MIXED PROBLEM FOR A 1D INHOMOGENEOUS WAVE EQUATION AT TWO COSTANT VELOCITIES WITH NONSTATIONARY CHARACTERISTIC OBLUQUE DERIVATIVES IN THE BOUNDARY MODES

​СМЕШАННАЯ ЗАДАЧА ДЛЯ ОДНОМЕРНОГО НЕОДНОРОДНОГО ВОЛНОВОГО УРАВНЕНИЯ ПРИ ДВУХ ПОСТОЯННЫХ СКОРОСТЯХ С НЕСТАЦИОНАРНЫМИ ХАРАКТЕРИСТИЧЕСКИМИ КОСЫМИ ПРОИЗВОДНЫМИ В ГРАНИЧНЫХ УСЛОВИЯХ

Ломовцев Фёдор Егорович

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

Беларусь, г. Минск

Устилко Екатерина Валерьевна

аспирант кафедры интеллектуальных методов моделирования, Белорусский государственный университет,

Беларусь, г. Минск

ABSTRACT

The unique and stable classical (twice continuously differentiable) solutions of the linear mixed problem for the one-dimensional two-velocities inhomogeneous wave equation with constant coefficients at the characteristic first skew derivatives in non-stationary boundary modes of a bounded string are found in explicit form. The non-stationarity of the skew derivatives is indicated by the time dependence of their coefficients. Boundary regimes for all values of time, the characteristic oblique derivatives are directed along one of the two critical characteristics of the wave equation. A Hadamard correctness criterion is derived from the necessary and sufficient smoothness requirements and agreement conditions of the boundary data with the initial data and the right-hand side of the wave equation. A global correctness theorem for this characteristic mixed problem, in which the longer the string oscillation time, the higher the smoothness requirements and the more matching conditions its input data.

АННОТАЦИЯ

Найдены в явном виде единственные и устойчивые классические (дважды непрерывно дифференцируемые) решения линейной смешанной задачи для одномерного двухскоростного неоднородного волнового уравнения с постоянными коэффициентами при характеристических первых косых производных в нестационарных граничных режимах ограниченной струны. На нестационарность косых производных указывает зависимость их коэффициентов от времени. В граничных режимах для всех значений времени характеристические косые производные направлены вдоль одной из двух критических характеристик волнового уравнения. Выведен критерий корректности по Адамару из необходимых и достаточных требований гладкости и условий согласования граничных данных с начальными данными и правой частью волнового уравнения. Доказана глобальная теорема корректности этой характеристической смешанной задачи, в которой чем больше время колебаний струны, тем выше требования гладкости и больше условий согласования ее входных данных.

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

Ключевые слова: Смешанная задача, характеристическая первая производная, классическое решение, критерий корректности, условия гладкости, глобальная теорема корректности.

