CC BY
1
Vladikavkaz Mathematical Journal 2024, Volume 26, Issue 2, P. 26-38
YAK 517.95
DOI 10.46698/x1302-5604-8948-x
EXISTENCE OF SOLUTIONS FOR A CLASS OF IMPULSIVE BURGERS EQUATION
S. G. Georgiev1'2 and A. Hakem3
1 Department of Mathematics, Sorbonne University, Paris 75005, France;
2 Department of Differential Equations, Sofia University "St. Kliment Ohridski", 15 Tzar Osvoboditel Blvd., Sofia 1504, Bulgaria; 3 Department of EBST, Djillali Liabes University, Sidi Bel Abbes 22000, Algeria E-mail: svetlingeorgiev1@gmail.com, hakemali@yahoo.com
Abstract. We study a class of impulsive Burgers equations. A new topological approach is applied to prove the existence of at least one and at least two nonnegative classical solutions. The arguments are based on recent theoretical results. Here we focus our attention on a class of Burgers equations and we investigate it for the existence of classical solutions. The Burgers equation can be used for modeling both traveling and standing nonlinear plane waves. The simplest model equation can describe the second-order nonlinear effects connected with the propagation of high-amplitude (finite-amplitude waves) plane waves and, in addition, the dissipative effects in real fluids. There are several approximate solutions to the Burgers equation. These solutions are always fixed to areas before and after the shock formation. For an area where the shock wave is forming no approximate solution has yet been found. Therefore, it is therefore necessary to solve the Burgers equation numerically in this area.
Keywords: Burgers equation, impulsive Burgers equation, positive solution, fixed point, cone, sum of operators.
AMS Subject Classification: 47H10, 35K70, 4G20.
For citation: Georgiev, S. G. and Hakem, A. Existence of Solutions for a Class of Impulsive Burgers Equation, Vladikavkaz Math. J., 2024, vol. 26, no. 2, pp. 26-38. DOI: 10.46698/x1302-5604-8948-x.
1. Introduction
The Burgers equation is a fundamental partial differential equation in fluid mechanics. It is also a very important model encountered in several areas of applied mathematics such as heat conduction, acoustic waves, gas dynamics and traffic flow. Analytical solutions of the partial differential equations modeling physical phenomena exist only in few of the cases. Therefore the need for the construction of efficient numerical methods for the approximate solution of these models always exists. Many of the analytical solutions to the Burgers equation involve Fourier series. There are several approximate solutions of the Burgers equation (see [1]). These solutions are always fixed to areas before and after the shock formation. For an area where the shock wave is forming no approximate solution has yet been found. It is therefore necessary to solve the Burgers equation numerically in this area (see [2, 3]). Numerical solutions themselves have difficulties with stability and accuracy.
© 2024 Georgiev, S. G. and Hakem, A.
Here, in this paper, we focus our attention on a class of Burgers equations and we will investigate it for existence of classical solutions. More precisely, consider the problem
ut + uux = 0, t € Jo, x € R,
u(0,x) = u0(x), x € R, (1.1)
u(tk+, x) = u(tk, x) + Ik(u(tk, x)), x € R, k € {1,... , m},
where
(H1) T> 0, 0 = to < ti < ... < tm < tm+1 = T, J = [0,T], Jo = J\{tk}m=1, m € N.
(H2) Ik € C([0,T] x R), |Ik(u)| ^ A|u|rk, u € R, rk > 0, k € {1,... ,m}, A is a positive constant.
(H3) u0 € C 1(R), |u0| ^ B on R, B is a positive constant.
Additional conditions for the constants A and B will be given below. Here
u(tk,x) = lim u(t,x), u(tk+,x) = lim u(t,x),x€ R. t^tfc - t^tfc+
Whereas impulsive differential equations are well studied, the literature concerning impulsive partial differential equations does not see to be very rich. To the best of our knowledge, there are no any references devoted on investigations of the impulsive Burgers equation for existence and uniqueness of classical solutions.
The paper is organized as follows. In the next section, we give some preliminary results. In Section 3, we prove existence of at least one solution for the problem (1.1). In Section 4, we prove existence of at least two nonnegative solutions of the problem (1.1). In Section 5, we give an example that illustrates our main results.
