(x) const and ψ(y) const.) Solution: f(x)=Aϕ(x)+B, g(y)=Aψ(y)–B + C, Q(z)=Az + C, where A, B,andC are arbitrary constants. 27. f(x)g(y) = Q(z), where z = ϕ(x) + ψ(y). Here, one of the two functionsf (x)andϕ(x) is prescribed and the other is assumed unknown; also one of the functions g(y)andψ(y) is prescribed and the other is unknown, and the function Q(z) is assumed unknown. (In similar equations with a composite argument, it is assumed that ϕ(x) const and ψ(y) const.) Solution: f(x)=ABe λϕ(x) , g(y)= A B e λψ(y) , Q(z)=Ae λz , where A, B,andλ are arbitrary constants. 28. f(x) + g(y) = Q(z), where z = ϕ(x)ψ(y). Here, one of the two functionsf (x)andϕ(x) is prescribed and the other is assumed unknown; also one of the functions g(y)andψ(y) is prescribed and the other is unknown, and the function Q(z) is assumed unknown. (In similar equations with a composite argument, it is assumed that ϕ(x) const and ψ(y) const.) Solution: f(x)=A ln ϕ(x)+B, g(y)=A ln ψ(y)–B + C, Q(z)=A ln z + C, where A, B,andC are arbitrary constants. 29. f(y) + g(x) + h(x)Q(z) + R(z) =0, where z = ϕ(x) + ψ(y). Equations of this type often arise in the functional separation of variables in partial differ- ential equations. 1 ◦ . Solution: f =– 1 2 A 1 A 4 ψ 2 +(A 1 B 1 + A 2 + A 4 B 3 )ψ – B 2 – B 1 B 3 – B 4 , g = 1 2 A 1 A 4 ϕ 2 +(A 1 B 1 + A 2 )ϕ + B 2 , h = A 4 ϕ + B 1 , Q =–A 1 z + B 3 , R = 1 2 A 1 A 4 z 2 –(A 2 + A 4 B 3 )z + B 4 , where the A k and B k are arbitrary constants and ϕ = ϕ(x)andψ = ψ(y) are arbitrary functions. 1448 FUNCTIONAL EQUATIONS 2 ◦ . Solution: f =–B 1 B 3 e –A 3 ψ + A 2 – A 1 A 4 A 3 ψ – B 2 – B 4 – A 1 A 4 A 2 3 , g = A 1 B 1 A 3 e A 3 ϕ + A 2 – A 1 A 4 A 3 ϕ + B 2 , h = B 1 e A 3 ϕ – A 4 A 3 , Q = B 3 e –A 3 z – A 1 A 3 , R = A 4 B 3 A 3 e –A 3 z + A 1 A 4 A 3 – A 2 z + B 4 , where A k and B k are arbitrary constants and ϕ = ϕ(x)andψ = ψ(y) are arbitrary functions. 3 ◦ . In addition, the functional equation has two degenerate solutions: f = A 1 ψ + B 1 , g = A 1 ϕ + B 2 , h = A 2 , R =–A 1 z – A 2 Q – B 1 – B 2 , where ϕ = ϕ(x), ψ = ψ(y), and Q = Q(z) are arbitrary functions; A 1 , A 2 , B 1 ,andB 2 are arbitrary constants; and f = A 1 ψ + B 1 , g = A 1 ϕ + A 2 h + B 2 , Q =–A 2 , R =–A 1 z – B 1 – B 2 , where ϕ = ϕ(x), ψ = ψ(y), and h = h(x) are arbitrary functions; A 1 , A 2 , B 1 ,andB 2 are arbitrary constants. 30. f(y) + g(x)Q(z) + h(x)R(z) =0, where z = ϕ(x) + ψ(y). Equations of this type often arise in the functional separation of variables in partial differ- ential equations. 1 ◦ . Solution: g(x)=A 2 B 1 e k 1 ϕ + A 2 B 2 e k 2 ϕ , h(x)=(k 1 – A 1 )B 1 e k 1 ϕ +(k 2 – A 1 )B 2 e k 2 ϕ , Q(z)=A 3 B 3 e –k 1 z + A 3 B 4 e –k 2 z , R(z)=(k 1 – A 1 )B 3 e –k 1 z +(k 2 – A 1 )B 4 e –k 2 z , (1) where B 1 , , B 4 are arbitrary constants and k 1 and k 2 are roots of the quadratic equation (k – A 1 )(k – A 4 )–A 2 A 3 = 0. In the degenerate case k 1 = k 2 ,thetermse k 2 ϕ and e –k 2 z in (1) must be replaced by ϕe k 1 ϕ and ze –k 1 z , respectively. In the case of purely imaginary or complex roots, one should extract the real (or imaginary) part of the roots in solution (1). The function f(y) is determined by the formulas B 2 = B 4 = 0 =⇒ f (y)=[A 2 A 3 +(k 1 – A 1 ) 2 ]B 1 B 3 e –k 1 ψ , B 1 = B 3 = 0 =⇒ f (y)=[A 2 A 3 +(k 2 – A 1 ) 2 ]B 2 B 4 e –k 2 ψ , A 1 = 0 =⇒ f(y)=(A 2 A 3 + k 2 1 )B 1 B 3 e –k 1 ψ +(A 2 A 3 + k 2 2 )B 2 B 4 e –k 2 ψ . (2) Solutions defined by (1) and (2) involve arbitrary functions ϕ = ϕ(x)andψ = ψ(y). T12.3. FUNCTIONAL EQUATIONS IN SEVERAL INDEPENDENT VARIABLES 1449 2 ◦ . In addition, the functional equation has two degenerate solutions, f = B 1 B 2 e A 1 ψ , g = A 2 B 1 e –A 1 ϕ , h = B 1 e –A 1 ϕ , R =–B 2 e A 1 z – A 2 Q, where ϕ = ϕ(x), ψ = ψ(y), and Q = Q(z) are arbitrary functions; A 1 , A 2 , B 1 ,andB 2 are arbitrary constants; and f = B 1 B 2 e A 1 ψ , h =–B 1 e –A 1 ϕ – A 2 g, Q = A 2 B 2 e A 1 z , R = B 2 e A 1 z , where ϕ = ϕ(x), ψ = ψ(y), and g = g(x) are arbitrary functions; and A 1 , A 2 , B 1 ,andB 2 are arbitrary constants. T12.3.2-3. Equations involving functions of two arguments. 