MIXED PROBLEM FOR A 1D INHOMOGENEOUS WAVE EGUATION AT TWO CONSTANT RATES WITH CHARACTERISTIC NONSTATIONARY SECOND DERIVATIVES IN A BOUNDARY MODE

​СМЕШАННАЯ ЗАДАЧА ДЛЯ ОДНОМЕРНОГО НЕОДНОРОДНОГО ВОЛНОВОГО УРАВНЕНИЯ ПРИ ДВУХ ПОСТОЯННЫХ СКОРОСТЯХ С ХАРАКТЕРИСТИЧЕСКИМИ НЕСТАЦИОНАРНЫМИ ВТОРЫМИ ПРОИЗВОДНЫМИ В ГРАНИЧНОМ РЕЖИМЕ

Ломовцев Фёдор Егорович

д-р физ.-мат.  наук, профессор кафедры интеллектуальных методов моделирования, профессор, Белорусский государственный университет,

Беларусь, г. Минск

Спесивцева Ксения Андреевна

аспирант кафедры интеллектуальных методов моделирования, Белорусский государственный университет,

Беларусь, г. Минск

ABSTRACT

By modifying the characteristic method, an explicit formula for a unique and stable classical solution to a linear mixed problem for a wave equation at two constant rates and with characteristic nonstationary second derivatives in the boundary mode on the first quarter of the plane was derived. Correctness criterion found for the first time according to Hadamard on the right-hand side of the equation, initial and boundary data for its unique and stable everywhere solvability in the set of classical solutions. These results are obtained without periodic extensions of the problem data (the right-hand side of the equation, the initial data and the boundary data). The correctness criterion includes the necessary and sufficient smoothness requirements on the problem data and two matching conditions for the boundary mode with the initial conditions and the equation. The smoothness requirements give twice continuous differentiability of its solution outside the critical characteristic, and two matching conditions are needed for smooth gluing of this solution on the critical characteristic of the equation.

АННОТАЦИЯ

Модификацией характеристического метода получена явная формула единственного и устойчивого классического решения линейной смешанной задачи для волнового уравнения с двумя постоянными скоростями и с характеристическими нестационарными вторыми производными в граничном режиме на первой четверти плоскости. Впервые найден критерий корректности по Адамару правой части уравнения, начальных и граничных данных для его единственной и устойчивой всюду разрешимости в множестве классических решений. Эти результаты получены без периодических расширений данных задачи (правой части уравнения, начальных данных и граничных данных). Критерий корректности включает необходимые и достаточные требования гладкости на данные задачи и два условия согласования граничного режима с начальными условиями и уравнением. Требования гладкости дают дважды непрерывную дифференцируемость ее решения вне критической характеристики, а для гладкой склейки этого решения на критической характеристике уравнения необходимы два условия согласования.

Keywords: Mixed problem, Characteristic second derivatives, Classical solution, Correctness criterion, Smoothness requirement, Global correctness theorem.

Ключевые слова: смешанная задача, характерные вторые производные, классическое решение, критерий корректности, требование гладкости, теорема глобальной корректности.

1. Introduction

Apart from the authors of this article, no one in the world is searching for classical solutions and studying the Hadamard correctness of the mixed problem for a two-rates wave equation with characteristic nonstationary second partial derivatives in boundary mode. For now we only do this in the case of two different constant rates  and  which does not exclude the case of equal rates  According to the Lomovtsev’s correction method for different rates  the known double integral over the characteristic triangle on the right-hand side  of wave equation requires correction, since this integral is not its classical solution in the first quarter plane, even with an even continuation of  over  to the second quarter plane.

The article [8] substantiates the regularity and stability of the solution to the mixed problem for a linear homogeneous wave equation with variable coefficients on  in bounded set of -dimensional Euclidean space. On one of the two parts of its boundary, the zero sum of partial derivatives of the solution up to the second order inclusive is indicated. This sum of second partial derivatives models the Wentzell effect of frictional damping. In the case of one-rate wave equation with a constant coefficient  at , these second partial derivatives in the Wentzell boundary condition (1.5) of paper [8] are characteristic for  by the definition 3 of our article.