I. Introduction

In this paper, using F.E. Lomovtsev’s method of auxiliary mixed problems for wave equations on the semi-axis, a unique and stable classical solution of a linear mixed problem for a two-velocity (for ) inhomogeneous (for ) wave equation with constant coefficients at the characteristic oblique derivatives in non-stationary boundary modes is found in explicit form. A correctness criterion (according to Hadamard) is also established: the existence, uniqueness and stability of the classical (twice continuously differentiable) solution of this characteristic mixed problem. The non-stationarity of the oblique derivatives means that the time dependence of their coefficients, and the characteristic property of the oblique derivatives is their direction along one of the two critical characteristics of the wave equation for all values of time. The correctness criterion consists of necessary and sufficient smoothness requirements and matching conditions for the problem data of the mixed problem (the right-hand side of the equation, the initial and boundary data), which guarantee its unique and stable solvability everywhere with respect to the problem data. In the present work, complete, final results of the study of the characteristic mixed problem in the case of a bounded string are obtained for classical solutions. We call theorems of correctness of mixed problems with necessary and sufficient smoothness requirements and matching conditions global. We obtained the minimum possible necessary (obligatory) conditions on the initial, boundary data and the right-hand side of the correctness criterion, which are sufficient for its Hadamard well-posedness, using the conditions proposed by E.V. Ustilko of new criterion values  of the initial data and  of the right-hand side of the wave equation in [1], (see Remark 1). The uniqueness of the correctness criteria (according to Hadamard) of global theorems lies in the fact that the input data of mixed problems for the oscillation equations of bounded strings are not explicitly continued outside the sets of their initial assignment, and sufficient assumptions on these data are simultaneously necessary, which guarantee their unambiguous and stable solvability everywhere. To justify the uniqueness of classical solutions and the need for correctness criteria, the input data of such mixed problems always admit smooth continuations from the half-strip of the plane to the first quarter of the plane without additional restrictions on these data (see Remark 3). In the works of other domestic and foreign authors, there are no global correctness theorems with correctness criteria (according to Hadamard) for all real-valued initial, boundary data and right-hand sides of wave equations of mixed (initial-boundary) problems and, especially, for non-stationary characteristic boundary regimes. In our works, in addition to proofs of global correctness theorems for mixed problems, explicit formulas for their classical (twice continuously differentiable) solutions are also derived. Sufficient conditions for the existence of a unique classical solution to a similar characteristic mixed problem for the simplest single-velocity (at ) homogeneous (at ) oscillation equation of a semi-bounded string were obtained in the article [2]. For the first time in this article, it was shown that a mixed problem with non-characteristic and characteristic oblique derivatives in the boundary regime has different classical (twice continuously differentiable) solutions and sufficient correctness conditions for the initial and boundary data. The characteristic oblique derivative in the boundary regime for a semi-bounded string causes higher smoothness and more matching conditions the initial and boundary data. Using special piecewise-smooth right-expanding continuations of the initial data from the segment  to the segments , the characteristic mixed problem was solved in [3]. Moreover, it turned out that in the case of a bounded string, for example, in the mixed problem from [3] for the simplest single-velocity homogeneous oscillation equation with a characteristic oblique derivative at the left and a boundary condition of the first kind at the right ends, the smoothness of the solutions and, consequently, the initial data of this problem increases with increasing oscillation time.

The global correctness theorem for a mixed problem for the simplest single-velocity (at ) inhomogeneous (at ) oscillation equation of a bounded string with non-characteristic oblique derivatives in the boundary conditions was first proved in the dissertation of E.N. Novikov [4] in Theorem 2.2, pp. 62–77. A global correctness theorem for a similar mixed problem for the simplest single-velocity inhomogeneous oscillation equation of a bounded string with characteristic oblique derivatives in boundary conditions was obtained in an article [5]. In these two works, the method of auxiliary mixed problems for wave equations on the half-line by F.E. Lomovtsev was also used, which does not require explicit continuations of the initial data outside the sets specified in the statements of the main mixed problems.

I.S. Lomov constructed a generalized solution to the mixed problem for an inhomogeneous telegraph equation with complex-valued potential  the right-hand side of the equation and the initial displacement at zero initial velocity and two-point boundary conditions [6]. In the general case, this equation does not allow separation of variables. The series of the formal solution may diverge. It continues the right-hand side of the equation and the initial displacement in a special way to the entire real line. Following the recommendations of L. Euler, to find the sum of the series of I.S. Lomov first uses the axiom of permutation in a series of integration and summation operations, and then presents a generalized solution in the form of a series that converges absolutely and uniformly with an exponential rate. He calls a generalized solution a function that satisfies the boundary and initial conditions in the usual sense, and the telegraph equation is almost everywhere. If the wave equations and boundary conditions allow separation of independent variables, then this technique is also used to solve mixed problems in the well-known separation method of variables (Fourier method) [7]. In our characteristic mixed problem (1)–(3) with non-stationary oblique derivatives, the independent variables  and  are not separated. There are more efficient methods than the Fourier method for explicitly solving mixed problems. Almost classical (generalized) solutions of a simple mixed problem with boundary conditions of the first kind for the string vibrations equation with one variable coefficient  are sought in the work of V.S. Rykhlov [8]. The Riemann formulas of explicit classical (twice continuously differentiable) solutions with correctness criteria (necessary and sufficient conditions) of the first and second mixed problems for a single-velocity telegraph equation with all variable coefficients are derived in articles [9] and [10], respectively. Riemann solved only the Cauchy problem for hyperbolic equations.

