NONLINEAR WAVE INTERACTIONS IN A MACROSCOPIC PRODUCTION MODEL

doi:10.1007/s10473-021-0309-8

数学物理学报(英文版) ›› 2021, Vol. 41 ›› Issue (3): 764-780.doi: 10.1007/s10473-021-0309-8

NONLINEAR WAVE INTERACTIONS IN A MACROSCOPIC PRODUCTION MODEL

MINHAJUL^1,2, T RAJA SEKHAR²

1. Center for Applicable Mathematics, Tata Institute of Fundamental Research, Bangalore, Karnataka-560065, India;
2. Department of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur-2, India

收稿日期:2019-10-08 修回日期:2020-07-21 出版日期:2021-06-25 发布日期:2021-06-07
通讯作者: MINHAJUL E-mail:minhajulbgh63@gmail.com
作者简介:T RAJA SEKHAR,E-mail:trajasekhar@maths.iitkgp.ac.in

NONLINEAR WAVE INTERACTIONS IN A MACROSCOPIC PRODUCTION MODEL

MINHAJUL^1,2, T RAJA SEKHAR²

1. Center for Applicable Mathematics, Tata Institute of Fundamental Research, Bangalore, Karnataka-560065, India;
2. Department of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur-2, India

Received:2019-10-08 Revised:2020-07-21 Online:2021-06-25 Published:2021-06-07
Contact: MINHAJUL E-mail:minhajulbgh63@gmail.com
About author:T RAJA SEKHAR,E-mail:trajasekhar@maths.iitkgp.ac.in

摘要/Abstract

摘要： In this article, we study the exhaustive analysis of nonlinear wave interactions for a 2× 2 homogeneous system of quasilinear hyperbolic partial differential equations (PDEs) governing the macroscopic production. We use the hodograph transformation and differential constraints technique to obtain the exact solution of governing equations. Furthermore, we study the interaction between simple waves in detail through exact solution of general initial value problem. Finally, we discuss the all possible interaction of elementary waves using the solution of Riemann problem.

关键词: Elementary waves, wave interactions, Riemann problem, simple wave, differential constraints, Hodograph transformation

Abstract: In this article, we study the exhaustive analysis of nonlinear wave interactions for a 2× 2 homogeneous system of quasilinear hyperbolic partial differential equations (PDEs) governing the macroscopic production. We use the hodograph transformation and differential constraints technique to obtain the exact solution of governing equations. Furthermore, we study the interaction between simple waves in detail through exact solution of general initial value problem. Finally, we discuss the all possible interaction of elementary waves using the solution of Riemann problem.

Key words: Elementary waves, wave interactions, Riemann problem, simple wave, differential constraints, Hodograph transformation

中图分类号:

35L45

MINHAJUL, T RAJA SEKHAR. NONLINEAR WAVE INTERACTIONS IN A MACROSCOPIC PRODUCTION MODEL[J]. 数学物理学报(英文版), 2021, 41(3): 764-780.

MINHAJUL, T RAJA SEKHAR. NONLINEAR WAVE INTERACTIONS IN A MACROSCOPIC PRODUCTION MODEL[J]. Acta mathematica scientia,Series B, 2021, 41(3): 764-780.

