The Existence of Fourier Coefficients and Periodic Multiplicity Based on Initial Values and One-Dimensional Wave Limits Requirements
DOI:
https://doi.org/10.26740/jpfa.v10n2.p146-157Keywords:
Partial Differential Equations, Fourier Analysis, Finite Difference, Wave EquationAbstract
Physical systems in partial differential equations can be interpreted in a visual form using a wave simulation. In particular, the interpretation of the differential equations used is in the nonlinear hyperbolic model, but in its completion, there are some limitations to the stability requirements found. The aim of this study is to investigate the analytical and numerical analysis of a wave equation with a similar unit and fractal intervals using the Fourier coefficient. The method in this research is to use the analytical solution approach, the spectral method, and the finite difference method. The hyperbolic wave equation's analytical solution approach, illustrated in the Fourier analysis, uses a pulse triangle. The spectral method minimizes errors when there is the addition of the same sample grid points or the periodic domain's expansion with a trigonometric basis. Meanwhile, different ways offer a more efficient solution. Based on the research results, the information obtained is that the Fourier analysis illustrates the pulse triangle use to solve the solution. These results are also suitable for adding sample points to the same spectra. Fourier analysis requires a relatively long time to solve one pulse triangle graph to need another solution, namely the finite difference method. However, its use is still limited in terms of stability when faced with more complex problems.References
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Zhang W and Jiang J. A New Family of Fourth-Order Locally One-Dimensional Schemes for The Three-Dimensional Wave Equation. Journal of Computational and Applied Mathematics. 2017; 311: 130-147. DOI: https://doi.org/10.1016/j.cam.2016.07.020.
Zhang W. A New Family of Fourth-Order Locally One-Dimensional Schemes for The 3D Elastic Wave Equation. Journal of Computational and Applied Mathematics. 2019; 348: 246-260. DOI: https://doi.org/10.1016/j.cam.2018.08.056.
Arioli G and Gazzola F. On a Nonlinear Nonlocal Hyperbolic System Modeling Suspension Bridges. Milan Journal of Mathematics. 2015; 83: 211-236. DOI: https://doi.org/10.1007/s00032-015-0239-9.
Bennour A, Khodja FA, and Teniou D. Exact and Approximate Controllability of Coupled One-Dimensional Hyperbolic Equations. Evolution Equations and Control Theory. 2017; 6(4): 487-516. DOI: https://doi.org/10.3934/eect.2017025.
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Korkmaz A. Explicit Exact Solutions to Some One-Dimensional Conformable Time Fractional Equations. Waves in Random and Complex Media. 2019; 29(1): 124-137. DOI: https://doi.org/10.1080/17455030.2017.1416702.
Xie J, Xu X, Li A, and Zhu Q. Experimental Validation of Frequency-Domain Finite-Difference Model of Active Pipe-Embedded Building Envelope in Time Domain by Using Fourier Series Analysis. Energy and Buildings. 2015; 99: 177-188. DOI: https://doi.org/10.1016/j.enbuild.2015.04.043.
Don WS, Gao Z, Li P, and Wen X. Hybrid Compact-WENO Finite Difference Scheme with Conjugate Fourier Shock Detection Algorithm for Hyperbolic Conservation Laws. SIAM Journal on Scientific Computing. 2016; 38(2): A691-A711. DOI: https://doi.org/10.1137/15M1021520.
Ji S. Periodic Solutions for One Dimensional Wave Equation with Bounded Nonlinearity. Journal of Differential Equations. 2018; 264(9): 5527-5540. DOI: https://doi.org/10.1016/j.jde.2018.02.001.
Ji S, Gao Y, and Zhu W. Existence and Multiplicity of Periodic Solutions for DirichletNeumann Boundary Value Problem of A Variable Coefficient Wave Equation. Advanced Nonlinear Studies. 2016; 16(4): 765-773. DOI: https://doi.org/10.1515/ans-2015-5058.