2. Preliminary Results
Below, assume that X is a real Banach space. Now, we will recall the definitions of compact and completely continuous mappings in Banach spaces.
Definition 2.1. Let K : M c X ^ X be a map. We say that K is compact if K(M) is contained in a compact subset of X. The map K is called a completely continuous map if it is continuous and it maps any bounded set into a relatively compact set.
Proposition 2.1 (Leray-Schauder Nonlinear Alternative [4]). Let C be a convex, closed subset of a Banach space E, 0 € U C C, where U is an open set. Let f : U —> C be a continuous, compact map. Then
(a) either f has a fixed point in U;
(b) or there exist x € dU, and A € (0,1), such that x = A/(x).
To prove our existence result we will use the following fixed point theorem which is a consequence of Proposition 2.1.
Theorem 2.1. Let E be a Banach space, Y a closed, convex subset of E, U be any open subset of Y with 0 € U. Consider two operators T and S, where
Tx = e x € U,
for e > 0 and S i U —y E be such that
(i) I — S : U —> Y continuous, compact and
(ii) {xeU : x = A(I — S)x, x € dU} = 0, for any A € (0, ±) .
Then there exists x* € U such that
Tx + Sx — X .
< We have that the operator \{I — S) :U —> Y is continuous and compact. Suppose that there exist x0 € dU and € (0,1), such that
x0 = Ho-(I - S)xo,
e
that is
xo — Ao (I - S)xo,
where Ao = ¿to \ G (0, • This contradicts the condition (ii). From Leray-Schauder nonlinear alternative, it follows that there exists x* € U, so that
a;* = -(I-S)x*,
e
or
ex + Sx — x ,
or
Tx * + Sx * — x *. >
Definition 2.2. Let X and Y be real Banach spaces. A map K : X ^ Y is called expansive if there exists a constant h > 1 for which one has the following inequality
||Kx - Ky||y ^ hyx - y\\x,
for any x, y € X.
Now, we will recall the definition for a cone in a Banach space.
Definition 2.3. A closed, convex set P in X is said to be cone if
1) ax € P for any a ^ 0 and for any x € P,
2) x, —x € P implies x — 0.
Denote P * — P\{0}. The next result is a fixed point theorem which we will use to prove existence of at least two nonnegative global classical solutions of the IVP (1.1). For its proof, we refer the reader to [5] and [6].
Theorem 2.2. Let P be a cone of a Banach space E; Q a subset of P and Ui, U2 and U3 three open bounded subsets of P, such that IJ\ C U2 C Us and 0 € U\. Assume that T : Q —> P is an expansive mapping, S : Us —> E is a completely continuous map and S(Us) C (/ - T)(Q). Suppose that (U2 \Ui) nfi/0, (C73 \ U2) flfi/0, and there exists u0 € P* such that the following conditions hold:
(i) Sx — (I - T)(x - Au0), for any A > 0 and x € dUi n (Q + Au0),
(ii) there exists e ^ 0, such that Sx — (I-T)(Ax), for any A ^ 1 + e, x € dU2 and Ax € Q,
(iii) Sx — (I - T)(x - Au0), for any A > 0 and x € dU3 n (Q + Au0).
Then T + S has at least two non-zero fixed points xi, x2 € P, such that
x1 € <9C/2 n Q and x2 € (Z73 \ F2) n Q,
or
xi € (f/2 \ f/i) n Q and x2 € (F3 \ U2) n Q.
