Handbook of mathematics for engineers and scienteists part 212 docx

Equations involving several unknown functions of a single argument.22.. Equations of this type often arise in the generalized separation of variables in partial differential equations..

15. f x2+ y2

= f (x)f (y).

Gauss’s equation.

Solution:

f (x) = exp(Cx2),

where C is an arbitrary constant In addition, the function f (x)≡ 0is also a solution

16.



f2(x) + f2(y)

2

1/2

= f

x2+ y2

2

1/2

.

Solution:

f (x) = (ax2+ 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)≡ 0is 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+ Cx2,

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

aexp

Cg(x) ,

where C is an arbitrary constant The function f (x)≡ 0is 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–1being the inverse of ϕ.

Solution:

f (x) = ϕ–1 aϕ(x) + b

,

where a and b are arbitrary constants.

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) = C1exp(C3x), g(y) = C2exp(C3y), h(z) = C1C2exp(C3z),

where C1, C2, and C3are 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) = C1x + C2, g(x) =1, h(x) = C1x (first solution);

f (x) = C1e ax + C2, g(x) = e ax , h(x) = C2(1– e ax) (second solution),

where a, C1, and C2are arbitrary constants

24. f1(x)g1(y) + f2(x)g2(y) + f3(x)g3(y) = 0.

Bilinear functional equation.

Two solutions:

f1(x) = C1f3(x), f2(x) = C2f3(x), g3(y) = –C1g1(y) – C2g2(y);

g1(y) = C1g3(y), g2(y) = C2g3(y), f3(x) = –C1f1(x) – C2f2(x),

where C1 and C2 are arbitrary constants, the functions on the right-hand sides of the solutions are prescribed arbitrarily

25. f1(x)g1(y) + f2(x)g2(y) + f3(x)g3(y) + f4(x)g4(y) = 0.

Bilinear functional equation.

Equations of this type often arise in the generalized separation of variables in partial differential equations

1◦ Solution:

f1(x) = C1f3(x) + C2f4(x), f2(x) = C3f3(x) + C4f4(x),

g3(y) = –C1g1(y) – C3g2(y), g4(y) = –C2g1(y) – C4g2(y).

It depends on four arbitrary constants C1, , C4 The functions on the right-hand sides of the solution are prescribed arbitrarily

2◦ The equation also has two other solutions,

f1(x) = C1f4(x), f2(x) = C2f4(x), f3(x) = C3f4(x), g4(y) = –C1g1(y) – C2g2(y) – C3g3(y);

g1(y) = C1g4(y), g2(y) = C2g4(y), g3(y) = C3g4(y), f4(x) = –C1f1(x) – C2f2(x) – C3f3(x),

involving three arbitrary constants

26. f (x) + g(y) = Q(z), where z = ϕ(x) + ψ(y).

Here, one of the two functions f (x) and ϕ(x) is prescribed and the other is assumed unknown, also one of the functions g(y) and ψ(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, and C are arbitrary constants.

27. f (x)g(y) = Q(z), where z = ϕ(x) + ψ(y).

Here, one of the two functions f (x) and ϕ(x) is prescribed and the other is assumed unknown; also one of the functions g(y) and ψ(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, and λ are arbitrary constants.

28. f (x) + g(y) = Q(z), where z = ϕ(x)ψ(y).

Here, one of the two functions f (x) and ϕ(x) is prescribed and the other is assumed unknown; also one of the functions g(y) and ψ(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, and C 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 = –12A1A4ψ2+ (A

