The Cauchy Problem for the General Telegraph Equation with

Variable Coefﬁcients under the Cauchy Conditions on a Curved

Line in the Plane

FEDOR LOMOVTSEV

Belarusian State University

Department of Mathematical Cybernetics

Independence Avenue, 4 Minsk

BELARUS

ANDREY KUKHAREV

Belarusian State University

Department of Differential Equations

Independence Avenue, 4 Minsk

BELARUS

Abstract: The Riemann method is used to prove the global correctness theorem to Cauchy problem for a general

telegraph equation with variable coefﬁcients under Cauchy conditions on a curved line in the plane. The global cor-

rectness theorem consists of an explicit Riemann formula for a unique and stable classical solution and a Hadamard

correctness criterion for this Cauchy problem. From the formulation of the Cauchy problem, the deﬁnition of its

classical solutions and the established smoothness criterion of the right-hand side of the equation, its correctness

criterion is derived. These results are obtained by Lomovtsev’s new implicit characteristics method which uses

only two differential characteristics equations and twelve inversion identities of six implicit mappings. If the right-

hand side of general telegraph equation depends only on one of two independent variables, then it is necessary and

sufﬁcient that it be continuous with respect to this variable. If the right-hand side of this equation depends on two

variables and is continuous, then in its integral smoothness requirements it is necessary and sufﬁcient the continu-

ity in one and continuous differentiability in the other variable. The correctness criterion represents the necessary

and sufﬁcient smoothness requirements of the right-hand side of the equation and the Cauchy data. From the

established global correctness theorem, the well-known Riemann formulas for classical solutions and correctness

criteria to Cauchy problems for the general and model telegraph equations in the upper half-plane are derived. In

the works of other authors, there is no necessary (minimally sufﬁcient) smoothness on the right-hand sides of the

hyperbolic equations of real Cauchy problems for the set of classical (twice continuously differentiable) solutions.

Key–Words: Generalized Cauchy problem, Telegraph equation, Implicit characteristic, Cauchy conditions on the

curve, Riemann formula, Correctness criterion, Smoothness requirement, Global correctness theorem

Received: March 21, 2023. Revised: November 12, 2023. Accepted: December 2, 2023. Published: December 20, 2023.

1 Introduction

In this paper, by modiﬁcation of the Riemann method

the Cauchy problem for a linear general inhomoge-

neous telegraph equation with variable coefﬁcients

and Cauchy data on a smooth curved line in the plane

has been explicitly solved in a set of classical solu-

tions. In this paper a Hadamard correctness crite-

rion has also been found for this Cauchy problem.

The Hadamard correctness of this Cauchy problem

means the existence, uniqueness and stability on the

right-hand side of the equation and Cauchy data of

its twice continuously differentiable solution. The ex-

plicit Riemann formula of its unique and stable clas-

sical solution is derived and its correctness criterion

is established (Theorem 3). The correctness crite-

rion of the Cauchy problem is the necessary and sufﬁ-

cient smoothness requirements to the right-hand side

of equation and the Cauchy data for existence of a

unique and stable classical (twice continuously dif-

ferentiable) solution. Previously, the necessary and

sufﬁcient smoothness requirements for the right-hand

sides of telegraph equations had been studied. The re-

sulting Hadamard correctness theorem of the Cauchy

problem is global. The concept of global correctness

theorems of linear boundary value problems is intro-

duced in the article, [1]. In it, with the help of Zorn’s

lemma, a theorem on the existence of global correct-

ness theorems is proved: every correctly posed lin-

ear boundary value problem for a partial differential

equation has a global theorem of its correct Hadamard

solvability in the corresponding pair of locally con-

vex topological vector spaces. Global theorems are

called correctness theorems of boundary value prob-

lems with correctness criteria for the existence of a

unique and stable classical (twice continuously differ-

entiable) solutions, i.e. with necessary and sufﬁcient

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conditions for their Hadamard correctness.

In this article, the well-known global correctness

theorems with explicit formulas of classical solutions

and Hadamard correctness criteria for Cauchy prob-

lems for the general telegraph equation (Theorem 8)

and the model telegraph equation (Theorem 11) in the

upper half-plane are derived from the proven global

theorem 3. The Riemann formula of the classical

solution and the Hadamard correctness criterion of

the Cauchy problem for the general telegraph equa-

tion in the upper half-plane are known. The article,

[2], contains an explicit formula for the classical so-

lution and the Hadamard correctness criterion of the

Cauchy problem for a model telegraph equation un-

der the critical characteristic g2(s, τ) = g2(0,0) of

the ﬁrst quarter of the plane, where there is no in-

ﬂuence of the boundary mode of the ﬁrst kind. The

latest results of Theorems 8 and 11 partially conﬁrm

the validity of Theorem 3. In this paper, the general-

ized Cauchy problem for the general telegraph equa-

tion with variable coefﬁcients is solved and studied in

a variety of classical solutions by a new implicit char-

acteristics method, which uses only two characteristic

differential equations and twelve inversion identities

from the deﬁnition of six implicit mappings. The re-

sults obtained in this article are needed to derive or-

dinary and Riemann formulas of classical solutions

of mixed problems for model and general telegraph

equations with variable coefﬁcients in rectilinear and

curvilinear domains of the plane (see remark 14).

In the article, [3], il obtained formula allows us

to explicitly express the classical solution, and at zero

initial velocity for the initial displacement, sufﬁcient

smoothness requirements are found to ensure its exis-

tence, uniqueness and continuous dependence on the

parameter λ > 0, the Cauchy problem for a homo-

geneous singular parameter Euler-Poisson-Darboux

wave equation with Dirac potential. The solution to

the Cauchy problem for the general Euler-Poisson-

Darboux equation is unique only for non-negative val-

ues of the parameter kin the Bessel operator with re-

spect to the time variable, [4]. The paper, [5], is de-

voted to the proof of the existence of a conformal scat-

tering operator for a nonlinear cubic wave equation

of defocusing on a nonstationary background. In it,

the proof is based on solving a characteristic problem

with an initial value by the Hermander method, which

consists in reducing the characteristic initial problem

to the standard Cauchy problem by slowing down the

wave propagation velocity. Solutions of the Dalem-

bert formula form of the Cauchy problem for linear

homogeneous partial differential equations with con-

stant coefﬁcients of the third order in paper, [6], are

obtained. Using the solutions obtained, some com-

puter tests were carried out on three different roots.

These tests indicate the dispersion dynamics of waves

with some initial proﬁle.