参考文献

[1] Curró C, Fusco D, Manganaro N. An exact description of nonlinear wave interaction processes ruled by 2×2 hyperbolic systems. Zeitschrift für angewandte Mathematik und Physik, 2013, 64(4):1227-1248
[2] Curró C, Fusco D, Manganaro N. Hodograph transformation and differential constraints for wave solutions to 2×2 quasilinear hyperbolic nonhomogeneous systems. Journal of Physics A:Mathematical and Theoretical, 2012, 45(19):195207
[3] Curró C, Fusco D, Manganaro N. A reduction procedure for generalized Riemann problems with application to nonlinear transmission lines. Journal of Physics A:Mathematical and Theoretical, 2011, 44(33):335205.
[4] Minhajul, Zeidan D, Raja Sekhar T. On the wave interactions in the drift-flux equations of two-phase flows. Applied Mathematics and Computation, 2018, 327:117-131
[5] Kuila S, Raja Sekhar T. Interaction of weak shocks in drift-flux model of compressible two-phase flows. Chaos, Solitons & Fractals, 2018, 107:222-227
[6] Minhajul, Raja Sekhar T. Interaction of elementary waves with a weak discontinuity in an isothermal drift-flux model of compressible two-phase flows. Quarterly of Applied Mathematics, 2019, 77:671-688
[7] Sun M. A note on the interactions of elementary waves for the AR traffic flow model without vacuum. Acta Mathematica Scientia, 2011, 31B(4):1503-1512
[8] Shen C, Sun M. Wave interactions and stability of the Riemann solutions for a scalar conservation law with a discontinuous flux function. Zeitschrift fr angewandte Mathematik und Physik (ZAMP), 2013, 64(4):1025-1056
[9] Sen A, Raja Sekhar T, Sharma V D. Wave interactions and stability of the Riemann solution for a strictly hyperbolic system of conservation laws. Quarterly of Applied Mathematics, 2017, 75(3):539-554
[10] Jeffrey A, Kawahara T. Asymptotic methods in nonlinear wave theory. Applicable Mathematics Series, Boston:Pitman Publishing, 1982
[11] Jeffrey A. Quasilinear hyperbolic systems and waves. London-San Francisco, Calif.-Melbourne, Pitman Publishing, Ltd.(Research Notes in Mathematics, No 5) 230 pp, 1976
[12] Whitham G B. Linear and nonlinear waves. Volume 42. John Wiley & Sons, 2011
[13] Raja Sekhar T, Minhajul. Elementary wave interactions in blood flow through artery. Journal of Mathematical Physics, 2017, 58(10):101502
[14] Raja Sekhar T, Sharma V D. Interaction of shallow water waves. Studies Appl Math, 2008, 121(1):1-25
[15] Courant R, Friedrichs K O. Supersonic flow and shock waves. Volume 21. Springer Science & Business Media, 1999
[16] Seymour B R, Varley E. Exact solutions describing soliton-like interactions in a nondispersive medium. SIAM Journal on Applied Mathematics, 1982, 42(4):804-821
[17] Currò C, Fusco D. On a class of quasilinear hyperbolic reducible systems allowing for special wave interactions. Zeitschrift für angewandte Mathematik und Physik, 1987, 38(4):580-594
[18] Lax P D. Development of singularities of solutions of nonlinear hyperbolic partial differential equations. Journal of Mathematical Physics, 1964, 5(5):611-613
[19] Li T T. Global classical solutions for quasilinear hyperbolic systems. John Wiley & Sons, 1994
[20] Coste L F, Gottlich S, Herty M. Data-fitted second-order macroscopic production models. SIAM Journal on Applied Mathematics, 2015, 75(3):999-1014
[21] Subhankar Sil, Raja Sekhar T. Nonlocally related systems, nonlocal symmetry reductions and exact solutions for one-dimensional macroscopic production model. The European Physical Journal Plus, 2020, 135(6):514
[22] Sun M. Singular solutions to the Riemann problem for a macroscopic production model. Zeitschrift für Angewandte Mathematik und Mechanik, 2017, 97(8):916-931
[23] Manganaro N, Meleshko S. Reduction procedure and generalized simple waves for systems written in Riemann variables. Nonlinear Dynamics, 2002, 30(1):87-102
[24] Sueet Millon Sahoo, Raja Sekhar T, Raja Sekhar G P. Exact solutions of generalized Riemann problem for rate-type material. International Journal of Non-Linear Mechanics, 2019, 110:16-20
[25] Fusco D, Manganaro N. A method for finding exact solutions to hyperbolic systems of first-order PDEs. IMA Journal of Applied Mathematics, 1996, 57(3):223-242
[26] Meleshko S V, Shapeev V P. The application of the differential-constraints method to the two-dimensional equations of gas dynanics. Journal of Applied Mathematics and Mechanics, 1999, 63(6):885-891
[27] Munteanu L, Donescu S. Introduction to Soliton Theory:Applications to Mechanics. Volume 143. Springer Science & Business Media, 2006
[28] Mentrelli A, Rogers C, Ruggeri T, Schief W K. On two-pulse and shock evolution in a class of ideally hard elastic materials. Asymptotic Methods in Nonlinear Wave Phenomena, 2007:132-143
[29] Rogers C, Schief W K. Bäcklund transformations and superposition principles in nonlinear elastodynamics. Studies in Applied Mathematics, 2010, 124(2):137-149
[30] Rogers C, Schief W K, Szereszewski A. Loop soliton interaction in an integrable nonlinear telegraphy model:reciprocal and Bäcklund transformations. Journal of Physics A:Mathematical and Theoretical, 2010, 43(38):385210
[31] Curró C, Fusco D, Manganaro N. Differential constraints and exact solution to riemann problems for a traffic flow model. Acta Applicandae Mathematicae, 2012, 122(1):167-178
[32] Smoller J. Shock waves and reaction-diffusion equations. Volume 258. Springer Science & Business Media, 2012

