For the first time, a correction method of test solutions
was proposed for a model wave equation with two constant
coefficients-rates for rectilinear first quarter in [1] and for
curvilinear first quarter in [2] of the plane. The developed
correction method is used to obtain classical (twice contin-
uously differentiable) solutions for which are twice contin-
uously differentiable on the critical characteristics for the
obtained classical solutions. A smoothness criterion of the
right-hand side of this equation in the first quarter of the
plane is derived. It is proved that if the right-hand side of
the two-rate model wave equation depends only on one of
two independent variables, then its continuity in one of these
variables is necessary and sufficient for these solutions to be
classical solutions of this equation in quarter of the plane.
Using the generalization of the correcting Goursat problem
from [1] to two variable coefficients-rates a1(x, t)and a2(x, t)
This work was carried out according to the SPNI ”Convergence-2025”, No.
11, research work 1.2.02.3.
by the new ”implicit characteristics method” it was calculated
classical solutions of new model wave equation
utt(x, t)+(a1−a2)utx(x, t)−a1a2uxx(x, t)−
−a−1
2(a2)tut(x, t)−a1(a2)xux(x, t) = f(x, t),
(x, t)∈˙
G∞=]0,+∞[×]0,+∞[,(1)
where f(x, t), a1(x, t), a2(x, t)are given real functions of
the variables xand t.
The characteristic equations dx = (−1)ia3−i(x, t)dt, i =
1,2,corresponds to equation (1). to the which have implicit
general integrals gi(x, t) = Ci, Ci∈R, i = 1,2.If the
coefficients a3−iare strictly positive, i. e. a3−i(x, t)≥
a(0)
3−i>0,(x, t)∈G∞,then variable ton the characteristics
g1(x, t) = C1, C1∈R, strictly decreases on characteristics
g2(x, t) = C2, C2∈R, and strictly increases with the
growth of the variable x. Therefore, the implicit functions
yi=gi(x, t) = Ci, x ∈R, t ≥0,have strictly mono-
tone implicit inverse functions x=hi{yi, t}, t ≥0,and
t=h(i)[x, yi], x ∈R, i = 1,2.By the definition of inverse
mappings on G∞, the following inversion identities hold:
gi(hi{yi, t}, t) = yi, t ≥0; hi{gi(x, t), t}=x, x ∈R,
i= 1,2,(2)
gi(x, h(i)[x, yi]) = yi, x ∈R;h(i)[x, gi(x, t)] = t, t ≥0,
i= 1,2,(3)
hi{yi, h(i)[x, yi]}=x, x ∈R;h(i)[hi{yi, t}, yi] = t, t ≥0,
i= 1,2.(4)
If a3−i(x, t)≥a(0)
3−i>0,(x, t)∈G∞, a3−i∈C2(G∞),
then the implicit functions gi, hi, h(i)are twice continuously
differentiable with respect to x, t, yi, i = 1,2,on G∞.
Let Ck(Ω) be a set of ktimes continuous differentiable
functions on the subset Ω⊂R2and C0(Ω) = C(Ω).The
critical characteristic g2(x, t) = g2(0,0) divides G∞into two
sets G−={(x, t)∈G∞:g2(x, t)> g2(0,0)}and G+=
{(x, t)∈G∞:g2(x, t)≤g2(0,0)}.
On the Correction Method of Test Solutions to the New Model Wave
Equation with Two Variable Rates in First Quarter of the Plane
FEDOR EGOROVICH LOMOVTSEV
Mechanics and Mathematics Faculty
Belarusian State University
Minsk, BELARUS
Abstract: A correction method of test solutions into classical solutions to a new inhomogeneous model wave
equation with variable two-rates in the first quarter of the plane was developed to derive the minimum
smoothness requirement of its right-hand side. In the case of different rates, it is not possible to derive the
minimum smoothness of right-hand side of the inhomogeneous model wave equation in the first quarter of the
plane without correcting both test generalized and test classical solutions. In this paper, classical solutions to
the inhomogeneous two-rate model wave equation and the smoothness criterion of their right-hand side are
obtained. We need them to find explicit unique and stable classical solutions and Hadamard correctness criteria
to mixed (initial-boundary) problems for this wave equation using developed by the author of the ”implicit
characteristics method”.
Keywords: Two-rate model wave equation, correction method, test solution, classical solution, implicit
characteristics method, smoothness criterion, correctness criterion
Received: November 19, 2022. Revised: June 13, 2023. Accepted: July 17, 2023. Published: August 28, 2023.
1. Introduction
2. Classical Solutions of a
Inhomogeneous Model Wave
Equation With Variable Two-rates
EQUATIONS
DOI: 10.37394/232021.2023.3.5
Fedor Egorovich Lomovtsev
E-ISSN: 2732-9976
41
Volume 3, 2023
Theorem 1. Let be a3−i(x, t)≥a(0)
3−i>0,(x, t)∈G∞=
[0,+∞[×[0,+∞[, a3−i∈C2(G∞), i = 1,2.If the function
f∈C(G∞),then on G+equation (1) has classical solutions
F()(x, t) =
g2(x,t)
Z
0
h1{g1(x,t),τ }
Z
h1{g1(−εg2(x,t),g2(x,t)),τ }
f(δ, τ)
a1(δ, τ) + a2(δ, τ)×
×exp (g1(x,t)
Z
g1(δ,τ )
E(e
δ, eτ)ds)dδdτ+
+
t
Z
g2(x,t)
h1{g1(x,t),τ }
Z
h2{g2(x,t),τ }
f(δ, τ)
a1(δ, τ) + a2(δ, τ)×
×exp (g1(x,t)
Z
g1(δ,τ )
E(e
δ, eτ)ds)dδdτ, ε = ˜ε+ 1 >0,(5)
where in the exponent is the integrand
E(˜
δ, ˜τ) =
a2
2(˜
δ, ˜τ)a1(˜
δ,˜τ)
a2(˜
δ,˜τ)˜
δ−a2(˜
δ, ˜τ)a1(˜
δ,˜τ)
a2(˜
δ,˜τ)˜τ
[a1(˜
δ, ˜τ) + a2(˜
δ, ˜τ)]2g1(˜
δ, ˜τ)˜
δ
.