Using the Lomov regularization method, a reg-

ularized asymptotic solution to the linear singularly

perturbed Cauchy problem in the presence of a spec-

tral singularity in the form of a weak turning point

for the limit operator was constructed in the article,

[7]. The main singularities of this Cauchy problems

are written out explicitly. In paper, [8], an asymp-

totic expansion in powers of a small parameter was

obtained to solve the Cauchy problem for a ﬁrst-order

differential equation with a small parameter at the

derivative. The conditions under which the boundary

layer phenomenon occurs have been found. The exis-

tence and destruction of weak generalized solutions

of two Cauchy problems for a wave equation with

two nonlinearities is studied, [9]. The Cauchy prob-

lem for a third-order model partial differential equa-

tion with power-law nonlinearity is studied in article,

[10]. For the linear part of the equation, analogues of

Green’s third formula for elliptic equations are con-

structed. An integral equation for classical solutions

of the Cauchy problem is obtained. It is proved that

every solution of the integral equation is a local-in-

time weak solution of the Cauchy problem.

A global correctness theorem (with necessary and

sufﬁcient conditions for the coefﬁcients of differen-

tial operators) of the Cauchy problem for linear com-

plex systems of ﬁrst-order differential equations in the

scales of Banach spaces of complex-valued vector-

exponential functions with an integral metric was ob-

tained in journal, [11]. It turned out that the neces-

sary and sufﬁcient conditions for the correctness of

his complex Cauchy problem in spaces of complex

functions with integral metrics and in spaces of com-

plex functions of vector-exponential type with supre-

mum norms from the monograph, [12], coincide. It is

proved the existence of local solution, global solution

and three conditions about the blow-up of solution to

the generalized damped Boussinesq equation, [13].

There are no global correctness theorems with

Hadamard correctness criteria for initial data and

right-hand sides of hyperbolic equations in real

Cauchy problems for classical solutions in the works

of other authors. In famous works, [3], [4], [5], [6],

[14], [15], [16], [17], [18], [19], and some others

there are theorems of existence, uniqueness and sta-

bility of classical solutions of real Cauchy problems

with their D’alembert, Riemann and others formulas

in some of them, but only if the right-hand sides of

hyperbolic equations have an overestimated sufﬁcient

smoothness. In these works the right-hand sides of

the wave equations do not have a necessary (minimum

sufﬁcient) smoothness. The minimum possible neces-

sary and sufﬁcient smoothness of the right-hand side

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of the general telegraph equation, respectively, is ex-

pressed by fundamentally new smoothness conditions

(8) and (32), which are absent in the works of other

authors.

Remark 1 For example, in theorem 2.1 of the work,

[20], with condition (32) on page 26 the replacement

of integration variables on page 27 proves the ne-

cessity and sufﬁciency of only the continuity of the

right-hand side fof the string oscillation equation

for the doubly continuous differentiability of solution

to the Cauchy problem when the function fdepends

only on the coordinates of the string points or time.

If fdepends on the coordinates of the string points

xand time t, then in, [20], on page 52 there is an

example of a continuous function f(x, t)=0for

x∈[0,+∞), t ∈[0,1) and f(x, t) = x(t−1) for

x∈[0,+∞), t ∈[1,+∞)that satisﬁes conditions

(32) with a discontinuous time derivative ∂f(x, t)/∂t.

In our article, the same is also stated in Corollar-

ies 5, 9, 12 and 6, 10, 13, respectively.

2 Statement of the generalized

Cauchy problem

Solve the Cauchy problem for a general telegraph

equation with real variable coefﬁcients under Cauchy

conditions on the curve lin the plane IR2:

Lu(x, t)≡utt(x, t)−a2(x, t)uxx(x, t)+

+b(x, t)ut(x, t) + c(x, t)ux(x, t)+

+q(x, t)u(x, t) = f(x, t),(x, t)∈IR2\l, (1)

u|l=ϕ(x), u~n|l=∂u

∂~nl=ψ(x), ~n(x, t)⊥l, (2)

where the coefﬁcients of the equation a, b, c, q are

real functions and the input data of the problem

f, ϕ, ψ are the given real functions of their inde-

pendent variables xand t,(∂u/∂~n)|lis the deriva-

tive of the normal to the curve lof the equation t=

χ(x), x ∈IR,IR = (−∞,+∞).Without excluding

the generality of the Cauchy problem in the plane IR2,

we study in detail this Cauchy problem (1), (2) only in

part of the plane G={(x, t)∈IR2:t≥χ(x), x ∈

IR}(see after the remark 4). By the number of sub-

scripts of functions, we denote the orders of their par-

tial derivatives.

Let Ck(Ω) be the set of ktimes continuously dif-

ferentiable functions on a subset Ω⊂IR2, C(Ω) be

the set of continuous functions on a subset Ω⊂IR2.

By the number of strokes over the functions of one

variable, we denote the orders of their ordinary deriva-

tives with respect to this variable.

Deﬁnition 1 Classical solutions of the Cauchy

problem (1), (2) on Gare called functions u∈C2(G),

satisfying equation (1) on ˙

G={(x, t)∈IR2:t >

χ(x), x ∈IR}in the usual sense, and the Cauchy

conditions (2) in the sense of the values of the limits

of the functions u( ˙x, ˙

t)and u~n( ˙x, ˙

t)at internal points

( ˙x, ˙

t)∈˙

Gwhen ˙x→xand ˙

t→t=χ(x).

Equation (1) has characteristic differential equa-

tions

dx = (−1)ia(x, t)dt, i = 1,2,(3)

which in Gcorrespond to two different families of im-

plicit characteristics

gi(x, t) = Ci, Ci∈IR, i = 1,2.(4)

If the coefﬁcient a(x, t)≥a0>0,(x, t)∈G,

then the characteristics g1(x, t) = C1, C1∈IR,are

strictly decreasing, and the characteristics g2(x, t) =

C2, C2∈IR,strictly increase with respect to the vari-

able xon the set Gof the plane Oxt, since by virtue

of equations (3) the derivative dx/dt =−a(x, t)≤

−a0<0for i= 1 and dx/dt =a(x, t)≥a0>0for

i= 2.Therefore, implicit functions yi=gi(x, t), t ≥

χ(x), x ∈IR,have explicit strictly monotone in-

verse functions x=hi{yi, t}, t ≥χ(x),and t=

h(i)[x, yi], x ∈IR, i = 1,2,for which, by deﬁnition

of inverse functions, the following conversion identi-

ties from the article, [2], are fulﬁlled:

gi(hi{yi, t}, t) = yi, t ≥χ(x),

hi{gi(x, t), t}=x, x ∈IR, i = 1,2,(5)

gi(x, h(i)[x, yi]) = yix∈IR,

h(i)[x, gi(x, t)] = t, t ≥χ(x), i = 1,2,(6)

hi{yi, h(i)[x, yi]}=x, x ∈IR,

h(i)[hi{yi, t}, yi] = t, t ≥χ(x), i = 1,2.(7)

If the function a∈C2(G),then the implicit func-

tions gi, hi, h(i)∈C2(G)by x, t, yi, i = 1,2,[2].