Define the spaces PC (J), PC 1(J) and PC 1(J, C1 (R)) by
PC (J) = {g : g € C (Jo), (3 g(t+), g(t-)) and g(t- )= g(j), j €{1,...,k}} ,
PC 1(J) = { g : g € PC (J) n C 1(Jo), (3 g'(t-), g'(t+)) and g'(t-) = g'(j), j € {1,..., k}}
and PC 1(J, C1
= {u : J x R ^ R : u(-,x) € PC 1(J), x € R and u(t, ■) € C 1(R), t € j}. (2.1) Suppose that X = PC 1(J, C 1(R)) is endowed with the norm
|u|| = sup ^ sup |u(t,x)|, sup |ux(t,x)|,
(t,x)e[tj ,tj+i]xR (t,x)e[tj ,tj+i]xR
sup |ut(t,x)|, j €{1,...,k} (t,x)e[tj ,tj+1]xR
provided it exists. Note that X is a Banach space. For u € X, define the operator
t
u(t,x) — uo(x) — > Ik(u(tk,x))+ I u(s,x) ux(s,x) ds, (t,x) €
t
S1u(t,x) = u(t, x) — uo(x) — ^ Ik(u(tk, x))+ / u(s,x) ux(s,x) ds, (t,x) € J x R.
o<tfc <t o
Lemma 2.1. Suppose that (H 1)-(H3) hold. If u € X satisfies the equation
S1u(t, x) = 0, (t, x) € J x R, (2.2)
then it is a solution of the problem (1.1).
< Let u € X be a solution of the equation (2.2). We differentiate the equation (2.2) with respect to t and we find
ut(t, x) + u(t, x)ux (t, x) = 0,
(t, x) € J x R, and u(0, x) = uo(x), x € R. Next, we put t = tj + and t = tj, j € {1,..., m}, in the equation (2.2) and we obtain
j
0 = u(tj+,x) — uo(x) — ^ Ik (u(tk, x))+y u(s,x) ux(s,x) ds, j €{1,...,m}, x € R,
and
0 = u(tj, x) — uo(x) — ^ Ik(u(tk, x)) + y u(s,x) ux(s,x) ds, j €{1,...,m}, x € R, o<tk <tj o
respectively. Therefore
u(tj+,x) = u(tj,x)+ Ij(u(tj,x)), j €{1,...,m}, x € R. Consequently u satisfies (1.1). This completes the proof. >
Lemma 2.2. Suppose that (H 1)-(H3) hold. If u € X, ||u|| ^ B, then
m
|S1u(t, x) | < 2B + A Y^ Brk + TB2, (t, x) € J x R.
k=1
< We have
|S1u(t, x)| =
^ lu
< u
(t,x) — uo(x) — ^ Ik(u(tk,x))+ / u(s,x)ux(s,x) ds o<tfc<t o
t
|u(t,x)| + |uo(x)| + ^ |Ik(u(tk,x))| + / |u(s,x)ux(s,x)| ds o<tfc<t o
t
|u(t,x)| + |uo(x)| + A ^ |u(tk,x)p + / |u(s,x)ux(s,x)| ds
o<tfc<t
^ 2B + A^Brk + TB2, (t, x) € J x R.
k=1
This completes the proof. >
(H4) Suppose that g € C(J x R) is a nonnegative function, such that
t x
216 (1+1 +12 +13)(1 + |x| + |x|2 + |x|3 + |x|4 + |x|5 + |x|6^ Jg(t1,x1)
t1 , x1 ) dx1
dt1 < D,
(t, x) € J x R, for some constant D > 0. In the last section, we will give an example for a function g and a constant D that satisfy (H4). For u € X, define the operator
t x
S2u(t, x) = J J (t — t1) (x — x1) g(t1,x1)S1u(t1,x1) dx1 dt1, (t, x) € J x R.
oo
Lemma 2.3. Suppose that (H 1)-(H4) hold. For u € X, ||u|| ^ B, we have ||S2u|| < D (2B + A ^ Brk + TB2 J , (t, x) € J x R.
