GENERAL INTEGRAL OF THE MODEL WAVE EQUATION WITH VARIABLE RATES a1(x,t) AND a2 ( x,t) IN THE UPPER HALF-PLANE
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Abstract
A new one-dimensional two-rate linear model wave equation utt(t,x)+(a1-a2)utx(t,x) - a1a2uxx(x,t)-a2-1(a2)tut(x,t)-a1(a2)xux(x,t) = ƒ(x,t) (1) a3-i(x,t)≥a3-i (0) >0, (x,t)∈G=]-∞,+∞ [x[0,+∞[, a3-i ∈ C2(G), i=1,2. A particular classical solution F of this two-rate model wave equation in the upper half-plane G is calculated. A double verification of this solution is made by substituting F into equation (1) and into the corresponding canonical form of equation (1), from which the function F was calculated. A smoothness criterion for the right-hand side ƒ of Eq. (1) for the classical solution F in the upper half-plane G is found. A smoothness criterion on ƒ for twice continuous differentiability F in the first quarter of the plane is discussed. With the help of the classical solution F, the general integral of equation (1) is derived from the set of all its classical solutions u ∈ C2(G), which is needed in solving the Cauchy problem and initial-boundary problems for equation (1). These results are obtained by applying the new "implicit characteristic method" of the equation developed earlier by the author.
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References
Lomovtsev, F. E. (2022). The Smoothness Criterion for the Classical Solution to Inhomogeneous Model Telegraph Equation at the Rate a(x,t) on the Half-Line. In Trudy 10-go mezhdunarodnogo nauchnogo seminara AMADE-2021 [Proc. 10th International Workshop AMADE-2021] (43–53). Minsk: BSU, ITC of the Ministry of Finance. (In Russ.).
Lomovtsev, F. E. (2022). Kriterii gladkosti chastnogo klassicheskogo resheniya neodnorodnogo model'nogo telegrafnogo uravneniya v pervoi chetverti ploskosti [Smoothness Criterion for a Particular Classical Solution of an Inhomogeneous Model Telegraph Equation in the First Quarter of the Plane]. Vestnik Polotskogo gosudarstvennogo universiteta. Seriya C, Fundamental'nye nauki [Herald of Polotsk State University. Series С. Fundamental sciences], (11), 99–116. DOI: 10.52928/2070-1624-2022-39-11-99-116. (In Russ., abstr. in Engl.).
Lomovtsev, F. E. (2017). Metod korrektirovki probnogo resheniya obshchego volnovogo uravneniya v pervoi chetverti ploskosti dlya minimal'noi gladkosti ego pravoi chasti [Correction method of test solutions of the general wave equation in the first quarter of the plane for the minimum smoothness of its right-hand side]. Zhurnal Belorusskogo gosudarstvennogo universiteta. Matematika. Informatika [J. of the Belarusian State University. Mathematics and informatics], (3), 38–52. (In Russ., abstr. in Engl.).
Lomovtsev, F. E., & Tochko, T. S. (2019). Smeshannaya zadacha dlya neodnorodnogo uravneniya kolebanii ogranichennoi struny pri kharakteristicheskikh nestatsionarnykh pervykh kosykh proizvodnykh na kontsakh [Mixed problem for an inhomogeneous vibration equation of a bounded string with characteristic non-stationary first oblique derivatives at the ends]. Vesnik Hrodzenskaha Dziarzhaunaha Universiteta imia Ianki Kupaly. Seryia 2. Matematyka. Fizika. Infarmatyka, Vylichal’naia Tekhnika i Kiravanne [Vesnik of Yanka Kupala State University of Grodno. Series 2. Mathematics. Physics. Informatics, Сomputer Technology and its Сontrol], 9(2), 56–75. (In Russ.).
Lomovtsev, F. E., & Ustilko, E. V. (2020). Smeshannaya zadacha dlya odnomernogo volnovogo uravneniya pri kharakteristicheskoi pervoi kosoi proizvodnoi v nestatsionarnom granichnom rezhime dlya gladkikh reshenii [A mixed problem for a one-dimensional wave equation with a characteristic first oblique derivative in a non-stationary boundary regime
for smooth solutions]. Vesnik Magileuskaga dzyarzhaunaga universiteta imya A. A. Kulyashova. Ser B. Pryrodaznauchyya navuki [Mogilev State A. Kuleshov Bulletin. Series B. Natural Sciences], 2(56), 21–36. (In Russ., abstr. in Engl.).
Lomovtsev, F. E., & Lysenko, V. V. (2019). Nekharakteristicheskaya smeshannaya zadacha dlya odnomernogo volnovogo uravneniya v pervoi chetverti ploskosti pri nestatsionarnykh granichnykh vtorykh proizvodnykh [A non-characteristic mixed problem for a one-dimensional wave equation in the first quarter of the plane with non-stationary boundary second derivatives]. Vesnіk Vіtsebskaga dzyarzhaunaga unіversіteta [Bulletin of the Vitebsk Dzyarzhaunaga University], 3(104), 5–17. (In Russ., abstr. in Engl.).
Lomovtsev, F. E., & Spesivtseva, K. A. (2021). Mixed Problem for a General 1D Wave Equation with Characteristic Second Derivatives in a Nonstationary Boundary Mode. Math Notes, 110(3), 329–338. DOI: 10.1134/S0001434621090030.
Lomovtsev, F. E. (2021). Pervaya smeshannaya zadacha dlya obshchego telegrafnogo uravneniya s peremennymi koeffitsientami na polupryamoi [The first mixed problem for the general telegraph equation with variable coefficients on the half-line]. Zhurnal Belorusskogo gosudarstvennogo universiteta. Matematika. Informatika [J. of the Belarusian State University. Mathematics and informatics], (1), 18–38. (In Russ., abstr. in Engl.).
Lomovtsev, F. E. (2022). Vtoraya smeshannaya zadacha dlya obshchego telegrafnogo uravneniya s peremennymi koeffitsientami v pervoi chetverti ploskosti [The second mixed problem for the general telegraph equation with variable coefficients in the first quarter of the plane]. Vesnik Hrodzenskaha Dziarzhaunaha Universiteta imia Ianki Kupaly. Seryia 2. Matematyka. Fizika. Infarmatyka, Vylichal’naia Tekhnika i Kiravanne [Vesnik of Yanka Kupala State University of Grodno. Series 2. Mathematics. Physics. Informatics, Сomputer Technology and its Сontrol], 12(3), 50–70. (In Russ., abstr. in Engl.).
Lomovtsev, F. E. (2015). Metod vspomogatel'nykh smeshannykh zadach dlya poluogranichennoi struny [Method of auxiliary mixed problems for a semi-bounded string]. In S. G. Krasovskii (Eds.), Shestye Bogdanovskie chteniya po obyknovennym differentsial'nym uravneniyam: materialy Mezhdunar. matem. konf.: v 2 ch. Ch. 2. [Sixth Bogdanov Readings on Ordinary Differential Equations: Proceedings of Intern. math. Conf. (in 2 part, Part 2)] (74–75). Minsk: BGU. (In Russ.).
Tikhonov, A. N., & Samarskii, A. A. (2004). Uravneniya matematicheskoi fiziki. Moscow: Nauka. (In Russ.).
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