The Cauchy problem (1), (2) is studied in a set of

classical solutions by Lomovtsev’s new implicit char-

acteristics method, which uses only differential equa-

tions (3) and conversion identities (5)–(7).

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Remark 2 In the case of a(x, t) = a=const >

0,(x, t)∈G, they are functions: g1(x, t) = x+at,

g2(x, t) = x−at, h1{y1, t}=y1−at, h2{y2, t}=

y2+at, h(1)[x, y1]=(y1−x)/a, h(2)[x, y2]=(x−

y2)/a, [20].

For unique solvability of the Cauchy problem, the

curve lmust be expressed by some, at least, con-

tinuously differentiable function on plane and sat-

isfy natural requirement that characteristics gi(x, t) =

Ci, Ci∈IR, i = 1,2,of equation (1) were not tan-

gent to curve land could intersect lno more than once.

3 The Riemann formula for the clas-

sical solution of the generalized

Cauchy problem

If the carrier lof the Cauchy data ϕ, ψ is given by the

equation t=χ(x), where χ∈C2(IR), then the Rie-

mann formula for the classical solution of this Cauchy

problem contains the following

Theorem 3 Let the coefﬁcients are a(x, t)≥a0>

0,∈G, a ∈C2(G), b, c, q ∈C1(G),and each of the

characteristics gi(x, t) = Ci, i = 1,2, is not tangent

to curve lof smoothness χ∈C2(IR) and intersects l

at most once. There is a unique and stable on f, ϕ, ψ

classical solution u∈C2(G)of the Cauchy problem

(1), (2) if and only if the input data has smoothness

f∈C(G), ϕ ∈C2(IR), ψ ∈C1(IR) and

Hi(x, t)≡

t+χ(x)

χ(s0)

f(hi{gi(x, t), τ}, τ)dτ ∈C1(G)

∀(s0, χ(s0)) ∈l, i = 1,2.(8)

The classical solution u∈C2(G)to the Cauchy

problem (1), (2) for (x, t)∈Gis the function

u(x, t) = (a u v)s1(x, t), χ(s1(x, t))

2a(x, t)+

+(a u v)s2(x, t), χ(s2(x, t))

2a(x, t)+

2a(x, t)

s1(x,t)

s2(x,t)hψ(s)1−χ02(s)a2(s, χ(s))+

+χ0(s)ϕ0(s)1 + a2(s, χ(s))i v(s, χ(s))

p1 + χ0(s)2−

−ϕ(s)vτ(s, τ)|τ=χ(s)−b(s, χ(s))v(s, χ(s)) +

+χ0(s)(a2(s, τ)v(s, τ))s|τ=χ(s)+

+c(s, χ(s))v(s, χ(s))ds +

2a(x, t)Z

4MP Q

f(s, τ)v(s, τ;x, t)ds dτ. (9)

It is here usi(x, t), χ(si(x, t))=ϕ(si(x, t)), i =

1,2,due to Cauchy conditions (2), si∈C2(G)are

solutions of equations gi(si, χ(si)) = gi(x, t), i =

1,2,triangle MP Q is a curved characteris-

tic triangle with vertex M(x, t)and vertices

Ps2(x, t), χ(s2(x, t)), Qs1(x, t), χ(s1(x, t))of

its curved base P Q and v(s, τ) = v(s, τ;x, t)is a

corresponding Riemann function.

Proof. First, we derive the formula for the formal

solution of the Cauchy problem. Equation (1) for any

functions u∈C2(G)multiply by any functions v∈

C2(G)and using obvious equalities

utt v= (utv)t−utvt=

= (utv)t−(u vt)t+u vtt,

a2uxx v= (uxa2v)x−ux(a2v)x=

= (uxa2v)x−(u(a2v)x)x+u(a2v)xx,

butv= (ubv)t−u(bv)t,

cuxv= (ucv)x−u(cv)x

we come to the identity

(Lu(x, t)) v(x, t)−u(x, t) (Mv(x, t)) =

=∂H(u(x, t), v(x, t))

∂t +∂K(u(x, t), v(x, t))

∂x (10)

for all u, v ∈C2(G).It is here

Mv=vtt −(a2v)xx −(bv)t−(cv)x+qv,

H(u, v) = utv−u vt+buv =

= (uv)t−u[2vt−bv],(11)

K(u, v) = −uxa2v+u(a2v)x+cuv =

=−(a2uv)x+u[2(a2v)x+cv].(12)

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Figure 1: Curvilinear characteristic triangle 4MP Q

in IR2.

The differential operator M,which is the conjugate

operator to the differential operator Lin the sense of

Schwartz distributions D0(˙

G), [21], is usually called

the formally conjugate operator to the operator L.In

view of the well-known Green formula, the double in-

tegral of the identity (10) over the curved characteris-

tic triangle 4MP Q in Gwith any vertex M(x, t)∈

Gand the vertices of its base P(s2(x, t), χ(s2(x, t)))

and Q(s1(x, t), χ(s1(x, t))) on the curve lis equal to

4MP Q

[(Lu(s, τ)) v(s, τ)−

−u(s, τ) (Mv(s, τ))]ds dτ =

4MP Q ∂H(u(s, τ), v(s, τ ))

∂τ +

+∂K(u(s, τ), v(s, τ ))

∂s ds dτ =

[K(u(s, τ), v(s, τ))dτ−

−H(u(s, τ), v(s, τ))ds],(13)

where l+=QM ∪MP ∪P Q is the contour of a

curved triangle 4MP Q with a positive bypass direc-

tion, due to the right orientation of the plane Osτ in

Fig. 1.

Since the curve lis given by the equation τ=

χ(s)∈C2(IR) on the plane Osτ, then the coordinates

of the points P(s2, τ2)and Q(s1, τ1)are solutions of

equation systems:

gi(si, τi) = gi(x, t),

τi=χ(si), i = 1,2.

Each of these two systems of equations has at

most one solution, since, according to the assump-

tions of Theorem 3, the strictly monotone charac-

teristics gi(x, t) = Ci, Ci∈IR, i = 1,2,inter-

sect the curve lno more than once. Solutions of

these equation systems are solutions of implicit equa-

tions si=hi{gi(x, t), τi}=hi{gi(x, t), χ(si)}, i =

1,2.This means that the functions si=si(x, t)∈

C2(G),i= 1,2, depend only on the coordinates

of the point M(x, t). Therefore, the coordinates of

the vertices Qand Pof the base of the triangle

4MP Q are expressed as Qs1(x, t), χ(s1(x, t))

and Ps2(x, t), χ(s2(x, t)).

