Mixed Problem for a Higher-Order Nonlinear Pseudoparabolic Equation
We examine the unique generalized solvability of the mixed problem for a higher-order nonlinear pseudoparabolic equation with two parameters in mixed derivatives. Using the Fourier variable separation method, we reduce the problem to a countable system of nonlinear integral equations whose unique so...
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| Published in | Journal of mathematical sciences (New York, N.Y.) Vol. 254; no. 6; pp. 776 - 787 |
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| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
New York
Springer US
02.05.2021
Springer Springer Nature B.V |
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| ISSN | 1072-3374 1573-8795 |
| DOI | 10.1007/s10958-021-05339-w |
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| Abstract | We examine the unique generalized solvability of the mixed problem for a higher-order nonlinear pseudoparabolic equation with two parameters in mixed derivatives. Using the Fourier variable separation method, we reduce the problem to a countable system of nonlinear integral equations whose unique solvability can be proved by the method of successive approximations. We prove the continuous dependence of a generalized solution to the mixed problem on the initial functions and the positive parameters. |
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| AbstractList | We examine the unique generalized solvability of the mixed problem for a higher-order nonlinear pseudoparabolic equation with two parameters in mixed derivatives. Using the Fourier variable separation method, we reduce the problem to a countable system of nonlinear integral equations whose unique solvability can be proved by the method of successive approximations. We prove the continuous dependence of a generalized solution to the mixed problem on the initial functions and the positive parameters. We examine the unique generalized solvability of the mixed problem for a higher-order nonlinear pseudoparabolic equation with two parameters in mixed derivatives. Using the Fourier variable separation method, we reduce the problem to a countable system of nonlinear integral equations whose unique solvability can be proved by the method of successive approximations. We prove the continuous dependence of a generalized solution to the mixed problem on the initial functions and the positive parameters. Keywords and phrases: pseudoparabolic equation, generalized derivative, method of successive approximations, parameter, unique solvability. AMS Subject Classification: 35A02, 35M10, 35S05 We examine the unique generalized solvability of the mixed problem for a higher-order non-linear pseudoparabolic equation with two parameters in mixed derivatives. Using the Fourier variable separation method, we reduce the problem to a countable system of nonlinear integral equations whose unique solvability can be proved by the method of successive approximations. We prove the continuous dependence of a generalized solution to the mixed problem on the initial functions and the positive parameters. Keywords and phrases: pseudoparabolic equation, generalized derivative, method of successive approximations, parameter, unique solvability. AMS Subject Classification: 35A02, 35M10, 35S05 We examine the unique generalized solvability of the mixed problem for a higher-order non-linear pseudoparabolic equation with two parameters in mixed derivatives. Using the Fourier variable separation method, we reduce the problem to a countable system of nonlinear integral equations whose unique solvability can be proved by the method of successive approximations. We prove the continuous dependence of a generalized solution to the mixed problem on the initial functions and the positive parameters. |
| Audience | Academic |
| Author | Yuldashev, T. K. Shabadikov, K. Kh |
| Author_xml | – sequence: 1 givenname: T. K. surname: Yuldashev fullname: Yuldashev, T. K. email: tursun.k.yuldashev@gmail.com organization: Reshetnev Siberian State University of Science and Technology – sequence: 2 givenname: K. Kh surname: Shabadikov fullname: Shabadikov, K. Kh organization: Fergana State University |
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| Keywords | pseudoparabolic equation 35A02 method of successive approximations parameter 35M10 generalized derivative 35S05 unique solvability |
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| References | Il’inVAMoiseevEIOptimization for an arbitrary sufficiently large time period for the boundary control of string oscillations by an elastic forceDiffer. Uravn.20064212169917112347125 YuldashevaAVOn a problem for a quasilinear equation of even orderItogi Nauki Tekhn. Sovr. Mat. Prilozh. Temat. Obzory201714043493799894 V. A. Il’in, “Solvability of mixed problems for hyperbolic and parabolic equations,” Usp. Mat. Nauk, 15, No. 2 (92), 97–154 (1960). PokhozhaevSIOn the solvability of quasilinear elliptic equations of arbitrary orderMat. Sb.198211722512656447720491.35020 T. K. Yuldashev, “Mixed problem for a nonlinear integro-differential equation with a higher-order parabolic operator,” Zh. Vychisl. Mat. Mat, Fiz., 52, No. 1, 112–123 (2012). AlexandrovVMKovalenkoEVProblems of Continuum Mechanics with Mixed Boundary Conditions [in Russian]1986MoscowNauka SkrypnikIVHigher-Order Nonlinear Elliptic Equations [in Russian]1973KievNaukova Dumka KoshanovBDSoldatovAPBoundary-value problem with normal derivatives for a higher-order elliptic equation on the planeDiffer. Uravn.201652121666168136046801368.35102 YuldashevTKInverse problem for a