31. f(x, y)f(y, z) = f (x, z). Cantor’s second equation. Solution: f(x, y)=Φ(y)/Φ(x), where Φ(x) is an arbitrary function. 32. f(x, y)f(u, v) – f(x, u)f(y, v) + f(x, v)f (y, u) =0. Solution: f(x, y)=ϕ(x)ψ(y)–ϕ( y)ψ(x), where ϕ(x)andψ(x) are arbitrary functions. 33. f f(x, y), z) = f f(x, z), f(y, z) . Skew self-distributivity equation. Solution: f(x, y)=g –1 g(x)+g(y) , where g(x) is an arbitrary continuous strictly increasing function. 34. f x + y 2 = G f(x), f(y) . Generalized Jensen equation. 1 ◦ . A necessary and sufficient condition for the existence of a continuous strictly increasing solution is the existence of a continuous strictly monotonic function g(x) such that G(x, y)=g –1 g(x)+g(y) 2 , where g(x) is an arbitrary continuous strictly monotonic function and g –1 (x)istheinverse of g(x). 2 ◦ . If condition 1 ◦ is satisfied, the general continuous strictly monotonic solution of the original equation is given by f(x)=ϕ(ax + b), where ϕ(x) is any continuous strictly monotonic solution, and a and b are arbitrary constants. 1450 FUNCTIONAL EQUATIONS References for Chapter T12 Acz ´ el, J., Functional Equations: History, Applications and Theory, Kluwer Academic, Dordrecht, 2002. Acz ´ el, J., Lectures on Functional Equations and Their Applications, Dover Publications, New York, 2006. Acz ´ el, J., Some general methods in the theory of functional equations with a single variable. New applications of functional equations [in Russian], Uspekhi Mat. Nauk, Vol. 11, No. 3 (69), pp. 3–68, 1956. Acz ´ el, J. and Dhombres, J., Functional Equations in Several Variables, Cambridge University Press, Cam- bridge, 1989. Agarwal, R. P., Difference Equations and Inequalities, 2nd Edition, Marcel Dekker, New York, 2000. Belitskii, G. R. and Tkachenko,V., One-Dimensional Functional Equations,Birkh ¨ auser Verlag, Boston, 2003. Castillo, E. and Ruiz-Cobo, R., Functional Equations in Science and Engineering, Marcel Dekker, New York, 1992. Czerwik, S., Functional Equations and Inequalities in Several Variables, World Scientific Publishing Co., Singapore, 2002. Doetsch, G., Guide to the Applications of the Laplace and Z-transforms [in Russian, translation from German], Nauka Publishers, Moscow, 1974, pp. 213, 215, 218. Elaydi, S., An Introduction to Difference Equations, 3rd Edition, Springer-Verlag, New York, 2005. Fikhtengol’ts, G. M., A Course of Differential and Integral Calculus, Vol. 1 [in Russian], Nauka Publishers, Moscow, 1969, pp. 157–160. Goldberg, S., Introduction to Difference Equations, Dover Publications, New York, 1986. Kuczma, M., An Introduction to the Theory of Functional Equations and Inequalities, Polish Scientific Pub- lishers, Warsaw, 1985. Kuczma, M., Functional Equations in a Single Variable, Polish Scientific Publishers, Warsaw, 1968. Kuczma, M., Choczewski, B., and Ger, R., Iterative Functional Equations, Cambridge University Press, Cambridge, 1990. Mathematical Encyclopedia, Vol. 2 [in Russian], Sovetskaya Entsikolpediya, Moscow, 1979, pp. 1029, 1030. Mathematical Encyclopedia, Vol. 5 [in Russian], Sovetskaya Entsikolpediya, Moscow, 1985, pp. 699, 700, 703, 704. Mirolyubov, A. A. and Soldatov, M. A., Linear Homogeneous Difference Equations [in Russian], Nauka Publishers, Moscow, 1981. Mirolyubov,A.A.andSoldatov,M.A.,Linear Nonhomogeneous Difference Equations [in Russian], Nauka Publishers, Moscow, 1986. Nechepurenko, M. I., Iterations of Real Functions and Functional Equations [in Russian], Institute of