In the curvilinear integral (13) using expressions

(11) and (12), the differential equation of the charac-

teristic from (3) for i= 1 and the obvious equalities

(u v)τa= (a u v)τ−aτu v,

(a2u v)s(1/a)=(a u v)s−a2uv(1/a)s=

= (a u v)s+asu v,

we calculate the integral along the characteristic QM

of the equation g1(s, τ) = g1(x, t):

[K(u(s, τ), v(s, τ))dτ −H(u(s, τ), v(s, τ))ds] =

[(a2u v)s(1/a)ds + (u v)τa dτ]+

Qu[2(a2v)s+cv]dτ +u[2vτ−bv]ds=

d(a u v) +

Qu[2(a2v)s+

+(c−aτ)v]dτ +u[2vτ+ (as−b)v]ds=

d(a u v) +

Qu[2(a2v)s+ (c−aτ)v]−

−au [2vτ+ (as−b)v]dτ =

d(a u v)+

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u{4aasv+ 2a2vs+ (c−aτ)v−

−2avτ−(as−b)av}dτ =

d(a u v) +

u{ − 4avτ+

+[4aas+ (c−aτ)−(as−b)a]v}dτ =

= (a u v)(M)−(a u v)(Q)−

χ(s1(x,t))

u{4a vτ−[ab −4aτ+c]v}dτ =

= (a u v)(x, t)−(a u v)s1(x, t), χ(s1(x, t))−

−

χ(s1(x,t))

u(s, τ){4a(s, τ)vτ(s, τ)−[a(s, τ)b(s, τ)−

−4aτ(s, τ) + c(s, τ)]v(s, τ)}dτ. (14)

Here on the characteristic QM for i= 1, we

used the well-known differential equation (3) of the

implicit characteristics from (4) and for the functions

w∈C1(G) a new representation from

ws(s, τ)=(−1)iwτ(s, τ)/a(s, τ), i = 1,2.(15)

Since on each of the characteristics QM and MP

variables s=s(τ), τ =τ(s)are simultaneously

mutually dependent, that is, respectively for i= 1

and i= 2 variables s=hi{gi(x, t), τ}and τ=

h(i)[s, gi(x, t)], according to the inversion formula

(5)–(7), then these representations follow from the ob-

vious formulas of the ﬁrst partial derivatives

ws(s, τ(s)) =

=ws(s, τ)|τ=τ(s)+wτ(s, τ)|τ=τ(s)τ0(s) =

=ws(s, τ)|τ=τ(s)+ (−1)iwτ(s, τ)|τ=τ(s)/a(s, τ),

wτ(s(τ), τ) =

=wτ(s, τ)|s=s(τ)+ws(s, τ)|s=s(τ)s0(τ) =

=wτ(s, τ)|s=s(τ)+ (−1)iws(s, τ)|s=s(τ)a(s, τ)

since τ0(s) = (−1)i/a(s, τ), s0(τ) = (−1)ia(s, τ),

i= 1,2,also due to the formulas (4).

In the previous equation from (14), to reduce a

curved integral of the second type along the charac-

teristic QM to an ordinary deﬁnite integral, we ap-

plied the parametric representation of the curve QM :

s=s1(τ) = h1{g1(x, t), τ}, τ0=τ, χ(s1)≤τ≤t,

and the differential equation from (3) for i= 1.Using

the inversion identities (5) – (7), we conclude that if

integration occurs over the variable s, then the vari-

able τ=h(1)[s, g1(x, t)], and if by the variable τ,

then the variable s=h1{g1(x, t), τ}.

Using the characteristic differential equation from

(3) for i= 2,we similarly calculate the integral

(13) along the characteristic MP with the equation

g2(s, τ) = g2(x, t) :

[K(u(s, τ), v(s, τ))dτ −H(u(s, τ), v(s, τ))ds] =

=−

[(u v)τa dτ + (a2u v)s(1/a)ds]+

Mu[2vτ−bv]ds +u[2(a2v)s+cv]dτ=

=−

d(a u v) +

Mu[2vτ−

−(as+b)v]ds +u[2(a2v)s+ (c+aτ)v]dτ=

= (a u v)(M)−(a u v)(P)−

−

χ(s2(x,t))

u{4a vτ−[ab −4aτ−c]v}dτ =

= (a u v)(x, t)−(a u v)s2(x, t), χ(s2(x, t))−

−

χ(s2(x,t))

u{4a(s, τ)vτ(s, τ)−[a(s, τ)b(s, τ)−

−4aτ(s, τ)−c(s, τ)]v(s, τ)}dτ. (16)

In the previous equation from (16), to reduce a

curved integral of the second type along the character-

istic MP to an ordinary deﬁnite integral, we applied

the parametric representation of the curve MP :s=

s2(τ) = h2{g2(x, t), τ}, τ0=τ, χ(s2(x, t)) ≤τ≤

t. In this previous equation, we also applied the differ-

ential equation from (3) and the above partial deriva-

tive representations from (15) for i= 2. In all ex-

pressions under the integral sign, all functions of two

variables depend on s, τ. Moreover, using formulas

(5)–(7), we conclude that if the integration occurs with

the variable s, then the variable τ=h(2)[s, g2(x, t)],

and if with τ, then s=h2{g2(x, t), τ}.

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Let the function v(s, τ) = v(s, τ;x, t)with pa-

rameters (x, t)be a classical solution of a homoge-

neous formally conjugate differential equation

Mv(s, τ) = 0,(s, τ)∈∆MP Q (17)

with conditions, respectively, on the characteristics of

QM and MP

4a(s, τ)vτ(s, τ)−[a(s, τ)b(s, τ)−

−4aτ(s, τ) + c(s, τ)]v(s, τ) = 0,

4a(s, τ)vτ(s, τ)−[a(s, τ)b(s, τ)−

−4aτ(s, τ)−c(s, τ)]v(s, τ)=0 (18)

in integrals (14) and (16) and the matching condition

v(M)=1 (19)

The conditions (18), (19) are obviously equivalent to

the two agreed Goursat conditions

v(s, τ) = exp (τ

k1(h1{g1(x, t), ρ}, ρ)dρ),

g1(s, τ) = g1(x, t), τ ∈[χ(s1(x, t)), t],

v(s, τ) = exp (τ

k2(h2{g2(x, t), ρ}, ρ)dρ),

g2(s, τ) = g2(x, t), τ ∈[χ(s2(x, t)), t],(20)

where the functions k1(s, τ) = {a(s, τ)b(s, τ)−

4aτ(s, τ) + c(s, τ)}/4a(s, τ)on the curve QM

and k2(s, τ) = {a(s, τ)b(s, τ)−4aτ(s, τ)−

c(s, τ)}/4a(s, τ)on the curve MP. It is well known

that the Goursat problem (17), (20) with coefﬁcients

a∈C2(G), b, c, q ∈C1(G)has a unique classical

solution v∈C2(∆MP Q),which is naturally called

the Riemann function of the Cauchy problem (1), (2)

on G. In the general case, the Riemann function is

uniquely found by the method of successive approxi-

mations, [19], p. 129–135.