NONLINEAR WAVE INTERACTIONS IN A MACROSCOPIC PRODUCTION MODEL

NONLINEAR WAVE INTERACTIONS IN A MACROSCOPIC PRODUCTION MODEL

PDF (PC)

摘要/Abstract

引用本文

使用本文

参考文献

相关文章 15

Metrics

本文评价

推荐阅读 0

[1]	刘见礼, 薛杰. GLOBAL SIMPLE WAVE SOLUTIONS TO A KIND OF TWO DIMENSIONAL HYPERBOLIC SYSTEM OF CONSERVATION LAWS[J]. 数学物理学报(英文版), 2020, 40(1): 261-271.
[2]	张婷, 盛万成. GLOBAL SOLUTIONS OF THE PERTURBED RIEMANN PROBLEM FOR THE CHROMATOGRAPHY EQUATIONS[J]. 数学物理学报(英文版), 2019, 39(1): 57-82.
[3]	王泽军, 张琦. FINITE TIME EMERGENCE OF A SHOCK WAVE FOR SCALAR CONSERVATION LAWS VIA[J]. 数学物理学报(英文版), 2019, 39(1): 83-93.
[4]	陈贵强, Matthew RIGBY. STABILITY OF STEADY MULTI-WAVE CONFIGURATIONS FOR THE FULL EULER EQUATIONS OF COMPRESSIBLE FLUID FLOW[J]. 数学物理学报(英文版), 2018, 38(5): 1485-1514.
[5]	陈宇, 周忆. SIMPLE WAVES OF THE TWO DIMENSIONAL COMPRESSIBLE FULL EULER EQUATIONS[J]. 数学物理学报(英文版), 2015, 35(4): 855-875.
[6]	唐平凡\|方北香\|王亚光. ON LOCAL STRUCTURAL STABILITY OF ONE-DIMENSIONAL SHOCKS IN RADIATION HYDRODYNAMICS[J]. 数学物理学报(英文版), 2015, 35(1): 1-44.
[7]	陈贵强, 肖长国, 张永前. EXISTENCE OF ENTROPY SOLUTIONS TO TWO-DIMENSIONAL STEADY EXOTHERMICALLY REACTING EULER EQUATIONS[J]. 数学物理学报(英文版), 2014, 34(1): 1-38.
[8]	王振, 张庆玲. THE RIEMANN PROBLEM WITH DELTA INITIAL DATA FOR THE ONE-DIMENSIONAL CHAPLYGIN GAS EQUATIONS[J]. 数学物理学报(英文版), 2012, 32(3): 825-841.
[9]	刘小民, 王振. THE RIEMANN PROBLEM FOR THE NONLINEAR DEGENERATE WAVE EQUATIONS[J]. 数学物理学报(英文版), 2011, 31(6): 2313-2322.
[10]	陈恕行. A NONLINEAR LAVRENTIEV-BITSADZE MIXED TYPE EQUATION[J]. 数学物理学报(英文版), 2011, 31(6): 2378-2388.
[11]	孙梅娜. A NOTE ON THE INTERACTIONS OF ELEMENTARY WAVES FOR THE AR TRAFFIC FLOW MODEL WITHOUT VACUUM[J]. 数学物理学报(英文版), 2011, 31(4): 1503-1512.
[12]	Gui-Qiang G. Chen, Weihua Ruan. A HYPERBOLIC SYSTEM OF CONSERVATION LAWS FOR FLUID FLOWS THROUGH COMPLIANT XISYMMETRIC VESSELS[J]. 数学物理学报(英文版), 2010, 30(2): 391-427.
[13]	王东红, 赵宁, 胡偶, 刘剑明. A GHOST FLUID BASED FRONT TRACKING METHOD FOR MULTIMEDIUM COMPRESSIBLE FLOWS[J]. 数学物理学报(英文版), 2009, 29(6): 1629-1646.
[14]	李杰权, 盛万成, 张同, 郑玉玺. TWO-DIMENSIONAL RIEMANN PROBLEMS: FROM SCALAR CONSERVATION LAWS TO COMPRESSIBLE EULER EQUATIONS[J]. 数学物理学报(英文版), 2009, 29(4): 777-802.
[15]	杨小舟. Multi-dimensional Riemann problem of scalar conservation law[J]. 数学物理学报(英文版), 1999, 19(2): 190-200.