1B1+ A2+ A4B3)ψ – B2– B1B3– B4,

g= 12A1A4ϕ2+ (A

1B1+ A2)ϕ + B2,

h = A4ϕ + B1,

Q = –A1z + B3,

R= 12A1A4z2– (A

2+ A4B3)z + B4,

where the A k and B k are arbitrary constants and ϕ = ϕ(x) and ψ = ψ(y) are arbitrary

functions

2◦ Solution:

f = –B1B3e–A3ψ+

A2– A A1A4

3



ψ – B2– B4– A1A4

A2 3 ,

g= A1B1

A3 e

A3ϕ+

A2– A A1A4

3



ϕ + B2,

h = B1e A3ϕ– A4

A3,

Q = B3e–A3z– A1

A3,

R = A4B3

A3 e

–A3z+

A1A4

A3 – A2



z + B4,

where A k and B k are arbitrary constants and ϕ = ϕ(x) and ψ = ψ(y) are arbitrary functions.

3◦ In addition, the functional equation has two degenerate solutions:

f = A1ψ + B1, g = A1ϕ + B2, h = A2, R = –A1z – A2Q – B1– B2,

where ϕ = ϕ(x), ψ = ψ(y), and Q = Q(z) are arbitrary functions; A1, A2, B1, and B2 are arbitrary constants; and

f = A1ψ + B1, g = A1ϕ + A2h + B2, Q = –A2, R = –A1z – B1– B2,

where ϕ = ϕ(x), ψ = ψ(y), and h = h(x) are arbitrary functions; A1, A2, B1, and B2 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) = A2B1e k1ϕ + A

2B2e k2ϕ, h(x) = (k1– A1)B1e k1ϕ + (k

2– A1)B2e k2ϕ, Q(z) = A3B3e–k1z + A

3B4e–k2z, R(z) = (k1– A1)B3e–k1z + (k

2– A1)B4e–k2z,

(1)

where B1, , B4are arbitrary constants and k1and k2are roots of the quadratic equation

(k – A1)(k – A4) – A2A3=0

In the degenerate case k1= k2, the terms e k2ϕ and e–k2z in (1) must be replaced by ϕe k1ϕand

ze–k1z, 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

B2= B4 =0 =⇒ f(y) = [A2 A3+ (k1– A1)2]B1B3e–k1ψ,

B1= B3 =0 =⇒ f(y) = [A2 A3+ (k2– A1)2]B2B4e–k2ψ,

A1=0 =⇒ f(y) = (A2 A3+ k12)B1B3e–k1ψ + (A

2A3+ k22)B2B4e–k2ψ.

(2)

Solutions defined by (1) and (2) involve arbitrary functions ϕ = ϕ(x) and ψ = ψ(y).

2◦ In addition, the functional equation has two degenerate solutions,

f = B1B2e A1ψ, g = A

2B1e–A1ϕ, h = B

1e–A1ϕ, R = –B

2e A1z – A

2Q,

where ϕ = ϕ(x), ψ = ψ(y), and Q = Q(z) are arbitrary functions; A1, A2, B1, and B2 are arbitrary constants; and

f = B1B2e A1ψ, h = –B

1e–A1ϕ – A

2g, Q = A2B2e A1z, R = B

2e A1z, where ϕ = ϕ(x), ψ = ψ(y), and g = g(x) are arbitrary functions; and A1, A2, B1, and B2

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),

whereΦ(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) and ψ(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) is the inverse

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.

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 and Soldatov, 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.

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arXiv.org (http://arxiv.org) A service of automated e-print archives of articles in the fields of

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Catalog of Mathematics Resources on the WWW and the Internet (http://mthwww.uwc.edu/

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EqWorld: World of Mathematical Equations (http://eqworld.ipmnet.ru) Extensive information

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1451

Computer Handbook of ODEs (http://www.scg.uwaterloo.ca/ ecterrab /handbook odes.html) An

online computer handbook of methods for solving ordinary differential...

Here, one of the two functions f (x) and ϕ(x) is prescribed and the other is assumed unknown; also one of the functions g(y) and ψ(y) is prescribed and the other is unknown, and the function... service of automated e-print archives of articles in the fields of< /small>

mathematics, nonlinear science, computer science, and physics.

Catalog of Mathematics