k=1
< We have
|S2u(t,x)| =
t x
J |(t — t1)2(x — x1)3g(t1, x1)S1u(t1, x1) dx1 dt1
oo
<
(t — t1) |x — x1| g(t1,x1)|S1u(t1,x1)| dx1
dt1 ^ ( 2B + A Brk + TB2
k=1
J J(t — t1)2|x — x1|3g(t1 ,x1)(1 +11)(1 + |x1|2 + |x1|3) dx1
dt1
t
t
x
t
x
x
< (2B + A ^ Br + TBA t2(1 + t)|x|3 (1 + |x|2 + |x|3 J I g(ti, xi) dxi
V k=i J 00
< 2B + A ]=J Brk + TB2j , (t,x) € J x R,
and
S2u(t, x)
^ 2
t x
2 J j (t - ti)(x - xi)3g(ti, xi)Siu(ti, xi) dxi dti 0 0
(t - ti)|x - xi|3g(ti, xi)|Siu(ti, xi)| dxi dti
(m
2B+A^Brk + TB 2 k=i
(t - 11)|x - xi|3g(ti,xi)(1 + ti)(1 + |x112 + |xi|3)dxi
(m \ t x 2B + A^Brfc + TB2) t(1+ t)|x|3(1 + |x|2 + |x|3) / f g(ti,xi) dxi k=i / 0 0
^ ^2B + A ]=J Brk + TB2j , (t, x) € J x R,
dti
and
d
—S2u(t,x)
t x
3 J |(t - ti)2(x - xi)2g(ti, xi)Siu(ti, xi) dxi dti 0 0
^ 9
(m
2B+A
k=1
< 3
Brk +TB;
(t - ti) (x - xi) g(ti, xi)|Siu(ti, xi)| dxi
dti
(t - ti)2(x - xi)2g(ti,xi)(1 + ti)(1 + |xi|2 + |xi|3) dxi
dti
< 108|2B + A^ Brk + TB2J t2(1+ t)|x|2(1 + |x|2 + |x|3) / f g(ti,xi) dxi
V k=i / 00
< ^2B + A ]=J Brk + TB2j , (t, x) € J
00 x
dti
Consequently
||S2u|| < ^2B + A Brk + TB2 j
This completes the proof. >
Lemma 2.4. Suppose that (H 1)-(H4) hold. If u € X satisfies the equation
S2u(t, x) — C, (t, x) € J x R, (2.3)
t
x
t
x
for some constant C, then u is a solution to the problem (1.1).
< We differentiate three times with respect to t the equation (2.3) and we get
x
2 y (x — x1)3g(t, x1 )S1u(t, x1) dx1 = 0, (t, x) € J x R, o
or
x
J(x — x1)3g(t, x1)S1u(t, x1) = 0, (t, x) € J x R. o
Now, we differentiate four times with respect to x the last equation and we find
6g(t, x)S1u(t, x) = 0, (t, x) € J x R,
or
g(t, x)S1 u(t, x) = 0, (t, x) € J x R,
whereupon
S1u(t, x) = 0, (t, x) € (0, T] x (R\{0}). Since S1u(-, ■) € C(J x R), we get
0 = lim S1u(t, x) = S1u(0, x) = lim S1u(t, x), (t, x) € J x R.
t^o x^o
Therefore
S1u(t,x)=0, (t, x) € J x R.
Hence and Lemma 2.1, we conclude that u is a solution to the problem (1.1). This completes the proof. >
3. Existence of at Least One Solution
Now, suppose that
(m \
2B + A Brk + TB2J < B.
(H6) e ^B + D ^2B + A f= Brk + TB2^ < B. For u € X, define the operators
Tu(t, x) = —eu(t, x),
Su(t, x) = (1 + e)u(t, x) + eS2u(t, x), (t, x) € J x R.
By Lemma 2.4, it follows that any fixed point of the operator T + S is a solution to the problem (1.1).
Lemma 3.1. Suppose that (H 1)-(H6) hold. For u € X, we have ||(I — S)u|| < B and ||((1+ e)I — S)u|| < eB.
< Applying Lemma 2.3, we get
||(I — S)u|| = || — eu — eS2u| < e||u|| + e||S2u|| < e ^B + D ^2B + ABrk + TB< B
and
||((1+ e)I — S)u|| = ||eS2u|| = e||S2u|| < eD ^2B + A Brfc + TB2j < eB.
This completes the proof. >
Our main result in this section is as follows.
Theorem 3.1. Suppose that (H 1)-(H6) hold. Then the problem (1.1) has at least one solution.
< Let Y denote the set of all equi-continuous families in X with respect to the norm || • ||. Let also, Y = Y be the closure of F,
U = {u € Y : ||u|| < B} .
