CC BY
7УДК: 517.95 MSC2010: 35E05
DIFFERENTIAL EQUATION ASSOCIATED WITH INVOLUTIONS
© F. N. Dekhkonov
National University of Uzbekistan, 100174, Tashkent, Uzbekistan e-mail: f.n.dehqonov@mail.ru
Differential equation associated with involutions. Dekhkonov F. N.
Abstract. In this paper, we consider with a class of system of differential equations whose argument transforms are involutions. In this an initial value problem for a differential equation with involution is reduced to an initial value problem for a higher order ordinary differential equation. Then either two initial conditions are necessary for a solution; the equation is then reduced to a boundary value problem for a higher order ODE.
Keywords: involution, linear differential equation, fixed point, boundary value problem
introduction
When studying the general properties of functional differential equations, it is always important to find and solve selected classes of equations as explicitly as possible, using methods that are capable of generalization. Differential equations with involutions is one of those classes.
The concept of involution is fundamental for the theory of groups and algebras, but, at the same time, being an object in mathematical analysis properties allow the obtaining of further information concerning this object. In order to be clear in this respect, let us define what we understand by involution in this analytical context. We follow the definitions of [1], [2] and [8].
Definition 1. Let A С R be a set containing more that one point and f : A ^ A a function such that f is not the identity Id. Then f is an involution if
f2 = f о f = Id
or, equivalently, if
f = f-1.
If A = R, we say that f is a strong involution (see, [2]).
Example 1. The following involutions are the most common examples:
1. f : R ^ R, f (x) = — x is an involution known as reflection.
2. f : R \ {0} ^ R \ {0}, f (x) = X known as inversion.
3. Let a, b, c G R, cb + a2 = 0, c = 0,
„ . (a i ^-f a i „. . ax + b
f : R \ a} ^ R \ - , f(x) = -,
c c cx — a
is a family of functions known as bilinear involutions.
1. Differential equations with involutions
Differential equations with involutions were introduced for the first time [6], [7], [9] and [10] and since then have become an important part in the general theory of functional differential equations, with applications to certain biomedical models [3], stability of motion [4], and the pantograph equation [5]. They can be transformed into ordinary differential equations and thus provide an abundant source of relations with analytic solutions, as well as heuristic ideas for equations of more general nature.
Definition 2. An expression of the form
F(x,y(fi(x)),...,y(f(x)),...,y(n)(fi(x)),...,y(n)(ffc(x))) = 0, x G R,
where f1, ...fk are involutions and F is a real function of nk + 1 real variables is called differential equations with involutions.
Example 2. The solution of the initial-value problem for the differential equation with reflection of the argument,
y'(x) = ay(—x), y(0) = yo.
Then, we get the following solution (see, [1])
y(x) = y0 (cos ax + sin ax).
Set a "new"involution function
f (x) = loga (b — ax), a > 0, a = 1, b> 0,
where, if a > 1, x < loga b, if 0 <a< 1, x > loga b. We consider the following problem
y'(x) = y(f (x)), y(0) = yo, f (x) = ln(4 — ex), x< ln4. (1)
If y(x) is a C1 solution then it is C2. By differentiation we have
y''(x) = f'(x) y'(f (x)),
than from f (f (x)) = x and (1), we get
y//(x) = f'(x) y(x) = —-7 y(x).
ex — 4
So we have (1) is equivalent to the ordinary Cauchy problem
y"(x) = 1-7 ^(x), y(°)= y'(0) = yo, x< ln4. (2)
ex 4
Obviously,
y(x) = Ci ((ex — 4) ln(4 — ex) — x (ex — 4) + ex) + 62 (ex — 4). Then from the initial conditions, we can write
yo = Ci(1 — 3ln3) — 3C2 jo = Ci(ln3 + 5) + C2
Consequently, we have
n yo n 1 + ln3
Ci = —, 62 = —yo
4, 2 4 ,
and
y(x) = f ((ex — 4) ln(4 — ex) — x (ex — 4) + ex) — ^lii+M
2. System of differential equations with involutions
In this section, we consider a system of differential equations with involution.
Theorem 1. Let the initial value problem
x'(t) = Fi (t,x(t),y(t),y(f (t))),
/ x(to) = xo, y(to) = Уo, (3)
y'(t) = F2(t,x(t),y(t),x(f (t))),
satisfy the following hypotheses:
(1) The function f (t) is a continuously differentiable strong involution with a fixed point to.
(2) The functions Fi, F2 are defined and are continuously differentiable in the whole space of its arguments.