According to the Cauchy conditions (2), the

values of the solution at the base vertices of

the triangle are known: usi(x, t), χ(si(x, t))=

ϕ(si(x, t)), i = 1,2. In formulas (13) we assume

Lu(s, τ) = f(s, τ),Mv(s, τ) = 0 on the triangle

∆MP Q and in by virtue of the relations (17)–(20)

and the equalities (14), (16) we obtain a formal solu-

tion of the Cauchy problem (1), (2) for all (x, t)∈G:

u(x, t) = (a u v)s1(x, t), χ(s1(x, t))

2a(x, t)+

+(a u v)s2(x, t), χ(s2(x, t))

2a(x, t)+

2a(x, t)

[H(u(s, τ), v(s, τ)) ds−

−K(u(s, τ), v(s, τ))dτ]+

2a(x, t)Z

4MP Q

f(s, τ)v(s, τ;x, t)ds dτ. (21)

If ~e is a tangent vector to the curve l, then from

(2) the derivative u~e|l=∂u/∂~e|l=ϕ0(x)and the

ﬁrst partial derivatives from uare calculated by the

formulas from, [19], p. 139:

us|l=u~e|lcos(~e, s) + u~n|lcos(~n, s) =

=ϕ0(s)−χ0(s)ψ(s)

p1 + χ0(s)2,(22)

uτ|l=u~e|lcos(~e, τ) + u~n|lcos(~n, τ) =

=ϕ0(s)χ0(s) + ψ(s)

p1 + χ0(s)2(23)

Substituting partial derivatives (22), (22) in (11), (12)

and under a curved integral of the second type along

the curve P Q in (21) and considering dτ =χ0(s)ds,

we ﬁnd

[H(u(s, τ), v(s, τ))ds −K(u(s, τ), v(s, τ))dτ] =

s1(x,t)

s2(x,t)nhuτ(s, τ)|τ=χ(s)v(s, χ(s))−

−u(s, χ(s)) vτ(s, τ)|τ=χ(s)+

+b(s, χ(s))u(s, χ(s))v(s, χ(s)) ids+

+hus(s, τ)|τ=χ(s)a2(s, χ(s))v(s, χ(s))−

−u(s, χ(s)) (a2(s, τ)v(s, τ))s|τ=χ(s)−

−c(s, χ(s))u(s, χ(s))v(s, χ(s))iχ0(s)dso=

s1(x,t)

s2(x,t)ϕ0(s)χ0(s) + ψ(s)

p1 + χ0(s)2v(s, χ(s))−

−u(s, χ(s)) vτ(s, τ)|τ=χ(s)+

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+b(s, χ(s))u(s, χ(s))v(s, χ(s))+

+ϕ0(s)−χ0(s)ψ(s)

p1 + χ0(s)2×

×a2(s, χ(s))v(s, χ(s))χ0(s)−

−u(s, χ(s)) (a2(s, τ)v(s, τ))s|τ=χ(s)χ0(s)−

−c(s, χ(s))u(s, χ(s))v(s, χ(s))χ0(s)ds =

s1(x,t)

s2(x,t)ϕ0(s)χ0(s) + ψ(s)

p1 + χ0(s)2v(s, χ(s))−

−ϕ(s)vτ(s, τ)|τ=χ(s)+

+b(s, χ(s))ϕ(s)v(s, χ(s))+

+ϕ0(s)−χ0(s)ψ(s)

p1 + χ0(s)2×

×a2(s, χ(s))v(s, χ(s))χ0(s)−

−ϕ(s) (a2(s, τ)v(s, τ))s|τ=χ(s)χ0(s)−

−c(s, χ(s))ϕ(s)v(s, χ(s))χ0(s)ds =

s1(x,t)

s2(x,t)hψ(s)1−χ02(s)a2(s, χ(s))+

+χ0(s)ϕ0(s)1 + a2(s, χ(s))i v(s, χ(s))

p1 + χ0(s)2−

−ϕ(s)vτ(s, τ)|τ=χ(s)−b(s, χ(s))v(s, χ(s))+

+χ0(s)(a2(s, τ)v(s, τ))s|τ=χ(s)+

+c(s, χ(s))v(s, χ(s))ds =

s1(x,t)

s2(x,t)hψ(s)1−χ02(s)a2(s, χ(s))+

+χ0(s)ϕ0(s)1 + a2(s, χ(s))i v(s, χ(s))

p1 + χ02(s)−

−ϕ(s)vτ(s, τ)|τ=χ(s)+

+χ0(s)a2(s, χ(s))vs(s, τ)|τ=χ(s)+

+χ0(s)(c(s, χ(s))+

+2a(s, χ(s))as(s, τ)|τ=χ(s))−

−b(s, χ(s))v(s, χ(s))ds. (24)

We substitute the expression (24) into the formula

(21). As a result, the ﬁnal formula for the formal solu-

tion of the Cauchy problem (1), (2) on the set Gtakes

the form (9).

Now we justify the necessity and sufﬁciency of

the smoothness for the right-hand side of the equa-

tion indicated in Theorem 3 and the initial data for

the doubly continuous differentiability of the func-

tion (9) on G. Directly from the telegraph equation

(1), Cauchy conditions (2) and deﬁnition 1 of clas-

sical solutions u∈C2(G)of this Cauchy problem

implies the need for smoothness of the input data

f∈C(G), ϕ ∈C2(IR), ψ ∈C1(IR).With the Rie-

mann function v∈C2(G)smoothness of the initial

data ϕ∈C2(IR), ψ ∈C1(IR) is obviously sufﬁcient

for twice continuous differentiability on Gof the ﬁrst

term and the curvilinear integral of the function (9) on

G. In the plane of Osτ by replacing variables

˜s=s, ˜τ=τ−χ(s)(25)

with a non-degenerate Jacobian J(s, τ) = ˜s0

s˜τ0

τ−

˜s0

τ˜τ0

s= 1 6= 0 characteristic triangle 4MP Q

with curved base P Q from Fig. 1 is reduced

to the corresponding characteristic triangle 4f

with vertex f

M(x, t),vertices e

P(˜

h2{˜g2(x, t),0},0),

Q(˜

h1{˜g1(x, t),0},0) of rectilinear base e

Qwith the

equation ˜τ= 0, where the functions ˜

hi,˜giare found

by replacing (25) from our functions hi, gi, i = 1,2.