Fourier method of variables separation and Parseval’s equality are applied to solving a mixed problem for linear homogeneous wave equation under some boundary conditions of the third kind on a finite interval in the work [2]. An equipartion principle of kinetic and potential energy for weak solution to this mixed problem is obtained.

On the right-hand side  of the simplest string vibration equation for  with real-valued continuous potential , Chernyatin V.A. proposed necessary and sufficient conditions that ensure the existence of a unique and stable classical solution to the first mixed problem on the segment [3]. In the case of zero boundary and zero initial conditions, these conditions contained equalities  which are not necessary (mandatory) for all time values  By virtue of Remark 2.4 of dissertation [11] for Hadamard correctness of this first mixed tasks must meet the necessary and sufficient coordination conditions  zero boundary data with zero initial data and the right-hand side  of the string vibration equation. In this Remark 2.4 of the work [11] explicit classical solutions of the first mixed problem for the simplest string vibration equation are approximated by explicit classical solutions of mixed problems with non-characteristic first partial derivatives in boundary conditions on the first quarter plane. The necessity and sufficiency of the last equalities is also confirmed by the works of F.E. Lomovtsev for inhomogeneous wave equations under boundary conditions with characteristic first oblique derivatives, in which nonstationary coefficients of the first partial derivatives can be zero.

Using the Fourier method, in article [5] necessary and sufficient conditions for the initial data of existence classical solution to linear first mixed problem for a wave equation with a complex-valued summable potential  are derived. An explicit representation of its classical solution is found in the form of a rapidly convergent series. In one of Khromov’s article, it is also proved only the sufficiency of equalities  for Hadamard correctness of first mixed problem in Chernyatin’s work [3].

In article [9] Lomov I.S. obtained a generalized solution to a linear mixed problem for telegraph equation with a complex-valued summable potential  on an interval under boundary conditions of the first kind, using the sequential and axiomatic methods of A.P. Khromov. In this mixed problem, one of boundary conditions is nonlocal, that is, with the solution value at some interior point of interval. These two methods of A.P. Khromov generalize the Fourier method, using the resolvent method, the Krylov’s idea on accelerating the convergence of Fourier series and Euler’s ideas on divergent series. In all A.P. Khromov’s works the right-hand sides of inhomogeneous wave equations have only an overestimated sufficient smoothness.

According to the Fourier method and the Krylov method accelerated convergence of Fourier series, a mixed problem is solved for a homogeneous wave equation with summable potential for a nonlocal two-point third boundary conditions [6]. Got her a classical solution without increasing the smoothness requirements for the initial speed. A generalized solution to this mixed problem has been found, which is the uniform limit of its classical solutions.

In [4] the proposed nonlinear problems were transformed into nonlinear ordinary differential equations. With the help of sine-Gordon expansion method, exact explicit solutions of nonlinear problems in terms of hyperbolic functions were obtained for the first time. Shown are 3D, 2D, and contour plots of solutions for a range of fractional order values. From these plots, one can identify the kink solution, the bell-shaped solitary wave solution, and the anti-bell-shaped soliton solution.

The work [7] contains an illustrative overview of the main mathematical methods and the names of their authors for solving the Cauchy problem and mixed problems for a one-rate linear and nonlinear wave equation during the 18th, 19th and 20th centuries. In 2017 F.E. Lomovtsev proposed a fundamentally new correction method test solutions into classical solutions of the two-rates linear wave equation in the first quarter of the plane, which was used to construct the classical solutions (12) of equation (5) in our article. In terms of value, naturalness and importance this correction method in mathematics is similar to A. Einstein’s special theory of relativity in physics.

In this paper, for the first time, by modifying the characteristic method a unique and stable classical solution is explicitly found and a correctness criterion is established to a linear mixed problem for factorized inhomogeneous wave equation with low-order terms in the wave equation and the characteristic nonstationary second derivatives in a boundary mode ofsemibounded string. These results are obtained without extensions of the problem data (right-hand side of equation, initial and boundary data). Nonstationarity of the boundary regime means the dependence of its real-valued coefficients on time. Characteristic nature of its second derivatives means their direction along the critical characteristic of wave equation. The proved theorem contains the Hadamard correctness criterion, that is, necessary and sufficient smoothness requirements on mixed problem data and their matching conditions. The correctness criterion ensures its unique and everywhere solvability in the set of twice continuously differentiable functions (see Remark 11).