Chambers DH, Chandrasekaran H, and Walston SE. Fourier Method for Calculating Fission Chain Neutron Multiplicity Distributions. Nuclear Science and Engineering. 2016; 184(2): 244-253. DOI: https://doi.org/10.13182/NSE15-109.
Zhang TW and Han SF. Multiplicity of Almost Periodic Oscillations in Delayed Harvesting Predator-Prey Model with Modified Leslie-Gower Holling-Type II Schemes. ScienceAsia. 2019; 45(5): 494-501. DOI: https://doi.org/10.2306/scienceasia1513-1874.2019.45.494.
Ambrosio V, Bartolo R, and Bisci GM. A Multiplicity Result for A Non-Local Parametric Problem with Periodic Boundary Conditions. Arkiv för Matematik. 2020; 58(1): 1-18. DOI: https://dx.doi.org/10.4310/ARKIV.2020.v58.n1.a1.
Kuznetsov DF. Expansion of Iterated Stratonovich Stochastic Integrals of Multiplicity 3 Based on Generalized Multiple Fourier Series Converging in The Mean: General Case of Series Summation. Preprint arXiv:1801.01564.2018; 2018. Available from: https://arxiv.org/abs/1801.01564.
Blömker D, Wacker P, and Wanner T. Probabilistic Estimates of the Maximum Norm of Random Neumann Fourier Series. Communications in Nonlinear Science and Numerical Simulation. 2017; 47: 348-369. DOI: https://doi.org/10.1016/j.cnsns.2016.11.023.
Gourevitch D, Gustafsson H, Kleinschmidt A, Persson D, and Sahi S. Fourier Coefficients and Small Automorphic Representations; 2018. Available from: http://hdl.handle.net/21.11116/0000-0002-82B6-D.
Liu ZJ, Adamu MY, Enoch S, and Ji-Huan H. Hybridization of Homotopy Perturbation Method and Laplace Transformation for the Partial Differential Equations. Thermal Science. 2017; 21(4): 1843-1846. DOI: https://doi.org/10.2298/TSCI160715078L.
Andrews U, Bonik G, Chen JP, Martin RW, and Teplyaev A. Wave Equation on One-Dimensional Fractals with Spectral Decimation and the Complex Dynamics of Polynomials. Journal of Fourier Analysis and Applications. 2017; 23: 994-1027. DOI: https://doi.org/10.1007/s00041-016-9494-6.
Dobrokhotov SY and Nazaikinskii VE. Characteristics with Singularities and the Boundary Values of the Asymptotic Solution of the Cauchy Problem for A Degenerate Wave Equation. Mathematical Notes. 2016; 100: 695-713. DOI: https://doi.org/10.1134/S0001434616110067.
Bilbao S and Hamilton B. Higher-Order Accurate Two-Step Finite Difference Schemes for the Many-Dimensional Wave Equation. Journal of Computational Physics. 2018; 367: 134-165. DOI: https://doi.org/10.1016/j.jcp.2018.04.012.
Murty AVSN, Srinivas MN, and Sastry DRVSRK. New Scheme for One Dimensional Wave Equation. International Journal of Civil Engineering and Technology (IJCIET). 2017; 8(8): 1028-1031. Available from: http://www.iaeme.com/ijciet/IJCIET_Paper.asp?sno=8807.
Szyszka B. A Nine-Point Finite Difference Scheme for One-Dimensional Wave Equation. AIP Conference Proceedings. 2017; 1863(1): 560078. DOI: https://doi.org/10.1063/1.4992761.
Agom EU, Ogunfiditimi FO, and Assi PN. Numerical Application of Adomian Decomposition Method to One Dimensional Wave Equations. Journal of Mathematical and Computational Science. 2017; 7(3): 554-563. Available from: http://scik.org/index.php/jmcs/article/view/3036.
Guo W and Guo BZ. Performance Output Tracking for A Wave Equation Subject to Unmatched General Boundary Harmonic Disturbance. Automatica. 2016; 68: 194-202. DOI: https://doi.org/10.1016/j.automatica.2016.01.041.