nonlinear integro-differential equation with a higher-order hyperbolic operatorVestn. Yuzhno-Ural. Univ. Ser. Mat. Mekh. Fiz.201351697530592931331.45007 ZamyshlyaevaAAHigher-order mathematical models of Sobolev typeVestn. Yuzhno-Ural. Univ. Ser. Mat. Model. Program.2014725281335.35149 KarimovSTOn a method for solving the Cauchy problem for a one-dimensional multiwave equation with a singular Bessel operatorIzv. Vyssh. Ucheb. Zaved. Mat.201782741 TodorovTGOn the continuity of bounded generalized solutions of higher-order quasilinear elliptic equationsVestn. Leningrad. Univ.1975195663402250 AlgazinSDKiykoIAFlutter of Plates and Shells [in Russian]2006MoscowNauka KarimovSTMultidimensional generalized Erd´elyi–Kober operator and its application to solving Cauchy problems for differential equations with singular coefficientsFract. Calc. Appl. Anal.2015184845861337739810.1515/fca-2015-0051 TK Yuldashev (5339_CR12) 2013; 5 BD Koshanov (5339_CR7) 2016; 52 AV Yuldasheva (5339_CR13) 2017; 140 VM Alexandrov (5339_CR1) 1986 VA Il’in (5339_CR4) 2006; 42 SI Pokhozhaev (5339_CR8) 1982; 117 5339_CR11 IV Skrypnik (5339_CR9) 1973 ST Karimov (5339_CR5) 2015; 18 ST Karimov (5339_CR6) 2017; 8 TG Todorov (5339_CR10) 1975; 19 AA Zamyshlyaeva (5339_CR14) 2014; 7 SD Algazin (5339_CR2) 2006 5339_CR3 |
| References_xml | – reference: V. A. Il’in, “Solvability of mixed problems for hyperbolic and parabolic equations,” Usp. Mat. Nauk, 15, No. 2 (92), 97–154 (1960). – reference: KoshanovBDSoldatovAPBoundary-value problem with normal derivatives for a higher-order elliptic equation on the planeDiffer. Uravn.201652121666168136046801368.35102 – reference: AlexandrovVMKovalenkoEVProblems of Continuum Mechanics with Mixed Boundary Conditions [in Russian]1986MoscowNauka – reference: Il’inVAMoiseevEIOptimization for an arbitrary sufficiently large time period for the boundary control of string oscillations by an elastic forceDiffer. Uravn.20064212169917112347125 – reference: PokhozhaevSIOn the solvability of quasilinear elliptic equations of arbitrary orderMat. Sb.198211722512656447720491.35020 – reference: SkrypnikIVHigher-Order Nonlinear Elliptic Equations [in Russian]1973KievNaukova Dumka – reference: ZamyshlyaevaAAHigher-order mathematical models of Sobolev typeVestn. Yuzhno-Ural. Univ. Ser. Mat. Model. Program.2014725281335.35149 – reference: KarimovSTOn a method for solving the Cauchy problem for a one-dimensional multiwave equation with a singular Bessel operatorIzv. Vyssh. Ucheb. Zaved. Mat.201782741 – reference: KarimovSTMultidimensional generalized Erd´elyi–Kober operator and its application to solving Cauchy problems for differential equations with singular coefficientsFract. Calc. Appl. Anal.2015184845861337739810.1515/fca-2015-0051 – reference: YuldashevaAVOn a problem for a quasilinear equation of even orderItogi Nauki Tekhn. Sovr. Mat. Prilozh. Temat. Obzory201714043493799894 – reference: TodorovTGOn the continuity of bounded generalized solutions of higher-order quasilinear elliptic equationsVestn. Leningrad. Univ.1975195663402250 – reference: T. K. Yuldashev, “Mixed problem for a nonlinear integro-differential equation with a higher-order parabolic operator,” Zh. Vychisl. Mat. Mat, Fiz., 52, No. 1, 112–123 (2012). – reference: AlgazinSDKiykoIAFlutter of Plates and Shells [in Russian]2006MoscowNauka – reference: YuldashevTKInverse problem for a nonlinear integro-differential equation with a higher-order hyperbolic operatorVestn. Yuzhno-Ural. Univ. Ser. Mat. Mekh. Fiz.201351697530592931331.45007 – volume: 7 start-page: 5 issue: 2 year: 2014 ident: 5339_CR14 publication-title: Vestn. Yuzhno-Ural. Univ. Ser. Mat. Model. Program. – ident: 5339_CR3 doi: 10.1070/RM1960v015n02ABEH004217 – volume-title: Flutter of Plates and Shells [in Russian] year: 2006 ident: 5339_CR2 – ident: 5339_CR11 – volume-title: Problems of Continuum Mechanics with Mixed Boundary Conditions [in Russian] year: 1986 ident: 5339_CR1 – volume: 117 start-page: 251 issue: 2 year: 1982 ident: 5339_CR8 publication-title: Mat. Sb. – volume-title: Higher-Order Nonlinear Elliptic Equations [in Russian] year: 1973 ident: 5339_CR9 – volume: 5 start-page: 69 issue: 1 year: 2013 ident: 5339_CR12 publication-title: Vestn. Yuzhno-Ural. Univ. Ser. Mat. Mekh. Fiz. – volume: 18 start-page: 845 issue: 4 year: 2015 ident: 5339_CR5 publication-title: Fract. Calc. Appl. Anal. doi: 10.1515/fca-2015-0051 – volume: 42 start-page: 1699 issue: 12 year: 2006 ident: 5339_CR4 publication-title: Differ. Uravn. – volume: 8 start-page: 27 year: 2017 ident: 5339_CR6 publication-title: Izv. Vyssh. Ucheb. Zaved. Mat. – volume: 19 start-page: 56 year: 1975 ident: 5339_CR10 publication-title: Vestn. Leningrad. Univ. – volume: 140 start-page: 43 year: 2017 ident: 5339_CR13 publication-title: Itogi Nauki Tekhn. Sovr. Mat. Prilozh. Temat. Obzory – volume: 52 start-page: 1666 issue: 12 year: 2016 ident: 5339_CR7 publication-title: Differ. Uravn. doi: 10.1134/S0374064116120074 |
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| Snippet | We examine the unique generalized solvability of the mixed problem for a higher-order nonlinear pseudoparabolic equation with two parameters in mixed... We examine the unique generalized solvability of the mixed problem for a higher-order non-linear pseudoparabolic equation with two parameters in mixed... |
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| SubjectTerms | Approximation Differential equations Integral equations Mathematics Mathematics and Statistics Parameters |
| Title | Mixed Problem for a Higher-Order Nonlinear Pseudoparabolic Equation |
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