In the first quarter of the plane a similar mixed problem was previously studied for the case of noncharacteristic nonstationary second derivatives. It is important to note that the correctness according to Hadamard of the mixed problem with characteristic second derivatives in the boundary mode, which we studied in this work, requires greater smoothness of the problem data and one more matching condition than for the same problem with noncharacteristic second derivatives in the boundary mode. For noncharacteristic nonstationary first oblique derivatives in boundary mode, correctness criterion and classical solution formula to the mixed problem for the simplest vibration equation of a semibounded string with  was established in, [11, pp. 23–51]. Similarly, Hadamard’s correctness of mixed problems with characteristic first oblique derivatives requires greater smoothness of the problem data and one matching condition more than for the same problems with noncharacteristic first oblique derivatives.

2. Statement of the main characteristic mixed problem. Preliminary concepts

In the first quarter of the plane , the mixed (initial-boundary value) problem is posed

            (1)

                    (2)

 (3)

where  ,   are first partial derivatives;  ,  ,  ,   are real constants; coefficients   are given real-valued functions of the variable ;  are given real-valued problem data of their variables  and . Here and below, the subscripts of functions denote their partial derivatives of the corresponding orders with respect to the variables indicated in the subscripts.

With the help of the physical interpretation for the general integral, just as in [12, pp. 54–59], it is proved that the equation (1) at  models forced oscillations of a homogeneous string. These oscillations are the result of the superimposition of a forward wave and a backward wave moving respectively in the positive and negative directions of the axis  with rates  , and also the impact of the driving force with the density . If the rates are equal , then the string oscillates in a medium at rest with no resistance, and Eq. (1) becomes the well-known equation for homogeneous string vibrations. If the rates are not equal to , then the string oscillates in a moving, elastically resistant to a homogeneous environment.

Equation (1) has two sets of real characteristics    

 Definition 1.  Characteristic , where the coefficient   is called  critical for equations (1) in the first quarter  of the plane .

We assume that in the boundary mode (3) the first partial derivatives are not characteristic, and the second partial derivatives are. Concepts of characteristic the first and second derivatives of problem (1)–(3) will be given in the Definition 3 of the next section. They can also depend on the coefficient  of the lower part of wave equation (1) in the Theorem 6.

Let   be the set of all  times continuously differentiable functions on a subset  of the plane  and .

Definition 2. The classical solution to the mixed problem (1)–(3) is the function   satisfying equation (1) on  in the usual sense, but the initial (2) and boundary (3) conditions in the sense of the limits for corresponding expressions from its values  and its derivatives at interior points  tending to corresponding boundary points  indicated in them.

Find classical solutions and a correctness criterion for the characteristic mixed problem (1)–(3).

3. Reduction to a simplified characteristic mixed problem. Preliminary results

Let us make the following replacement of the unknown function

               (4)

 which will allow us to reduce the original mixed problem (1)–(3) to an equivalent simple mixed problem

                                (5)

                                                     (6)

 (7)

 with new coefficients and problem data

                       (8)

For boundary conditions of mixed problems, the concept of characteristic first partial derivatives was introduced by Yurchuk N.I. in 2009, [1], and characteristic second partial derivatives were introduced by Lomovtsev F.E. in 2019.

Definition 3. In the boundary mode (7), the first and second partial derivatives are called  characteristic if, respectively,  and   for all , i.e. they are directed along the critical characteristic  of the equation (5) for all .

For the Definition 3 of classical solutions  of the mixed problem (5)–(7) directly follow obvious necessary smoothness requirements

                               (9)

Additional nonobvious necessary smoothness from  and  will be specified later.

In the boundary regime (7) we set  calculate the values of the terms of its left side using the initial conditions (6) for  and equation (5) for  and get the first matching condition between them

   (10)

According to Definition 2 of the classical solutions to the mixed problem (5)–(7), here we have used the smoothness (9) and the following values of partial derivatives

                                (11)

The second matching condition for them will be derived in the proof of Theorem 1 below.