The double integral of the triangle 4MP Q from

(9) after replacing (25) becomes the corresponding

double integral of the triangle 4f

Q. These tri-

angles are simultaneously twice continuously differ-

entiable by x, t, since for χ∈C2(IR) replacement

(25) is non-degenerate and twice continuously differ-

entiable on G.

Moreover, this triangle 4f

Qeven has the

form of a triangle from Fig. 2, because the transfor-

mation (25) preserves the ﬁrst variable sand has the

type of rotation by the second variable τ.

Therefore, additional to the image ˜

f∈C(e

G)of

the right-hand side fafter the change (25), the neces-

sary and sufﬁcient requirements

Hi(x, t)≡

f(˜

hi{˜gi(x, t),˜τ},˜τ)d˜τ∈C1(e

G),

i= 1,2,(26)

after the reverse to (25) replacement of variables (see

below (32))

s= ˜s, τ = ˜τ+χ(˜s)(27)

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Figure 2: Characteristic triangle 4f

Qin IR2.

become additional to f∈C(G)necessary and sufﬁ-

cient smoothness requirements (8) on the double inte-

gral of (9) in Theorem 3.

If in the reverse substitution (27) the variable

˜τ= 0,then the variable τ=χ(˜s) = χ(s)for all

points (s, χ(s)) ∈lof the curve in the lower limits of

integrals (8). If in (27) the variable ˜τ=t, then the

variable τ=t+χ(˜s) = t+χ(x)in the upper limits

of integrals (27), since ˜s=s=x. Indeed, if in a tri-

angle 4MP Q the second coordinate is τ=t, then it

is obvious that the ﬁrst coordinate must necessarily be

s=x. This is also the second inversion identities (5)

conﬁrms when one variable τ=t, then the other vari-

able s=hi{gi(x, t), τ}=hi{gi(x, t), t}=xand

vice versa. In integrals (26), the Jacobian of the in-

verse substitution to (25) is ˜

J(˜s, ˜τ) = s0

˜sτ0

˜τ−s0

˜ττ0

˜s= 1

and the upper half-plane e

Gis the image of the task set

Gof the Cauchy problem (1), (2) after replacement

(25).

The uniqueness of the classical solution (9) of the

Cauchy problem (1), (2) can be justiﬁed in the same

way as in the textbook, [19], p. 139. Its stability with

respect to f, ϕ, ψ ensures the established existence

and uniqueness of the classical solution (9) by Ba-

nach’s closed graph theorem or Banach’s open map-

ping theorem. To conclude the proof of Theorem 3,

we say that stability also follows from the formula (9).

Remark 4 The Riemann formula for the classical so-

lution of the Cauchy problem (1), (2) on Gwith the

coefﬁcient a(x, t)≡1is given in, [19], p. 139.

For the lower half-plane IR2\Gin (13) the ori-

entation of the contour l+of the characteristic tri-

angle 4MP Q, does not change if, as before, at the

points Pand Qrespectively intersect the characteris-

tics g2(s, τ) = g2(x, t)and g1(s, τ) = g1(x, t)with

the curve l, since the points Pand Qare swapped in

IR2\G. Thus, the unique and stable classical solution

of the Cauchy problem (1), (2) on the lower half-plane

IR2\Gis also given by the formula (9).

Corollary 5 If the right-hand side fof the equa-

tion (1) depends only on xor tand is continuous

f∈C(IR),then Theorem 3 is true without smooth-

ness requirements (8).

Proof. It is easier to verify the continuous dif-

ferentiability of the integrals from (26) on ˜

G, which

is equivalent to the continuous differentiability of the

integrals from (8) on G. If the function ˜

f=˜

f(˜τ)does

not depend on ˜s, then the integrals from (26) are equal

to the integral

Hi(t)≡

f(˜τ)d˜τ, i = 1,2,(28)

which is continuously differentiable with respect

to tfor continuous functions ˜

f, since the ﬁrst deriva-

tive ∂˜

Hi(t)/∂t =˜

f(t)is continuous. If the func-

tion ˜

f=˜

f(˜s)does not depend on ˜τ, then inte-

grals from (26) after change of integration variables

˜si=˜

hi{˜gi(x, t),˜τ}, i = 1,2,are equal to integrals

Hi(x, t)≡

f(˜

hi{˜gi(x, t),˜τ})d˜τ=

hi{˜gi(x,t),0}

f(˜si)

(∂˜

hi{˜gi(x, t),˜τ}/∂˜τ)˜τ=˜

h(i)[˜gi(x,t),˜τ]d˜si

∈C1(e

G), i = 1,2.(29)

Here the functions ˜

Hi(x, t), i = 1,2,are indeed

continuously differentiable with respect to xand tfor

a continuous function ˜

f=˜

f(˜s),because in the last in-

tegrals, it is explicitly independent of variables xand

t. Corollary 5 is proved.

Corollary 6 If the right-hand side fof the equation

(1) depends on xand t, then in Theorem 3 the in-

tegrals (8) belong to the set C1(G)is equivalent to

their belonging to the set C(0,1)(G)or C(1,0)(G).The

sets C(0,1)(G)and C(1,0)(G)are sets of continuous or

continuously differentiable with respect to xand con-

tinuously differentiable or continuous with respect to

tfunctions on G.

Proof. Firstly for smoother functions ˜

f∈

C1(e

G), by changing the integration variables, we de-

rive special equalities for the ﬁrst partial derivatives

of the integrals (26), which are equivalent to the in-

tegrals (8). Then the equalities of their ﬁrst and last

parts, which do not contain explicit derivatives of the

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functions ˜

f∈C1(e

G).are extended by passing to the

limit along ˜

ffrom smoother functions ˜

f∈C1(e

to continuous functions ˜

f∈C(e

G),satisfying the in-

tegral smoothness requirements (26). Corollary 6 is

proved.

We apply the derived Riemann formula of the

classical solution and the correctness criterion of the

Cauchy problem (1), (2) on Gin global theorem 3 to

its two important special cases, which partially con-

ﬁrm the validity of the results obtained.

4 The Cauchy problem for the gen-

eral telegraph equation in the up-

per half-plane

A special case of the Cauchy problem (1), (2) in the

plane is the following Cauchy problem in the upper

half-plane:

utt(x, t)−a2(x, t)uxx(x, t) + b(x, t)ut(x, t)+

+c(x, t)ux(x, t) + q(x, t)u(x, t) = f(x, t),

(x, t)∈˙

G∞= IR ×(0,+∞),(30)

u|t=0 =ϕ(x), ut|t=0 =ψ(x),

x∈IR = (−∞,+∞),(31)

where the coefﬁcients a, b, c, q are real functions

and the input data f, ϕ, ψ are given real functions of

their variables xand t. By the lower indices of the

functions we indicate the variables of partial deriva-

tives of these functions.