In the works [3, 7, 8, 11] and the works of other authors, there are no complete correctness criteria for Hadamard (necessary and sufficient smoothness requirements and matching conditions for the consistency of the right-hand sides of the equations, initial and boundary data) even in the case of non-characteristic mixed problems for the oscillation equations of a bounded string in the set of classical (twice continuously differentiable) solutions. In global Theorem 2 of this paper, we derive and prove a correctness criterion with formulas for classical solutions of the characteristic mixed problem (1)–(3).

2. The main characteristic mixed problem for the inhomogeneous oscillation equation of a bounded string with non-stationary characteristic first derivatives at the ends

The following characteristic mixed problem is investigated:

 (1)

                    (2)

       (3)

where the initial data of the mixed problem , , ,  are given real functions of their variables   are coefficients of the boundary conditions ,  are real functions of variable  and constant wave velocities  We denote partial derivatives of the corresponding orders of functions by subscripts with respect to the specified variables. The first partial derivative  with respect to the time variable  in the boundary conditions (3) denotes the action of dynamic forces at the ends  and  strings. The mixed problem (1)–(3) is characteristic, because in the boundary conditions (3) for  and  for all  the oblique derivatives from the first partial derivatives  and  are directed along the characteristics   of equation (1).

Definition 1. We will call the classical solution of the characteristic mixed problem (1)–(3) in a half-strip of the plane  a twice continuously differentiable function  satisfying equation (1) at the interior points  of the set  and initial conditions (2) and boundary conditions (3) in the sense of the limits of the corresponding differential expressions of the values  at the interior points  for   for all specified boundary points  of the set

It is required to find classical solutions  and establish a correctness criterion according to Hadamard, i.e. necessary and sufficient conditions on the problem data  for the characteristic mixed problem (1)–(3) on  to have a unique solution, continuous in

To solve problem (1)–(3) using the method of auxiliary mixed problems for a semi-bounded string, the upper half-strip  we replace the time-expanding set of closed rectangles  where  Then we divide these rectangles  into smaller rectangles   with height  The resulting rectangles are divided by characteristics into the following three triangles:

At the first stage of proof, solutions and correctness conditions of the main mixed problems for the org are derivedlimited segment  in the first rectangle  by narrowing the solutions and correctness conditions of the auxiliary mixed problems to the corresponding two trapezoids of the rectangle . At the second stage, recurrent formulas for solutions and correctness conditions of the main mixed problems for the limited segment  in the remaining rectangles  are constructed and their proof is carried out by the method of mathematical induction.

3. Auxiliary characteristic mixed problem

The classical solution and the well-posedness criterion of problem (1)–(3) can be derived from the classical solution and the correctness criterion of the auxiliary mixed problem in the first quadrant of the plane   :

                                 (4)

                                             (5)

                                  (6)

where the coefficients ,  are  times continuously differentiable real functions of the variable , the input data of the mixed problem , , ,   are given real functions of the variables   and constant wave velocities  In the boundary mode (6) the first oblique derivative is directed along the critical characteristic  of equation (4) for

We denote by  the set of all  times continuously differentiable functions on the subset  of the plane .

Definition 2. A smooth solution from the class of functions  of the auxiliary mixed problem (4)–(6) on  is a function ,   satisfying equation (4) in the usual sense on  and the initial conditions (5) and the boundary regime (6) in the sense of the limits of the corresponding differential expressions of its values  at the interior points   for   for all boundary points  of the set  specified in them.

It is required to find smooth solutions  explicitly and establish a correctness criterion by Hadamard, i.e. necessary and sufficient conditions on the input data  so that the mixed problem (4)–(6) on  has a unique smooth solution , continuous in .