Critical characteristic  divides the first quarter  of the plane into sets  and

From the well-known correction method test solutions into classical solutions by Lomovtsev F.E. in the first quarter of the plane, Theorem 1 uses the necessary smoothness (14), (15) of the right-hand side , additional to the required  and the following particular classical solutions of the inhomogeneous equation (5) in  and  

                   (12)

where , ,  in ,  in . Solutions  in ,  in  and the necessary smoothness (14), (15) on  are taken from Theorems 1–3 of the above article with the correction method by F.E. Lomovtsev in 2017.

We introduce the following notations

   (13)

Theorem 1.  Let in the boundary mode (7) be the coefficients      , noncharacteristic first oblique derivatives, i.e.  for all  and characteristic second partial derivatives, i.e.  for all  . Then the characteristic mixed problem (5)–(7) on  has a unique and stable on  classical solution  for those and only those , which have smoothness (9),  and

(14)

                     (15)

                      (16)

                                                   (17)

            (18)

 satisfy the matching conditions (10) and

                                     (19)

Here the function  is the value of the derivative with respect to the vector  from function  for ,  and  is the length of this vector.

Classical solution  to the characteristic mixed problem (5)–(7) on  is the function:

                         (20)

      (21)

 Proof. Particular solutions of the inhomogeneous equation (5) given by formulas (12) make it possible to pass from the solution of the inhomogeneous equation (5) in the domain  to the solution of the homogeneous equation (5) in the domain . General integral of inhomogeneous equation (5) can be represented as classical solutions given by the sum of the general integral   of the homogeneous equation (5) and solutions (12) of the inhomogeneous equation in the domain :

                            (22)

 Here  and  are any twice continuously differentiable functions of their variables , function  is equal to  in  and  in . Adding and subtracting on the right-hand part of this equality a finite number , we represent the general integral (22) in another form

Then the general integral (22) takes the equivalent form

                             (23)

 where   are any twice continuously differentiable functions of the form

                       (24)

 According to the well known immersion method in solutions with fixed values by F.E. Lomovtsev, the use of the general integral (23) instead of (22) simplifies the solving of differential equations systems.

1). On the set  , the classical solution to the mixed problem (5)–(7) is solution of the Cauchy problem (5), (6), which is expressed by the generalized d’Alembert formula (Euler) (20) from the dissertation, [11], on page 85 at the constants . This solution is unique and stable in    with known sufficient smoothness requirements (9), (14), (15) at  due to Theorem 3 of the above article with the Lomovtsev’s correction method. The necessity of conditions (14), (15) at  for  from (9) was proved in above article with the Lomovtsev’s correction method. Due to formula (35) of Theorem 3 of this above article at  for , our integral smoothness requirements (14) and (15) at  on  are equivalent to the requirements

The stability of solution (20) with respect to    follows from the Banach closed graph theorem or the Banach open mapping theorem due to the existence and uniqueness classical solution  of the Cauchy problem (5), (6). In addition, directly from formula (20) for any  we derive a continuous dependence of the solution  in the Banach space  from the problem data    in the Cartesian product of the Banach spaces  with norms

2). Solutions to problem (5)–(7) on the set  will be sought as solutions to the Picard problem for equation (5) on  with the equality  on the critical characteristic   and boundary mode equality (7). Substituting the general integral (23) into the first equality, we obtain the first equation of the system

      (25)

 Having expressed the function  from here, we make the change  and, taking into account the equality  from (24) we have the function

                                                      (26)

 which coincides with the function  from (13).

The second equation of the system is found by substituting the general integral (23) into the boundary mode (7), from which we deduce that

                         (27)

Taking into account the characteristicity of the second derivatives, we transform the left-hand side of this equality and reduce to a first-order ordinary differential equation

Solving this equation for , we obtain the function

with an arbitrary constant . From equalities (24) we have that  and hence

                                           (28)

Thus, a unique solution (26), (28) of the equation system (25), (27) is found. We substitute the functions  from (26) for  and  from (28) for  into the general integral (23), we change the integration variables ,  and have the unique formal solution (21) to the problem (5)–(7) on .