As above, the characteristic equations dx −

(−1)ia(x, t)dt = 0 give the characteristics gi(x, t) =

Ci, i = 1,2.If a(x, t)≥a0>0,then they decrease

strictly by tat i= 1 and increase at i= 2 with the

growth of x, since dx/dt =−a(x, t)≤ −a0<0

at i= 1 and dx/dt =a(x, t)≥a0>0for

i= 2.Therefore, the functions yi=gi(x, t)have

inverse functions x=hi{yi, t}, t =h(i)[x, yi], i =

1,2.Moreover, each of the characteristics gi(x, t) =

Ci, i = 1,2,intersects the curve lof the equation

t= 0 only once. If the coefﬁcient is a∈C2(G∞),

then the functions gi, hi, h(i), i = 1,2,are twice

continuously differentiable by their variables in G∞,

[2]. By deﬁnition of inverse functions, the conversion

identities (5)–(7) are fulﬁlled.

Deﬁnition 7 Classical solutions of the Cauchy prob-

lem (30), (31) on G∞are called functions u∈

C2(G∞), G∞= IR ×[0,+∞),satisfying equation

(30) on ˙

G∞= IR ×(0,+∞)in the usual sense and

the initial conditions (31) in the sense of the values of

the limits of the functions u( ˙x, ˙

t)and u˙

t( ˙x, ˙

t)at inter-

nal points ( ˙x, ˙

t)∈˙

G∞when ˙x→xand ˙

t→0.

Theorem 8 Let in equation (30) the coefﬁcients be

a(x, t)≥a0>0,(x, t)∈G∞, a ∈C2(G∞),

b, c, q ∈C1(G∞).The Cauchy problem (30), (31)

on G∞has a unique and stable on f, ϕ, ψ classi-

cal solution u∈C2(G∞)if and only if when the

right-hand side of the equation and the initial data

are f∈C(G∞), ϕ ∈C2(IR), ψ ∈C1(IR) and

f(hi{gi(x, t), τ}, τ)dτ ∈C1(G∞), i = 1,2.(32)

The classical solution u∈C2(G∞)of the Cauchy

problem (30), (31) for all (x, t)∈G∞is the function

u(x, t) = (auv)(h2{g2(x, t),0},0)

2a(x, t)+

+(auv)(h1{g1(x, t),0},0)

2a(x, t)+

2a(x, t)

h1{g1(x,t),0}

h2{g2(x,t),0}

[ψ(s)v(s, 0)−

−ϕ(s)vτ(s, 0) + b(s, 0)ϕ(s)v(s, 0)] ds+

2a(x, t)

dτ

h1{g1(x,t),τ }

h2{g2(x,t),τ }

f(s, τ)v(s, τ;x, t)ds. (33)

By virtue of (31) u(h2{g2(x, t),0},0) =

ϕ(h2{g2(x, t),0}), u(h1{g1(x, t),0},0) =

ϕ(h1{g1(x, t),0}),Riemann function v(s, τ) =

v(s, τ;x, t)is solution of Goursat problem:

vτ τ (s, τ )−(a2(s, τ )v(s, τ ))ss−

−(b(s, τ)v(s, τ))τ−(c(s, τ)v(s, τ))s+

+q(s, τ)v(s, τ)=0,(s, τ)∈∆MP Q, (34)

v(s, τ) = exp (τ

k1(h1{g1(x, t), ρ}, ρ)dρ),

g1(s, τ) = g1(x, t),

v(s, τ) = exp (τ

k2(h2{g2(x, t), ρ}, ρ)dρ),

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g2(s, τ) = g2(x, t), τ ∈[0, t],(35)

with functions k1(s, τ) = {a(s, τ)b(s, τ)−

4aτ(s, τ) + c(s, τ)}/4a(s, τ)on the curve QM

and k2(s, τ) = {a(s, τ)b(s, τ)−4aτ(s, τ)−

c(s, τ)}/4a(s, τ)on the curve MP.

Proof. In the Cauchy problem (30), (31), the ini-

tial conditions are given at t= 0.Therefore, in Theo-

rem 3, the function t=χ(x) = 0 and hence its deriva-

tive χ0(x) = 0.Therefore, we have the functions

si(x, t) = hi{gi(x, t),0}, i = 1,2,and hence the

coordinates of the points Q(h1{g1(x, t),0},0) and

P(h2{g2(x, t),0},0).In addition, the smoothness re-

quirements for the right-hand side fof the equation

(30) and the initial data (31), including integral re-

quirements (8) for G, from Theorem 3 for χ(x)=0

turn into equivalent smoothness requirements, includ-

ing integral requirements (32), on G∞from Theorem

8. The Goursat problem (17), (20) on the triangle

4MP Q with a curved base P Q at χ(x)=0be-

comes the Goursat problem (34), (35) on the trian-

gle 4MP Q with a rectilinear base P Q. Formulas (9)

of the classical solution u∈C2(G)from Theorem 3

takes the form of formula (33) of the classical solu-

tion u∈C2(G∞)from Theorem 8, since the double

repeated integral of (33) is equal to the double integral

of (9) for χ(x) = 0.Theorem 8 is proved.

Corollaries 9, 10 follow from Corollaries 5, 6.

Corollary 9 If the right-hand side fof the equa-

tion (30) depends only on xor tand is continuous

f∈C(IR),then Theorem 8 is true without smooth-

ness requirements (32).

Corollary 10 If the right-hand side fof the equation

(30) depends on xand t, then in Theorem 8 the be-

longing of the integrals (32) to the set C1(G∞)is

equivalent to their belonging to the set C(0,1)(G∞)

or C(1,0)(G∞).The sets C(0,1)(G∞)and C(1,0)(G∞)

are sets of continuous or continuously differentiable

with respect to xand continuously differentiable or

continuous with respect to tfunctions on G∞.

5 The Cauchy problem for the model

telegraph equation in the upper

half-plane.

In the half-plane G∞= IR ×(0,+∞), to solve the

Cauchy problem for the model telegraph equation:

Lu(x, t)≡utt(x, t)−a2(x, t)uxx(x, t)−

−a−1(x, t)at(x, t)ut(x, t)−

−a(x, t)ax(x, t)ux(x, t) = f(x, t),(x, t)∈˙

G∞(36)

with the initial conditions

u|t=0 =ϕ(x), ut|t=0 =ψ(x), x ∈IR.(37)

The correctness criterion and the formula of the

classical solution of this problem Cauchy gives

Theorem 11 Let the coefﬁcient be a(x, t)≥a0>

0,(x, t)∈G∞, a ∈C2(G∞).The Cauchy problem

(36), (37) has a unique and stable on f, ϕ, ψ classical

solution u∈C2(G∞)if and only if f∈C(G∞),

ϕ∈C2(IR), ψ ∈C1(IR) and (32) is true.