For u € U and e > 0, define the operators
T (u)(t, x) = eu(t, x),
S(u)(t,x) = u(t,x) — eu(t,x) — eS2(u)(t,x), (t,x) € J x R.
For u € U, we have
||(I — S)u|| = ||eu + eS2u|| ^ e||u|| + e||S2u|| ^ eB1 + eAB1.
Thus, S : U —> X is continuous and (I — S)(U) resides in a compact subset of Y. Now, suppose that there is a u € dU, so that
u = A(I — S)u,
or
u = Ae (u + S2u), (3.1)
for some A € (0, Then, using that S2u(0, x) = 0, we get
u(0, x) = Ae(u(0, x) + S2u(0, x)) = Aeu(0, x), x € R,
whereupon Ae = 1, which is a contradiction. Consequently
{uGU : u = Ai (I - S)u, u e dU} = 0
for any Ai € (0, i). Then, from Theorem 2.1, it follows that the operator T + S has a fixed point u* € Y. Therefore
u*(t,x) = Tu*(t,x) + Su*(t,x) = eu*(t,x) + u*(t,x) — eu*(t,x) — eS2u*(t,x), (t,x) € J x R, whereupon
S2u*(t, x) = 0, (t, x) € J x R.
From here, u* is a solution to the problem (1.1). From here and from Lemma 2.4, it follows that u is a solution to the IVP (1.1). This completes the proof. >
4. Existence of at Least Two Nonnegative Solutions
Suppose:
(H7) Let m > 0 be large enough and A, B, r, L, R1 be positive constants that satisfy the following conditions
r<L<Rh e > 0, Ei > ( -A- + 1 ) L,
V 5m 1
D\2Rl + AY^Rrlk+TR\ \ <-.
\ k=i
For x € X, define the operators
T\u(t, x) = (1 + me)u(t, x) — ey^,
S3u(t, x) = —eS2u(t, x) — meu(t, x) — e^, (t, x) € J x R.
Our main result in this section is as follows.
Theorem 4.1. Suppose that (H 1)-(H4) and (H7) hold. Then the problem (1.1) has at least two nonnegative solutions. < Define
U1 = P~ = {v € P : ||v|| < r },
U = Pl = {v € P : ||v|| < L},
U3 = Pr1 = {v € P : ||v|| < Ri},
D ( m \ L
R2 = Ri + — \2Rl+AYRrlk+TR2l \ + —, m \ ^^ 1 / 5m
\ k=i /
Q = Pr2 = {v € P : ||v|| < R2}.
1) For v1,v2 € Q, we have
||Tivi - Tiv2| = (1 + me)|vi - v21|,
whereupon T\ : Q —>• E is an expansive operator with a constant 1 + me.
2) For v € P_rx , we get
||S3v|| ^4S2v\\+me\\v\\+e^ ^ e (^D + A^J R[k + + mEi + .
Therefore Ss(Pr1) is uniformly bounded. Since S3 : —> E is continuous, we have that ^(Pbi) is equi-continuous. Consequently S3 : Pr1 E is a 0-set contraction.
3) Let vi € PBl. Set
1L
= v\ H--S2vi + -—.
m 5m
Note that S2v 1 + f > 0 on J x R. We have ^ 0 on J x R and
1 L D / m \ T
\\v2W < INI + — ll^i|| + — < + - 2Ei + A V R\k + TR\ + — = R2. m 5m m \ i / 5m
\ k=i /
Therefore v2 € Q and
c L L
-emv2 = -emvi - eS2vi - e— - e—,
or
(/ - Ti)v2 = -emv2 + e— = S3V1. Consequently S3(PBl) C (/ - Ti)(Q).
4) Assume that for any uo € P* there exist A ^ 0 and u € dPr n (Q + Auo) or u € dPRl n (Q + Au0), such that
Sau = (I — T0(u — Auo).
Then
or
—eS2U — meu — e— = —me(u — Xu,q) + e—,
—S2u = Amu0 +
L
Hence,
II« =
L
Xmuo + — 5
>
L 5"'
This is a contradiction.