Let us now prove the sufficiency of requirements (9), (14), (15) at , (16)–(18) for a doubly continuous differentiability of the function (21) on . Due to requirements (9) on , it is obvious twice continuous differentiability of the first term of the function  from (21). Since for the functions  and  we obtain the equalities

                      (29)

 from them, for a continuous function   with properties (14), (15) at , we have the inclusion . To verify the doubly continuous differentiability of the second term of the function (21), it suffices to show once the continuous differentiability of its integrand. Firstly, for smooth coefficients     , the exponent is obviously continuously differentiable

Secondly, by the hypothesis of the Theorem 1, the given  and therefore it remains to prove the continuous differentiability of the function  from (13). For  is true the equality

                      (30)

The first term of this sum is obviously equal to

For coefficients      continuous differentiability of terms from this sum is guaranteed by smoothness requirements (9), (16), (17) on

Let us analyze the smoothness of the second term in the sum (29). From the identity  should  With the help of the first formula from (29) we deduce the equality

and its differentiation with respect to , we find the value of the mixed derivative

Expressing the derivative with respect to  in terms of the derivative with respect to , using the first formula from (29) derive the value of the second derivative

Based on these formulas and (29), we calculate the value of the second term in the sum (30)

    (31)

 Here, the continuous differentiability of the first two terms follows from the condition (18), and the last two terms from the conditions (14) and (15) at  for . Thus, the smoothness requirements (9), (14), (15) at , (16)–(18) from the Theorem 1 are sufficient for doubly continuous differentiability of the function  on the set .

From our solving of the Picard problem above, we know that twice continuously differentiable functions  on  and  on  are equal on the critical characteristic . In Theorem 1, the matching conditions contain the values of the problem data     and their derivatives at the origin , in which, by definition, they are limits of the corresponding expressions on the values of the doubly continuous differentiable functions  at interior points  Therefore, for smooth gluing, the function  must be twice continuously differentiable in some neighborhood of the characteristic , which in Theorem 1 is provided by the required smoothness of the problem data. Based on the terms of matching conditions (10), (19) we can talk about the continuity of partial derivatives for the first and second orders of the solutions  and   on the line :

These equalities are first calculated for smoother    and then extend by passage to the limit in    to less smooth    with properties (14)–(18). This is possible because these smoother problem data obviously satisfy such properties.

The uniqueness of the classical solution  to the mixed problem (5)–(7) in  follows from the uniqueness of the classical solution  of the Cauchy problem (5), (6) in , the uniqueness classical solution  of the above Picard problem in  and their doubly continuous differentiability on the characteristic .

3). The necessity of smoothness (9) and the matching condition (10) were established by us before formulation of Theorem 1. The necessity of smoothness (14), (15) on  is established in Theorems 2 and 3 from the above article with the Lomovtsev’s correction method, where it is actually derived from the equalities

                                                      (32)

 since the smooth functions are , . Equalities (32) correspond to derivatives of  and  along the characteristics of equation (5), as well as solutions of the equalities system (29).

Let us prove the necessity of the remaining smoothness requirements , (16), (17), (18) and matching conditions (19).

Let us derive the need for continuous differentiability . According to the Definition 2 the general integral (23) does not lose its smoothness when it is substituted into equation (5) and boundary mode (7). Therefore, we substitute the general integral of the homogeneous equation (5) from the form  for  and   in (7) and taking into account the characteristic nature of the second derivatives, we arrive at the equality

From the smoothness of  in Theorem 1 and the twice continuous differentiability of  on , hence we have .

Let us verify the need for additional smoothness (16), (17) of the initial data  and , already having the necessary smoothness from (9). Since the classical solutions (23) should not lose their smoothness when substituting into boundary mode (7), in it we take ,  ,  . For any  solutions  of the homogeneous equation (5) we substitute into the boundary mode (7), we take into account the characteristic nature of the second derivatives, and we have

In view of the smoothness     and  hence we have the smoothness

                                                   (33)

Having calculated the derivative with respect to   of the left-hand side of this equality, we establish the smoothness

To obtain the requirement from (17) on , in the general integral (23) we put

Then for any  solution

of the homogeneous equation (5) we substitute into the boundary mode (7) and, on the basis of the characteristicity of the second partial derivatives, we arrive at the equality

From the smoothness     the smoothness of the left-hand side of this equality follows

                                                   (34)

 Here we take the derivative with respect to  from left-hand side and have the smoothness of the difference