The classical solution u∈C2(G∞)of the

Cauchy problem (36), (37) on G∞is the function

u(x, t) = ϕ(h2{g2(x, t),0}+ϕ(h1{g1(x, t),0})

2a(x, t)+

h1{g1(x,t),0}

h2{g2(x,t),0}

ψ(s)

a(s, 0) ds +

dτ

h1{g1(x,t),τ }

h2{g2(x,t),τ }

f(s, τ)

a(s, τ)ds, (x, t)∈G∞.(38)

Proof. Equation (36) of this Cauchy problem is

a special case of the general telegraph equation (30)

considered above for b(x, t) = −a−1(x, t)at(x, t),

c(x, t) = −a(x, t)ax(x, t), q(x, t)=0.Firsly we de-

rive the formula (38) of the Cauchy problem (36), (37)

of Theorem 11 from the formula (33) of the Cauchy

problem (30), (31) of Theorem 8. For all (x, t)∈G∞

the Riemann function is well known

v(s, τ) = v(s, τ;x, t) = a(x, t)

a(s, τ),(s, τ)∈G∞,(39)

which is a solution of the Goursat problem (34), (35)

for the conjugate equation to the model telegraph

equation (36). For (x, t)∈G∞, using the Riemann

function (39) and the initial conditions (37), we cal-

culate the ﬁrst term of the formula (33)

(auv)(h2{g2(x, t),0},0)

2a(x, t)+

+(auv)(h1{g1(x, t),0},0)

2a(x, t)=

=(au)(h2{g2(x, t),0},0)a(x, t)

2a(x, t)a(h2{g2(x, t),0},0) +

+(au)(h1{g1(x, t),0},0)a(x, t)

2a(x, t)a(h1{g1(x, t),0},0) =

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=u(h2{g2(x, t),0},0) + u(h1{g1(x, t),0},0)

=ϕ(h2{g2(x, t),0}) + ϕ(h1{g1(x, t),0})

2.(40)

For all (s, τ)∈G∞, we similarly calculate the

second term of the formula (33)

vτ(s, τ)|τ=0 −v(s, 0)b(s, 0) =

=a(x, t)

a(s, τ)ττ=0 +a(x, t)

a(s, 0)

aτ(s, τ)|τ=0

a(s, 0) =

=a(x, t)−aτ(s, τ)|τ=0

a2(s, 0) +at(s, τ)|τ=0

a2(s, 0) = 0,(41)

ψ(s)v(s, 0)

a(x, t)=ψ(s)a(x, t)

a(x, t)a(s, 0) =ψ(s)

a(s, 0).(42)

We calculate the third term of the formula (33)

2a(x, t)Z

4MP Q

f(s, τ)v(s, τ :x, t)ds dτ =

4MP Q

f(s, τ)

a(s, τ)ds dτ =

dτ

h1{g1(x,t), τ }

h2{g2(x,t), τ }

f(s, τ)

a(s, τ)ds, (x, t)∈G∞.(43)

Since the smoothness on f, ϕ, ψ of the Cauchy

problems (30), (31) and (36), (37) coincide, then

from the equalities (40)–(43) we conclude that for the

Cauchy problem (36), (37) the formula of the classi-

cal solution (33) is equal to the formula of the classical

solution (38). Theorem 11 has been proved.

By Corollaries 9, 10, Corollaries 12, 13 hold.

Corollary 12 If the right-hand side fof the equa-

tion (36) depends only on xor tand is continuous

f∈C(IR),then Theorem 11 is true without integral

smoothness requirements (32).

Corollary 13 If the right-hand side fof the equa-

tion (36) depends on xand t, then in Theorem 11,

the belonging of integrals (32) to the set C1(G∞)is

equivalent to their belonging to the sets C(0,1)(G∞)

or C(1,0)(G∞).

Remark 14 The results of this work are needed to

solve mixed problems for wave equations in curvilin-

ear domains. The ﬁrst and second mixed problems for

model and general telegraph equations with variable

coefﬁcients have been solved by Lomovtsev F.E. us-

ing his new methods: implicit characteristics method

from, [2], and generalization of Riemann method to

these mixed problems on a half-line and a segment.

6 Conclusion

The generalized Riemann formula (9) of the classical

solution and the Hadamard correctness criterion of the

generalized real Cauchy problem (1), (2) for a general

linear inhomogeneous telegraph equation with vari-

able coefﬁcients under Cauchy conditions on a dou-

bly continuously differentiable and noncharacteristic

curve lof the plane are obtained. In global Theorem

3, the Riemann formula (9) of its unique and continu-

ous on right-hand side fof the equation and Cauchy

data ϕ, ψ of the classical solution u∈C2(G)is es-

tablished by a modiﬁcation of the Riemann method

and new implicit characteristics method. From the

formulation of the Cauchy problem and the deﬁnition

of classical solutions, its Hadamard correctness crite-

rion is found: f∈C(G), ϕ ∈C2(IR), ψ ∈C1(IR)

and new integral smoothness requirements (8). If the

right-hand side fof the equation (1) depends on one

of two variables xand t, then it is necessary and sufﬁ-

cient that it be continuous with respect to this variable.

If the right-hand side fof the equation (1) depends

on two variables and is continuous with respect to x

and t, then in its integral smoothness requirements it

is necessary and sufﬁcient to have continuity in one

and continuous differentiability in the other variable.

From Theorem 3, the already known Riemann formu-

las of classical solutions and the correctness criteria

of Cauchy problems in Theorem 8 for general and in

Theorem 11 for model telegraph equations in the up-

per half-plane are derived. These Riemann formulas

contain implicit functions of characteristics for tele-

graph equations. They partially conﬁrm the correct-

ness of the main results of this work in Theorem 3.

In this work, the necessary and sufﬁcient smoothness

of the right-hand side of the telegraph equation with

real variable coefﬁcients under Cauchy conditions on

a curved line of the plane for twice continuously dif-

ferentiable solutions was proven for the ﬁrst time.

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Contribution of Individual Authors to the

Creation of a Scientific Article

(Ghostwriting Policy)

Fedor Egorovich Lomovtsev derived the Riemann

formula and the correctness criterion of the

generalized Cauchy problem under Cauchy

conditions on a curved line of a plane. He

formulated and proved Theorem 2, Theorem 7 and

Corollaries 4, 5, 8, 9. Andrey Leonidovich

Kukharev derived Theorem 10 from Theorem 7 and

translated this article into English.

Sources of Funding for Research Presented

in a Scientific Article or Scientific Article

Itself

The work was funded by the BRFFI of the Republic

of Belarus (grant No. F22KI-001).

Conflict of Interest

The authors have no conflicts of interest to declare

that are relevant to the content of this article.

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