Theorem 2. Let the assumptions of Theorem 1 be true. Then
the functions (5) are classical solutions of equation (1) on G+
for necessary smoothness
f∈C(G∞),
t
Z
0
fh1{g1(x, t), τ}, τdτ ∈C1(G+),
h2{g2(x,t),g2(x,t)}
Z
−εg2(x,t)
fδ, g2(x, t)dδ−
−
g2(x,t)
Z
0
fh1{g1−εg2(x, t), g2(x, t), τ}, τdτ −
−
t
Z
g2(x,t)
fh2{g2(x, t), τ}, τdτ ∈C1(G+).(6)
Theorem 3. In the requirements of Theorem 1, equation (1)
has classical solutions
F(ϑ)(x, t) =
g2(x,t)
Z
0
h1{g1(x,t),τ }
Z
h1{g1(ϑg2(x,t),g2(x,t)),τ }
f(δ, τ)
a1(δ, τ) + a2(δ, τ)×
×exp (g1(x,t)
Z
g1(δ,τ )
E(˜
δ, ˜τ)ds)dδdτ +
+
t
Z
g2(x,t)
h1{g1(x,t),τ }
Z
h2{g2(x,t),τ }
f(δ, τ)
a1(δ, τ) + a2(δ, τ)×
×exp (g1(x,t)
Z
g1(δ,τ )
E(˜
δ, ˜τ)ds)dδdτ, ϑ =˜
ϑ−1≥1,(7)
on G−under the necessary smoothness conditions
f∈C(G−),
t
Z
0
fh1{g1(x, t), τ}, τdτ ∈C1(˜
G−),
h2{g2(x,t),g2(x,t)}
Z
ϑg2(x,t)
fδ, g2(x, t)dδ −
−
g2(x,t)
Z
0
fh1{g1ϑg2(x, t), g2(x, t), τ}, τdτ −
−
t
Z
g2(x,t)
fh2{g2(x, t), τ}, τdτ ∈C1(G−).(8)
Corollary 1. Under the assumptions of Theorem 1, the
functions F()from (5) on G+and F(ϑ)from (7) on G−
with ˜
ϑ= ˜ε+ 2 are classical solutions of Eq. (1) on the first
quarter of the plane G∞with smoothness criterion (6) and
(8) for ˜
ϑ= ˜ε+ 2 of the right-hand side fon G∞.
Corollary 2. Let the assumptions of Theorem 1 hold and the
right-hand side fof Eq. (1) does not depend on xor tin G∞.
Then the continuity of f∈[0,+∞[in tor x, respectively, is
necessary and sufficient for the functions F()from (5) and
F(ϑ)from (7) with ˜
ϑ= ˜ε+ 2 are classical solutions of the
inhomogeneous equation (1) in G∞.
Corollary 3. Let the requirements of Theorem 1 be true and
the function fdepends on xand t. Then for f∈C(G∞)the
requirement that the integrals from (6) and ( 8) for ˜
ϑ= ˜ε+ 2
bilong to the space C1(G∞)it is equivalent that they belong
to the spaces C(1,0)(G∞)or C(0,1)(G∞).Here C(1,0)(G∞)
or C(0,1)(G∞)are respectively, the spaces of continuously
differentiable with respect to xor tand continuous with
respect to tor xfunctions on G∞.
Notes. In the rectilinear [1] and curvilinear [2] first quarter
of the plane for the wave equation (1) with constant coeffi-
cients a3−i(x, t) = a(0)
3−i>0, i = 1,2,its particular classical
solution is constructed on G∞for the parameters ˜ε= 0 on
G+and ˜
ϑ= 2 on G−,which satisfy the equality ˜
ϑ= ˜ε+ 2
from our Corollary 1. Calculation of classical solutions with
minimal smoothness the right-hand side of the wave equations
in a quarter plane by correction method in mathematics is a
kind of analogue of the relativity theory in physics.
[1] F.E. Lomovtsev, “Correction method of test solutions to the general wave
equation in the first quarter of the plane for the minimum smoothness of
its right-hand side,“ Zhurnal Belorussky state university. Mathematics.
Computer science. No. 3. pp.38–52. 2017.
[2] F.E. Lomovtsev, “In the curvilinear first quarter of the plane, the correc-
tion method of test solutions for the minimum smoothness of the right-
hand side of the wave equation with constant coefficients,“ Kulyashov.
No. 2 (60). Seryya B. Pryrodaznauchyya sciences (mathematics, physics,
biology). pp.7–22. 2022.
References
EQUATIONS
DOI: 10.37394/232021.2023.3.5
Fedor Egorovich Lomovtsev
E-ISSN: 2732-9976
42
Volume 3, 2023
Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
The author contributed in the present research, at all
stages from the formulation of the problem to the
final findings and solution.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interest
The author has no conflict of interest to declare that
is relevant to the content of this article.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en
_US
EQUATIONS
DOI: 10.37394/232021.2023.3.5
Fedor Egorovich Lomovtsev
E-ISSN: 2732-9976
43
Volume 3, 2023