The solution   is a classical solution of the inhomogeneous equation (5) in the domain  , therefore, after its substitution into the boundary mode (7), we obtain

Since the right-hand side of this equality is once continuously differentiable due to the need for the given  to be continuously differentiable and the solution , to be twice continuously differentiable, then, according to (31), we have the inclusion

                           (35)

From this, the inclusion (35) implies the necessity of the smoothness requirement (18) on  due to the necessary requirement (14) for x = 0 and the smoothness of the coefficients

In Theorem 1, it remains to prove the necessity of the matching condition (18). Let us preliminarily establish the existence of the derivative with respect to the vector present in this condition  from the right-hand side  at the origin of the plane for any continuous function   satisfying requirements (14), (15), (18). With this goal for all smoother  we differentiate with respect to  and arrive at the identity

The right-hand side of this equality at  after applying the characteristicity condition  is equal to the expression

              (36)

Since the existence of a finite value of  is provided by proven necessary condition (18), then this equality (36) extends by passing to the limit in  from smoother functions  to all continuous functions  , which satisfy the smoothness requirement (14), (15), (18). The latter means the existence of a derivative with respect to the direction  of all continuous right-hand sides of equation  satisfying conditions (14), (15), (18).

The derivation of the necessity for the second matching condition (19) is similar to the derivation of the first matching conditions (10). For smoother  we find the third-order derivatives

         (37)

Then for any smoother  and  we differentiate equality (7) one times in , we set , apply the derived values (11), (37) and find the equality (19). Then, by passing to the limit in , we extend this equality to the real data    of problem (5)–(7) with smoothness (9), (14)–(18) due to proven existence of a bounded derivative .

The continuous dependence of solution (21) on     to the mixed problem (5)–(7) on  follows from the Banach closed graph theorem or the Banach open mapping theorem due to existence and uniqueness of the classical solution  of the indicated Picard problem.

Let us introduce bounded trapezoids   and triangle  For any , formula (21) can also be the stability of the solution  in the Banach space  according to the problem data     in the Cartesian product of the Banach spaces  with norms

Theorem 1 is proved.

Corollary 2.  If the right-hand side  depends only on  or , then the statement of Theorem 1 is true without the integral smoothness requirements (14), (15) on .

 Proof. The changes of integration variables are the same as on pages 86, 87 in, [11], it can be proven that for continuous functions  which depend only on , the smoothness requirements (14), (15) on  are always satisfied. Only in our Theorem 1 for  we must additionally use the twice continuous differentiability of the functions  from (12), since they are already corrected classical solutions of the inhomogeneous equation (5) in the set  by virtue of the above Lomovtsev’s article. If continuous functions  depend only on , then the smoothness requirements (14), (15) on  are obvious. Corollary 2 is proved.

Remark 3.  Formula (20) for  becomes the well-known d’Alembert (Euler) formula for solving the Cauchy problem (5), (6) in  [12].

Corollary 4.  Let the right-hand side  depend on  and  In the integral smoothness requirements (14), (15), membership integrals to the set  is equivalent to their membership in the sets  or . Here  and  be the sets respectively continuous in  or  and functions continuously differentiable with respect to  or  on .

 Proof of Corollary 4 is similar to the proof on page 103 of, [11], using the relevant representation of the first partial derivative with respect to  of the functions  from (14), (15) through the partial derivative with respect to  of these functions. Here for  we also need to take advantage of the fact that the functions (12) are already corrected classical solutions of the inhomogeneous equation (5) in  by virtue of the Lomovtsev correction method. The brief proof of Corollary 4 is complete.

Remark 5.  In Theorem 1 of present paper, the smoothness requirements (16), (17) and the matching condition (18) may have a different form of recording. But due to the inclusions (33)–(35) they are equivalent to similar smoothness requirements and matching condition for this article. In the present paper, this writing form of the smoothness requirements (16), (17) and the matching condition (18) is more correct, since the derivatives  can not exist for those  for which .

4. Main characteristic mixed problem. Main results

On the basis of Theorem 1 and its proof given above, the following theorem can be formulated.