Equation (4) in the plane  of variables   has two different families of characteristics:  

To solve the inhomogeneous equation (4) on , a particular classical solution  on  is used, given by the formulas

                                                         (7)

Using a modified characteristic method, the following theorem was proved in [1].

Theorem 1. Let in the boundary regime (6) the coefficients ,     In order for the characteristic mixed problem (4)–(6) in  to have a unique and stable in     solution , it is necessary and sufficient that the following smoothness requirements be satisfied

                                          (8)

                                                      (9)

matching conditions

                                             (10)

                                          (11)

                                   (12)

and integral smoothness requirements

                                                   (13)

                                                              (14)

                                                    (15)

  where all partial derivatives of  and  up to and including order  must be continuous on the characteristic   Then the smooth solution  of the mixed problem (4)–(6) in  is the function

                               (16)

                                                    (17)

The article [1] defines the following criterial values of the initial data  and the right-hand side  under the matching conditions (12) of Theorem 1 for the auxiliary characteristic mixed problem (4)–(6).

Definition 3. Finite real numbers    values of derivatives of order  of functions    from (9) for  under the matching conditions (12) for  are called the criterion values of the characteristic mixed problem (4)–(6) on , respectively, for the initial data  and the right-hand side  of the equation.

Example 1. Let the coefficients and input data in the mixed problem (4)–(6) be the following values      

Then for the Cauchy problem (4), (5) with initial displacements  on the set , according to formula (16) of Theorem 1, we obviously have smooth solutions

In the case of characteristic boundary conditions (6), when solving the mixed problem (4)–(6) on the set , we have following solutions for smoother initial displacements  and smoother boundary data  using formula (17) of Theorem 1

because the function  at   is equal to  In this case, on , there exist smooth first partial derivatives of the solutions  at  

Therefore, for these solutions  the boundary regime (6) is satisfied

The resulting identity  is then extended by limiting with respect to  and  to all initial displacements  with the additional smoothness requirement  from (9) of Theorem 1 and boundary data

Here it is important to say right away that for all less smooth data   the solutions  from (17) of the mixed problem (4)–(6) on the set  are smooth, i.e.  due to the smoothness  from the additional smoothness requirements (9).

Remark 1.  With the criterion values    E.V. Ustilko from [1] simplifies the notation of the matching conditions for  times continuously differentiable solutions of our auxiliary characteristic mixed problem (4)–(6). The disadvantage of these criterion values is that they also contain smooth partial derivatives of lower orders of the initial data and the right-hand side of the equation, the existence of which is beyond doubt. This disadvantage is absent in the above-mentioned article with T.S. Tochko [5], where other criterion values      for  for  proposed by F.E. Lomovtsev for the single-speed wave equation with . These criterion values contain partial derivatives of the initial data and the right-hand side of the wave equation of only higher orders and terms of dubious smoothness caused by the characteristic nature of the oblique derivative in the boundary condition (18) of the article [5]. For example, in it the criterial values  are the limits of the corresponding sums of partial derivatives of orders  only for  Remark 2.3 in [5] states that for  for all even  the criterial values  are

where  and the symbol  denotes the value of the derivative with respect to the vector  at  of the sums of partial derivatives of orders  of the function  satisfying the smoothness requirements (16), (17) of the article [5]. In this article, T.S. Tochko derived the compatibility conditions (6) with her functions  for even  and odd  from (7), and also proved her auxiliary theorem 2.1.

Remark 2.  The authors of the article [1] checked the validity of the matching conditions (10)–(12) and the classical solutions (16), (17) from the above formulated theorem 1 on a personal computer in the Wolfram Mathematica computer algebra system.