(3) The given equations are uniquely solvable with respect to y(f (t)), x(f (t)):
y(f (t)) = Gi(t,x(t),y(t),x'(t)), (4)
x(f (t)) = G2(t,x(t),y(t),y'(t)). (5)
Then the solution of the system of ordinary differential equations <9Fi d Fi d Fi d Fi
x''(i>=ir+SMx' (t)+dy-(t) y' «+ff' W F1f «,*<' (i»,y(f «>,»«>,(6)
and
dF2 dF2 dF2 d F2
y(i»=ir+dF)x (t)+dyicDv (t)+dycTM)f'(t) F2(f (t),(t)), y(f (t)),x(t)),(7)
(where y(f (t)) and x(f (t)) are given by expression (4) and (5)) with the initial conditions
x(to) = xo, x' (to) = Fi(to,xo,yo ,yo), (8)
and
y(to) = yo, y' (to) = F2 (to, xo, yo, xo). (9)
Proof. Equations (6) and (7) are obtained by differentiating (3). Indeed, we can write ,,, , dFi dFi ,. . dFi ,. . dFi „,. .
x <*> = "aT + a*) (t) + y(t> + f) f'(t) (t)),
and
d F2 d F2 dF2 d F2
y' '(t>=1F2+mx'(t)+-my (t)+ff (t) y (f (t)),
than from (3) and relation f (f (t)) = t its follows that
x' (f (t)) = Fi(f (t),x(f (t)),y(f (t)),y(t)),
and
y' (f (t)) = F2(f (t),x(f (t)),y(f (t)),x(t)).
The second of the initial conditions (8), (9) are compatibility condition and is found from (3), with regard to (3) initial condition and f (to) = to. □
Example 3. We consider the following initial value problem
x'^ y(f ^ x(0) = xo, y(0) = yo, f (t) = —t. (10)
V (t)= x(f (t)),
We can write
'x''(t) = f'(t) y'(f (t)), y'(t) = f'(t) x'(f (t)). Than from f(f(t)) = t and (10) we get
x''(t) = f'(t) x(t), y''(t) = f'(t) y(t).
So we have (10) is equivalent to the boundary value problem x''(t) + x(t) = 0,
x(0) = xo, y(0) = yo, x' (0) = yo, y' (0) = xo
y' '(t)+ y(t) = 0, Obviously,
x(t) = 6i cos t + 62 sint, y(t) = 63 cos t + 64 sin t. Then from the boundary conditions, we can write
'x(t) = xo cost + yo sint,
y(t) = yo cos t + xo sin t.
Theorem 2. Let the initial value problem
x' (t) = Fi(t,x(t),y(t),x(f (t))),
x(to) = xo, y(to) = yo, (11)
y' (t) = F2(t,x(t),y(t),y(f (t))),
satisfy the following hypotheses:
(1) The function f (t) is a continuously differentiable strong involution with a fixed point t0.
(2) The functions Fi, F2 are defined and are continuously differentiable in the whole space of its arguments.
(3) The given equations are uniquely solvable with respect to y(f (t)), x(f (t)):
x(f (t)) = Gi(t,x(t),y(t),x' (t)), (12)
y(f (t)) = G2(t,x(t),y(t),y' (t)). (13)
Then the solution of the system of ordinary differential equations
... , dFi dFi . dFi .. .
x' '(t) = St + SM x' (t) + y (i)+
d Fi
+ If) f'(t) Fi(f (t), x(f (t)), y(f (t)), x(t)), (14)
and
dF2 d F2 d F2 d F2
y''« = IF+mx'W+y'(i>+f)f (t) F2(f(t),x(f(t)),y(f(i)),y(i)),(15)
(where x(f(t)) and y(f (t)) are given by expression (12) and (13)) with the initial conditions
x(to) = xo, x' (to) = Fi(to,xo,yo,xo), (16)
and
y (to) = yo, y' (to) = F2 (to, xo, yo, yo). (17)
Proof. This theorem proof similar to theorem 1. Equations (14) and (15) are obtained by differentiating (11). Indeed, we can write
,,, , dFi dFi ,, , dFi ,, , dFi „,, ,
x''(i) = rn + x (i) + m y(t) + axiT(t))f'(t) x (f (t)),
and
d F2 d F2 dF2 d F2
v*4'=i^2+mx' w+didy (i)+dyiTMf (i) y (f (i)),
than from (11) and relation f (f (t)) = t its follows that
x' (f (t)) = Fi(f (t),x(f (t)),y(f (t)),y(t)),
and
y' (f (t)) = F2(f (t),x(f (t)),y(f (t)),x(t)). The second of the initial conditions (16), (17) are compatibility condition and is found from (11), with regard to (11) initial condition and f(t0) = t0. □
Example 4. We consider the following problem
x'(t) = a x( f (t)) 1 1
,(()) a Cf;)))), x(1)= xo, y(1)= yo, f (t) = -, a,p > -. (18)
y' (t)= p y(f (t)), t 2
We can write the following system
x (t) = a f (t) x (f(t)), y'(t)= p f'(t) y'(f (t)).
Than, according to f(f(t)) = t and (18) we get
x (t) = a2 f (t) x(t), y'(t)= p2 f'(t) y(t).
Then we have (18) is equivalent to the boundary value problem t2 x''(t) + a2 x(t) = 0,
x(1) = ^ y(1)= Vo, x (1) = a xo, y (1) = p Vo.
t2 y' '(t)+ p2 y(t) = 0,
Obviously,
(x(t) = Vt (c1 cos lnt + 62 sin ^f-1 lnt),
|y(t) = Vt (63 cos ^ 4f-1 ln t + 64 sin v/4f~T ln t). Then from the boundary conditions, we can write the following solution ix(t) = xo cos ln t + sin ln t),
|y(t) = yo V ( cos ln t + V== sin ln t).
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