Theorem 6.  Let in the boundary mode (3) the coefficients be      , first partial derivatives are noncharacteristic:    , and the second partial derivatives are taken along the critical characteristic of equation (1):  . Then the characteristic mixed problem  (1)–(3) in the area  has a unique and stable on   classical solution  for those and only those , which are smoothness requirements (9),  and (14)–(18) for all  instead of  and satisfy the matching conditions:

(38)

                                (39)

 where the functions are

This classical solution to the characteristic mixed problem (1)–(3) on  is the function

   (40)

           (41)

In this theorem, we use the following functions:

Proof. The correctness criterion for main characteristic problem (1)–(3) formulated in the Theorem 6 follows from the correctness criterion for the simplified characteristic problem (5)–(7), which is indicated in Theorem 1. The correctness of the main characteristic mixed problem (1)–(3) depends on the condition of noncharacteristicity of the first partial derivatives:   , which involves the coefficient  of the lower part of equation (1). In the smoothness requirements (9) and also (14), (15), the presence of infinitely differentiable and invertible exponentials from formulas of solutions (4), coefficients and problem data (8) does not affect the smoothness of the input data     in infinitely small neighborhoods of internal endpoints . That is obvious for the functions  from (16), (17), (18). For the functions  this follows from (14), (15) due to relations of the form (28), because  correspond to derivatives along the characteristics of equation (1) of functions (12), which are expressed through double integrals of  along the corresponding trapezoids. Therefore, in the formulation of the Theorem 6, smoothness (9), (14)–(18) is required from the problem data  instead of data . Matching conditions (38) and (39) are obtained by substituting problem data from (8) into matching conditions (10) and (19), respectively. Formulas (40) and (41) for the unique and stable classical solution of the main characteristic mixed problem (1)–(3) are derived by substituting the values of right-hand side, initial data, boundary data and boundary coefficients from (8) to the solution formulas (20) and (21) of the simplified characteristic mixed problem (5)–(7).

By Corollaries 2, 4, Corollaries 7, 9 hold.

Corollary 7.  If the right-hand side  depends only on  or , then the statement of the theorem 6 is true without the integral smoothness requirements (14), (15) on .

Remark 8.  The solution formula (40) of the Cauchy problem (1), (2) on  is a generalization of the well known d’Alembert (Euler) formula to a more general one dimensional two-rates wave equation.

Corollary 9.  Let the right-hand side  depend on  and  In the integral smoothness requirements (14), (15), membership integrals to the set  is equivalent to their membership in the sets  or . Here  and  be the sets respectively continuous functions in  or  and continuously differentiable functions with respect to  or  on .

Remark 10. In the case of equation coefficients   and boundary mode with   , Theorem 6 gives a unique and stable classical solution and correctness criterion to the mixed problem for equation (1) with noncharacteristic nonstationary first oblique derivative in the boundary mode (3).

Remark 11.  A mixed problem with characteristic or noncharacteristic boundary conditions has different solutions, [1]. It is well known that in mixed problems for a bounded string with characteristic first oblique derivatives in the boundary conditions, as the string oscillation time increases, the smoothness of the problem data and the number of matching conditions increases. This pattern is also valid for characteristic second partial derivatives in the boundary conditions. Therefore, in the future, the smoothness criteria and the matching conditions to the problem data for classical solutions of mixed problem obtained in this work will be generalized to spaces of higher orders smoothness. The authors of this work have already obtained some sufficient matching conditions on the data of an auxiliary mixed problem with the (sufficient) smoothness of its solutions being overestimated by one in spaces of higher orders of smoothness, [10].

5. Conclusion

In the present paper, we derive an explicit formula for the unique and stable classical solution to main mixed problem with characteristic second derivatives in the boundary mode with time-dependent coefficients. For the unique and stable everywhere solvability of the problem posed the necessary and sufficient smoothness requirements on the right-hand side of the equation, the initial data and the boundary data, and two matching conditions the boundary mode with the initial conditions and the equation are found. These smoothness requirements of the problem data serve as a criterion for the doubly continuous differentiability of two formulas (40) and (41) of one solution to the main characteristic mixed problem on the sets  and , respectively, and the two matching conditions are also on the critical characteristic . The correctness of solutions (20), (21), (40), (41) and second matching conditions (19), (39) is verified us in the Mathematica Wolfram computer algebra system.

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