5) Suppose that for any e\ ^ 0 small enough there exist a u\ € <9Pl and Ai ^ 1 + e\, such that X\Ui € P_rx and
S3U1 = (/ — Ti}(AiUi). (4.1)
In particular, for e\ > -g^, we have u\ € <9Pl, Ai«i € P_r1; \\ ^ 1 + e\ and (4.1) holds. Since u\ € <9Pl and \\u\ € P_r1; it follows that
Moreover,
or
^ + 1)L<AiL = AI||UI|KEI.
LL —eboUi — meu 1 — e— = —\\meui + e—, 10 10
¿>2^1 + t" = (Ai — l)mu\. 5
From here,
and
5
S2U1 +
L
= (Ai — 1)m||ui|| = (Ai — 1)mL
2
5m
+ 1 ^ Ai,
which is a contradiction.
Therefore all conditions of Theorem 2.2 hold. Hence, the problem (1.1) has at least two solutions ui and u2 so that
= L < ||U2|| < Ri,
or
r <
<L<
< Ri.
This completes the proof. >
5. An Example
Let T = 1, n = 1, m = 3, t\ = 4, t2 = ¿3 = Consider the problem
+ uux = 0, t €
"■ïMMMi'sMi1
x € R,
5
5
Here Then
Take Then
and
u(0,x) = —iéI, 1 + x2
u(ifc+, x) = u(ifc, x) + (u(ifc, x))4, x € R, k € {1, 2, 3}. A = 1, B = 1, C = 1, r = 1.
m
2B + A ^ Brk + TB2 = 2 + 3 + 1 = 6. k=i
1 3 ~ 11
D = e = ^r, r = -, Ri = l, r = -, L = -, m = 1050.
10 2' ' 8' 2'
-D + A ^ BTk + TB2j = y^J <1 = B
B + D 2B + A^B^+TB2 ) ) = 1 (i+ 6 ) <1=5_
k=i
Thus, (H5) and (H6) hold. Hence and Theorem 3.1, it follows that the considered problem has at least one solution. Moreover,
and
D(2Rl+Aj2R?+TRi)=-^<i- =
1 TIIlll ~~ 1050 < 10 ~ 5' k=1 /
Consequently (H7) holds. By Theorem 4.1, it follows that the considered problem has at least two nonnegative solutions.
Now, we will construct a function g for arbitrary n. Let
1 + sn\/2 +
7 / \ i 1 i S "y 2 | S , , % S v 2 __
(s)= gT^TV2+^' Ks) = arctanYT^22' SGR-
Then
22\/2s (1 — s ) ;// , ll\/2s (1 + s )
h (s) =--i-'—=-, '(s) = —---s € R.
yJ (l-sllV2 + s22)(l+sllV2 + s22) yJ 1 + s40
Therefore
-œ < lim (1 + |s| +-----h s6)h(s) < to,
-to < lim (1 + |s| +-----h s6)l(s) < to.
Hence, there exists a positive constant C1 so that
1 , l + s11v/2 + ^22 1 'ai-
e
s € R. Note that by [7, p. 707, Integral 79], we have
dz
1 + z4
4\/2
1 + z\/2 + z2
1 -zV2 + z2
+
2\/2
arctan ■
Let
.10
Q(s) =
(1 + s2)4(1 + s44)(1 + s + s2)2 Then there exists a positive constant C2 so that
t x
216(1 + t + t2 +t3) (1 + |x| + ■ ■ ■ + x6)/ Q(s) ds J Q(y) dy
0 0
Take #(i,x) = ^Q(t)Q(x), (i,x) e J x R. Hence,
t x
216(1 +1 +12 + t3) (1 + |x| + ■ ■ ■ + x6) y Jg(s, y) dy
1z
, s € R.
2 •
< C2, (t,x) € J x R.
00 References
ds < D, (t,x) € J x R.
1. Konicek, P., Bednarik, M. and Cervenka, M. Finite-Amplitude Acoustic Waves in a Liquid-Filled Tube, Hydroacoustics (Editors: E. Kozaczka, G. Grelowska), Gdynia, Naval Academy, 2000, pp. 85-88.