4. Correctness criterion for the main characteristic mixed problem

We will prove the global correctness theorem for the mixed problem (1)–(3) using the auxiliary mixed problem (4)–(6). The theorem uses particular classical solutions  of the non-homogeneous two-speed () of wave equation (1)

and the following functions

Theorem 2.  Let in the boundary regimes (3) the real coefficients ,       In order for the characteristic mixed problem (1)–(3) in  had a unique and stable classical solution  for the problem data , , , ,  necessary and sufficient requirements must be met:

                                       (18)

                                                                                         (19)

                                                     (20)

                                                   (21)

                      (22)

                                    (23)

                        (24)

                                                             (25)

  By this solution  of the mixed problem (1)–(3) on  is the function

                                             (26)

                                                        (27)

                 (28)

Here    are restrictions of the solution  respectively to triangles     with recurrent initial data:

                                                            (29)

 Proof. At the first step of mathematical induction we find the formulas for the solution, necessary and sufficient conditions for its existence, stability and uniqueness in the rectangle  for problem (1)–(3).

For  formulas (26), (27) coincide with the restrictions to the trapezoid , respectively, of formulas (16) , (17) for     and . Necessary and sufficient conditions (18)–(25) for   coincide with conditions (8)–(15) for     and  from Theorem 1 for  and

Smoothness requirements (18)–(22) for  ensure twice continuous differentiability of the function  in the triangles  and . In turn, the matching conditions (23)–(25) for   guarantee the continuity of the function  itself and its first and second partial derivatives on the common side of the triangles  and , which is located on the critical characteristic .

To find the classical solution of the original problem consisting of equation (1), initial conditions (2), and the second boundary condition at  from (3) in the trapezoid , we make a change of variable :

                                 (30)

                                                  (31)

[                                  (32)

The resulting mixed problem (30)–(32) is equivalent to the original problem (1)–(3) for the new function  with new problem data: ,  . The trapezoid  consists of two triangles    The formula for the unique and a stable classical solution  and the correctness criterion of problem (30)–(32) are obtained from Theorem 1, for  , , using the problem data specified above. We also replace  with  and  with .

As a result of these substitutions from formula (16) for triangle  we find the function

      (33)

 In the resulting formula, we return to the original data  from the data  and make the inverse substitution . After performing these actions, the function presented in formula (33) becomes a function  of the form (26) for . Projecting formula (17) onto the triangle  with , ,  and replacing  with  and  with , we similarly obtain the function

 which by replacing  with  and  is reduced to a function  of the form (28) for

If in the restriction of the integral smoothness requirements (13)–(15) for   on the trapezoid  for problem (30)–(32) take  instead of  then after the transition from  to  and the change of variable  they are transformed into the integral smoothness requirements (20)–(22) for  on the trapezoid

In the problem (30)–(32) on the trapezoid  for      the matching conditions (10)–(12) for  and  as a result of the change of variable  and the transition from  to  obviously become the matching conditions (23)–(25) for

Therefore, the equality of the functions  on  obtained above, together with the smoothness requirements (18)–(22) and the matching conditions (23)–(25) for  indicate the twice continuous differentiability of the functions  and  not only in the triangles  and  but also on their common side, which is on the characteristic  Thus, Theorem 2 is valid in the rectangle .

At the next stage of the proof of the Theorem 2 using the method of mathematical induction, we assume that Theorem 2 is valid in the rectangle  Then we will show its validity on the rectangle

In Theorem 2 on rectangles , the smoothness of the classical solution  from (26)–(28) changes  together with the smoothness of the recurrent initial data  from (29), the right-hand side  of the equation and the boundary data   from (18). Therefore, under the agreement conditions (25) of Theorem 2, according to Theorem 1 with Definition 3 on  there exist finite real criterial values      of the characteristic mixed problem (1)–(3) on  These criterial values    are the values of the derivatives with respect to  of order  from functions    from (19) of Theorem 2 for .