Com- putational Mathematics and Mathematical Geophysics, Novosibirsk, 1997. Polyanin, A. D., Functional Equations, From Website EqWorld —The World of Mathematical Equations, http://eqworld.ipmnet.ru/en/solutions/fe.htm. Polyanin, A. D. and Manzhirov, A. V., Handbook of Integral Equations: Exact Solutions (Supplement. Some Functional Equations) [in Russian], Faktorial, Moscow, 1998. Polyanin, A. D. and Zaitsev, V. F., Handbook of Nonlinear Partial Differential Equations (Supplements S.4.4 and S.5.5), Chapman & Hall/CRC Press, Boca Raton, 2004. Supplement Some Useful Electronic Mathematical Resources arXiv.org (http://arxiv.org). A service of automated e-print archives of articles in the fields of mathematics, nonlinear science, computer science, and physics. Catalog of Mathematics Resources on the WWW and the Internet (http://mthwww.uwc.edu/ wwwmahes/files/math01.htm). CFD Codes List (http://www.fges.demon.co.uk/cfd/CFD codes p.html). Free software. CFD Resources Online (http://www.cfd-online.com/Links). Software, modeling and numerics, etc. Computer Handbook of ODEs (http://www.scg.uwaterloo.ca/ ecterrab/handbook odes.html). An online computer handbook of methods for solving ordinary differential equations. Deal.II (http://www.dealii.org). Finite element differential equations analysis library. Dictionary of Algorithms and Data Structures—NIST (http://www.nist.gov/dads/). The diction- ary of algorithms, algorithmic techniques, data structures, archetypical problems, and related definitions. DOE ACTS Collection (http://acts.nersc.gov). The Advanced CompuTational Software (ACTS) Collection is a set of software tools for computation sciences. EEVL: Internet Guide to Engineering, Mathematics and Computing (http://www.eevl.ac.uk). Cross-search 20 databases in engineering, mathematics, and computing. EqWorld: World of Mathematical Equations (http://eqworld.ipmnet.ru). Extensive information on algebraic, ordinary differential, partial differential, integral, functional, and other mathemat- ical equations. FOLDOC—Computing Dictionary (http://foldoc.doc.ic.ac.uk/foldoc/index.html). The free on- line dictionary of computing is a searchable dictionary of terms from computing and related fields. Free Software (http://www.wseas.com/software). Download free software packages for scientific- engineering purposes. FSF/UNESCO Free Software Directory (http://directory.fsf.org). GAMS: Guide to Available Mathematical Software (http://gams.nist.gov). A cross-index and virtual repository of mathematical and statistical software components of use in computational science and engineering. Google—Mathematics Websites (http://directory.google.com/Top/Science/Math/). A directory of more than 11,000 mathematics Websites ordered by type and mathematical subject. Google— Software (http://directory.google.com/Top/Science/Math/Software). A directory of software. Mathcom—PDEs (http://www.mathcom.com/corpdir/techinfo.mdir/scifaq/q260.html). Partial differential equations and finite element modeling. Mathematical Atlas (http://www.math-atlas.org). A collection of short articles designed to provide an introduction to the areas of modern mathematics. 1451 . (http://arxiv.org). A service of automated e-print archives of articles in the fields of mathematics, nonlinear science, computer science, and physics. Catalog of Mathematics Resources on the WWW and the Internet. p.html). Free software. CFD Resources Online (http://www.cfd-online.com/Links). Software, modeling and numerics, etc. Computer Handbook of ODEs (http://www.scg.uwaterloo.ca/ ecterrab /handbook odes.html) ψ(y). Here, one of the two functions f(x )and (x) is prescribed and the other is assumed unknown, also one of the functions g(y )and (y) is prescribed and the other is unknown, and the function