CC BY
0MODELING OF NONLINEAR PROCESSES BY SOLVING AN EQUATION OF HAMILTON JACOBI TYPE
Rakhmonova M
Economic and pedagogic institute https://doi.org/10.5281/zenodo.11211470
Annotation. The qualitative properties of the problem Cauchy to the second order degenerate type parabolic equation is established using the solution of the corresponding to the equation Hamilton- Jacoby equation. It is solved the problem choosing of an appropriate initial approximation for the iteration process keeping properties of localization of solutions, a finite speed of perturbation of distribution. The results of numerical experiments are discussed.
Key words: non-linear problems, blow-up solutions, self-similar solutions, Hamilton-Jacoby equation, semi-infinite interval, numerical solution, self-similar equations.
One of the most remarkable properties that distinguish non-linear problems from linear ones is the possibility of a singularity, even with absolutely smooth data, or more precisely, in a data class for which the theory of existence, uniqueness, and continuous dependence can be established in small time intervals. In non-linear problems, even with smooth coefficients, unbounded solutions (regimes with peaking) can arise [1].
The present work continues the cycle of studies of blow-up solutions of nonlinear parabolic equations.
For the first time such a class of blow up solution interest in with exacerbation is caused by their unusual properties localization and related with it the emergence of non-stationary dissipative structures. Detailed reviews can be found in many works [1-5]. We note that self-similar solutions play an important role in the study of these problems. They describe all types of structures and waves that can arise in a given nonlinear medium [4]
In this paper, we study in the domain Qt ~{ (x)' 0 < t < T' x e R+} the qualitative properties of solutions on the basis of a self-similar analysis to the following problem for the heat equation with double nonlinearity
r
L(u ) = -
du d
dt dx
u
duk
dx
p-2 , \ p duk
dx
= 0 (1)
u |i=0 = u0 (*) > 0, u |x=0 = (T - tya, 0 < t < T, a> 0 (2) where m, p e R are numerical parameters characterizing the property of a nonlinear medium. The first boundary condition is called the blow-up regime.
Qualitative properties of the problem is established using the solution of corresponding to the equation (1) the Hamilton- Jacoby equation (the first order double nonlinear equation)
P-2
du dt
= kp u
—1 m+k—3
d u
dx
d) u
(—)2, u(0, x) = u (x) > 0, x g R+ (3)
dx
This circumstance indicates the localization of combustion: the temperature increases in the exacerbation in the contracting region near the center of symmetry, while outside this region [1]
It tends to the limiting, time-constant temperature distribution, although the self-similar solution in the LS-mode exists on a semi-infinite interval, described the regime is realized only in a restricted area. Using comparison theorems, the strict localization of combustion processes in this case and "circumcision of the infinite tail for a self-similar solution [1]
Consider the following self-similar solution of the equation (1)
u(t,x) = (T-1Vf (i), i = x[r(t)]-1/p,T(t) = (T-1)
1-(m+k ( p-1)-2))a
(4)
where function f (ç) satisfy to the self-similar equation
d / rm-1
dfk
f ) - [1 - (k(p-1) + m - 2)a] i f + af = 0 dç p dq
(5)
Subtituting (4) to (3) lead Hamilton Jacoby equation (3) to the following self-similar
Hamilton Jacoby equation
f
m+k ( p-1)-2
dfk
dÇ
P-2
f 2 (1 - (m + k(p-1) - 2)g)e df
dç
ii~ + af = 0 (6) dç
Note that the function
&(£) = (b-i )+*, where (n)+ = max(0,n), yx =-P-1
P
m + k ( p -1) - 2 p -1
is approximately solution of the Hamilton Jacoby equation (6)
f i) = 0(Ç)œ(T), r(t ) = (T -1 )
1-(m+k ( p-1))a
, Oi) = (b-iY1,b >0
At the numerical solution of a problem the equation was approximated on a grid under the in a combination to the method of balance. Iterative processes were constructed basing on the method Picard, Newton and a special method. Results of computational experiments shows, that all listed iterative methods are effective for the solution of considered nonlinear problems and leads to the nonlinear effects if we will use as initial approximation the solutions of Hamilton Jacoby self-similar equations constructed by the method of nonlinear splitting and the method of standard equation [4].
For computation the following numerical scheme were used
s s+1 s s+1 s s+1 s
A y,-1 - C, y, + Bi yi+! =- F,
1
a+1 = '
( y/+1 )m-1
( y^f - ( yj+l)k
h
p - 2
+
( j )m-1
( yj+l)k - ( y,-1)k
h
p-2
a =
2
( j )m-1
( y+')k - ( yj+1)k
h
p - 2
+
( yj+1 )m 1
( yj+11)k - ( yj^
h
p-2
m+k ( p-1)- p
A =2 -kp-2 ■ yi y-1
,p-2
h
h
B = 2 • kp
-2 y
m+k( p-1)-p
i+1
yi+1 - y
P-2
C = A + B +1 F = -*• yß + y
l l l i s i s i
REFERENCES
1. A.A Samarskii., V.A Galaktionov., S.P.Kurdyomov, A.P.Mikhailov Blow-up in quasilinear parabolic equations. Berlin, 4, Walter de Grueter, (1995), 535.
2. A. S. Kalashnikov Some problems of the qualitative theory of nonlinear degenerate parabolicequations of second order," Russian Mathematical Surveys, 1987, vol. 42, pp. 169222,.
3. P. Cianci, A. V. Martynenko, and A. F. Tedeev, "The blow-up phenomenon for degenerate parabolic equations with variable coefficients and nonlinear source," Nonlinear Analysis: Theory, Methods &Applications A, vol. 73, no. 7, pp. 2310-2323, 2010.
4. M.Aripov // Method of standard equations to solving of nonlinear boundary value problems. Tashkent FAN, (Monography), 1988, p. 137.