The mixed problem (1)–(3) in the rectangle  for the function  by changing the variable  is reduced to the next mixed problem for the function  

                               (34)

                                                                                    (35)

                                     (36)

 in the rectangle  with the right-hand side  of the equation and boundary data   

By the hypothesis of mathematical induction, there exists a unique and stable classical solution  of the auxiliary mixed problem (34)–(36) in  for , given by formulas (26)–(28) from Theorem 2 for  and with  instead of

                                                     (37)

                                                                (38)

                                                (39)

In formulas (37)–(39) we perform the reverse transition from   ,  to     for  and make the inverse substitution  As a result, we obtain formulas (26)–(28) for

Formulas (37)–(39) for  and with the variable  instead of the variable , define a twice continuously differentiable function  with respect to  and  on the rectangle  by the induction hypothesis, due to the twice continuous differentiability of  in , and also smoothness of the initial data  and , which are expressed in (29) through the function  Considering the sufficient smoothness of the replacement  we conclude from here that formulas (26)–(28) for  from Theorem 2 also give a twice continuously differentiable function  with respect to  and  on the rectangle

Necessary and sufficient requirements for smoothness and consistency (18)–(25) for  from Theorem 2 for the mixed problem (1)–(3) in  are easily derived by replacing  from the necessary and sufficient requirements of smoothness and consistency (18)–(25) for  and with the variable  instead of the variable  for the mixed problem (34)–(36) in .

It remains to verify that functions (26)–(28) are twice continuously differentiable for  on the common side  of rectangles  and  On the common side  of these rectangles the equalities are obviously true

                              (40)

 for all  Differentiating these equalities (40) the appropriate number of times with respect to  and  using equation (1), we obtain the values of the partial derivatives

 From these equalities we conclude that functions (26)–(28) are twice continuously differentiable on the common side  of rectangles  and  Theorem 2 is proved.

Remark 3. The uniqueness of classical solutions and the necessity of correctness criteria for our mixed problem (1)–(3) require a proof of the possibility of smooth continuations of its classical solutions outside the upper half-strip of the plane  without additional restrictions on the right-hand side of the equation, initial and boundary data. It is illustrated by the example of the first mixed problem at the end of the article [12]. In the mixed problem (1)–(3) of this paper, the rectangles  for  become a half-strip  of the first quarter of the plane  Therefore, we need continuations of its initial data with  on  for which the unique solutions of our auxiliary mixed problem (4)–(6) belong to the sets   There are infinitely many such smooth extensions of the right-hand side  to  and initial data  for . For example, to have smooth continuations  of the right-hand side  of the wave equation (1) from  to the first quadrant plane  it is sufficient to first smoothly continue the values  along  to some half-strip  of points  for  to functions from  by the method of continuation by reflection and then by cutting off an infinitely differentiable function to functions   from [11], pp. 24–31. Thus, in Theorem 2, our new continuation in the form of a recurrence of the initial data and the right-hand side along the positive time coordinate  axis in the explicit formulas of solutions is preferable to any continuation of the initial data and the right-hand side of the equation along the  axis.

Remark 4. The authors of this article checked the validity of the matching conditions (23)–(25) and classical solutions (26)–(28) of the proven Theorem 2 on a personal computer in the Wolfram Mathematica computer algebra system.

Corollary 1.  If the right-hand side  in  depends only on  or  and  then the assertion of Theorem 2 is valid without the integral smoothness requirements (20)–(22).

Proof. When the right-hand side  in  depends only on  or  then in Theorem 2 the integral smoothness requirements (20)–(22) are satisfied. This is proven by replacing the integration variables in the same way as in Corollary 5 of the article [13].

Corollary 2.  If the hight-hand side  of the equation (1) in  depends on  and  then under the smoothness conditions (20)–(22) of Theorem 2, the membership of expressions with integrals of the function  in the sets  is equivalent to their membership in the sets  or , respectively, for   and   Here  and  are sets of  times continuously differentiable functions with respect to  or continuous functions with respect to  and continuous functions with respect to  or  times continuously differentiable functions with respect to  on .

 Proof. The proof of our Сorollary 2 is similar to the proof of Сorollary 6 in the above-mentioned article [13].