2. Mitome, H. An Exact Solution for Finite-Amplitude Plane Sound Waves in a Dissipative Fluid, Journal of the Acoustical Society of America, 1989, vol. 86, no. 6, pp. 2334-2338.
3. Ochmann, M. Nonlinear Resonant Oscilllations in Closed Tubes — An Application of the Averaging Method, Journal of the Acoustical Society of America, 1985, vol. 77, no. 1, pp. 61-66. DOI: 10.1121/1.391901.
4. Agarwal, R. P., Meehan, M. and O'Regan, D. Fixed Point Theory and Applications, Cambridge Tracts in Mathematics, Vol. 141, Cambridge University Press, 2001. DOI: 10.1017/CB09780511543005.
5. Djebali, S. and Mebarki, K. Fixed Point Index for Expansive Perturbation of fc-Set Contraction Mappings, Topological Methods in Nonlinear Analysis, 2019, vol. 54, no. 2, pp. 613-640. DOI: 10.12775/TMNA.2019.055.
6. Mouhous, M., Georgiev, S. and Mebarki, K. Existence of Solutions for a Class of First Order Boundary Value Problems, Archivum Mathematicum, 2022, vol. 58, no. 3, pp. 141-158. DOI: 10.5817/AM2022-3-141.
7. Polyanin, A. and Manzhirov, A. Handbook of Integral Equations, CRC Press, 1998.
Received October 13, 2023 Svetlin G. Georgiev
Department of Mathematics, Sorbonne University,
Paris 75005, France,
Professor
Department of Differential Equations, Sofia University "St. Kliment Ohridski", Faculty of Mathematics and Informatics, 15 Tzar Osvoboditel Blvd., Sofia 1504, Bulgaria, Professor
E-mail: svetlingeorgiev1@gmail. com Ali Hakem
Djillali Liabes University, Laboratory ACEDP,
Sidi Bel Abbes 22000, Algeria,
Professor
E-mail: hakemali@yahoo.com
https://orcid.org/0000-0001-6145-4514
Владикавказский математический журнал 2024, Том 26, Выпуск 2, С. 26-38
СУЩЕСТВОВАНИЕ РЕШЕНИЙ ДЛЯ ОДНОГО КЛАССА ИМПУЛЬСНЫХ УРАВНЕНИЙ БЮРГЕРСА
Георгиев С. Г.1'2, Хакем А.3
1 Университета Сорбонны, Франция, 75005, Париж, 2 Софийский университет им. Св. Климента Охридского, Болгария, 1504, София, Бульвар Царя-Освободителя, 15;
3 Джиллали Лиабес университет, Алжир, 22000, Сиди Бел Аббес E-mail: svetlingeorgiev1@gmail.com, hakemali@yahoo.com
Аннотация. Мы изучаем класс импульсных уравнений Бюргерса. Для доказательства существования хотя бы одного и хотя бы двух неотрицательных классических решений применяется новый топологический подход. Обоснования опираются на недавние теоретические результаты. Основное внимание уделяется классу уравнений Бюргерса и вопросу существования классических решений. Уравнение Бюр-герса можно использовать для моделирования как бегущих, так и стоячих нелинейных плоских волн. Простейшее модельное уравнение способно описать нелинейные эффекты второго порядка, связанные с распространением плоских волн большой амплитуды (волн конечной амплитуды), а также дисси-пативные эффекты в реальных жидкостях. Существует несколько приближенных решений уравнения Бюргерса. Эти решения всегда фиксируются до и после образования ударной волны. Для области формирования ударной волны приближенное решение пока не найдено. Поэтому в этой области необходимо численное решение уравнения Бюргерса.
Ключевые слова: уравнение Бюргерса, импульсное уравнение Бюргерса, положительное решение, неподвижная точка, конус, сумма операторов.
AMS Subject Classification: 47H10, 35K70, 4G20.
Образец цитирования: Georgiev, S. G. and Hakem, A. Existence of Solutions for a Class of Impulsive Burgers Equation // Владикавк. мат. журн.—2024.—Т. 26, № 2.—C. 26-38 (in English). DOI: 10.46698/x1302-5604-8948-x.