 5. Conclusion. In this paper, explicit formulas (26)–(28) of the unique and stable classical solution  are derived without continuations along the  axis of the initial data and the right-hand sides of the wave equation of the mixed problem (1)–(3) with time-dependent real coefficients in the boundary condition (3) with characteristic first oblique derivatives at the ends of a bounded string. For the unique and stable everywhere solvability of this mixed problem, necessary and sufficient smoothness requirements (18)–(22) are found for the right-hand side  of the equation, the initial data  and the boundary data  and the agreement conditions (23)–(25) of the boundary conditions (3) with the initial conditions (2) and equation (1). The advantage of the method of auxiliary mixed problems is the projection of their results onto rectangles and the continuation of the input data of mixed problems along the positive time coordinate  axis. They are preferable to traditional periodic and any other continuations of the initial data and right-hand sides of the wave equations along the  axis.

References:

1. Lomovtsev, FE., Ustilko, EV. Mixed Problem for the Two-Velocity Wave Equation with Characteristic Oblique Derivative at the Endpoint of a Semibounded String. // Mathematical Notes. 2024. Vol. 116, No. 3. pp. 498–513.
2. Baranovskaya, S.N., Yurchuk, N.I. Mixed problem for string vibration equation with time dependence of an oblique derivative in boundary mode. //  Differential equations.  2009.  Vol. 45. No. 8.  pp. 1188–1191.
3. Shlapakova, T.S., Yurchuk, N.I. A mixed problem for the finite string vibration equation with a time dependent oblique derivative in the boundary condition which is directed along the characteristics. // Vesnik of the Belarusian State University. Seriya 1. 2013. No. 2. pp. 84–90.
4. Novikov, E.N. Mixed problems for forced vibration equation of a bounded string under nonstationary boundary conditions with first and second oblique derivatives. Abstract dis. : Candidate of Physics and Mathematics, Sciences (01.01.02), IM NAS of Belarus. 2017. 285 p.
5. Lomovtsev, F.E., Tochko, T.S. A mixed problem on forced vibrations of a bounded string with nonstationary characteristic oblique derivatives in boundary conditions. // Applied Mathematics and Physics. 2024. Vol. 56. No. 2. pp. 97–113. DOI 10.52575/2687-0959-2024-56-2-97-113
6. Lomov, I.S. Construction of a generalized solution of a mixed problem for a telegraph equation: sequential and axiomatic approaches. // Differential equations. 2022. Vol. 58. No. 11. pp. 1471–1483.
7. Tikhonov, A.N., Samarsky, A.A. Equations of mathematical physics. Moscow. 2004. 798 p.
8. Rykhlov, V.S. Classical solution of the initial-boundary value problem for the wave equation with mixed derivative. // Journal of Mathematical Sciences. 2024. Vol. 287. No. 4. pp. 630–663.
9. Lomovtsev, F.E. Riemann formulas of the first mixed problem for the general telegraph equation with variable coefficients in the first quadrant of the plane. I. // Vesnik of A.A. Kulyashov Mogilev State University. Ser. B. 2023. No. 2(62). pp. 16–31.
10. Lomovtsev, F.E., Liu, Z. Riemann Formulas of the Classical Solution and a Correctness Criterion to the Second Mixed Problem for the General Telegraph Equation with Variable Coefficients on a Segment. // Bulletin of the Iranian Mathematical Society. 2025. 51:45: pp. 1–32. https://doi.org/10.1007/s41980-025-00968-2
11. Lyons, J-L., Maggenes, E. Inhomogeneous boundary value problems and their applications. Moscow. 1971. 371 p.
12. Lomovtsev, F.E. The global Hadamard correctness theorem for the first mixed problem for the wave equation in a half-strip of the plane. // Vesnik of the Yanka Kupala State University of Grodno, Ser. 2, Mathematics, Physics, Informatics, Computing and Control. 2021. Vol. 11. No. 1. pp. 68–82.
13. Lomovtsev, F.E., Kukharev, A.L. The Cauchy Problem for the General Telegraph Equation with Variable Coefficients under the Cauchy Conditions on a Curved Line in the Plane. // WSEAS Transaction on Mathematics. 2023. Vol. 22. No. 1. pp. 936–949. DOI 10.37394/23206.2023.22.103