Guo W, Zhou HC, and Krstic M. Adaptive Error Feedback Regulation Problem for 1D Wave Equation. International Journal of Robust and Nonlinear Control. 2018; 28(15): 4309-4329. DOI: https://doi.org/10.1002/rnc.4234.
Leukauf A, Schirrer A, and Talic E. Fourier Galerkin Approach to Wave Equation with Absorbing Boundary Conditions. World Academy of Science, Engineering and Technology. 2017; 11(8): 380-385. DOI: https://doi.org/10.5281/zenodo.1131844.
Lissy P and Rovenţa I. Optimal Filtration for the Approximation of Boundary Controls for the One-Dimensional Wave Equation Using A Finite-Difference Method. Mathematics of Computation. 2019; 88: 273-291. DOI: https://doi.org/10.1090/mcom/3345.
An Y. Uniform Dispersion Reduction Schemes for the One-Dimensional Wave Equation in Isotropic Media. Journal of Computational Physics. 2017; 341: 13-21. DOI: https://doi.org/10.1016/j.jcp.2017.04.015.
Gerdt VP, Robertz D, and Blinkov YA. Strong Consistency and Thomas Decomposition of Finite Difference Approximations to Systems of Partial Differential Equations. Preprint arXiv:2009.01731.2020; 2020. Available from: https://arxiv.org/abs/2009.01731.
Taghizadeh N and Noori SRM. Reduced Differential Transform Method for Solving Parabolic-Like and Hyperbolic-Like Equations. SeMA Journal. 2017; 74(4): 559-567. DOI: https://doi.org/10.1007/s40324-016-0101-1.
Jufriansah A, Pramudya Y, Hermanto A, and Khusnani A. Surface Wave Topography using the 4 Point FDM Simulator. Science and Technology Indonesia. 2020; 5(4): 117-120. DOI: https://doi.org/10.26554/sti.2020.5.4.117-120.
Peng B, Luo CH, Sinha N, Tai CC, Xie X, and Xie H. Fourier Series Analysis for Novel Spatiotemporal Pulse Waves: Normal, Taut, and Slippery Pulse Images. Evidence-Based Complementary and Alternative Medicine. 2019; 2019: 5734018. DOI: https://doi.org/10.1155/2019/5734018.
Berchio E, Ferrero A, and Gazzola F. Structural Instability of Nonlinear Plates Modelling Suspension Bridges: Mathematical Answers to Some Long-Standing Questions. Nonlinear Analysis: Real World Applications. 2016; 28: 91-125. DOI: https://doi.org/10.1016/j.nonrwa.2015.09.005.
Ming CY. Solution of Differential Equations with Applications to Engineering Problems. Dynamical Systems - Analytical and Computational Techniques; 2017: 233-264. DOI: http://dx.doi.org/10.5772/67539.
Zhukovsky KV. A Method of Inverse Differential Operators Using Orthogonal Polynomials and Special Functions for Solving Some Types of Differential Equations and Physical Problems. Moscow University Physics Bulletin. 2015; 70: 93-100. DOI: https://doi.org/10.3103/S0027134915020137.
Feng L, Liu F, Turnerra I, Yang Q, and Zhuang P. Unstructured Mesh Finite Difference/Finite Element Method for The 2D Time-Space Riesz Fractial Diffusion Equation on Irregular Convex Domains Applied Mathematical Modelling. 2018; 59: 441-463. DOI: https://doi.org/10.1016/j.apm.2018.01.044.
Jufriansah A, Hermanto A, Toifur M, and Prasetyo E. Investigation of The Physics Phenomena of Weakly Damped Wave Equations with Forced Force: Theory and Simulation. Journal of Physics: Conference Series. 2020; 1567: 022012. DOI: https://doi.org/10.1088/1742-6596/1567/2/022012.
Zhang W and Jiang J. A New Family of Fourth-Order Locally One-Dimensional Schemes for The Three-Dimensional Wave Equation. Journal of Computational and Applied Mathematics. 2017; 311: 130-147. DOI: https://doi.org/10.1016/j.cam.2016.07.020.
Zhang W. A New Family of Fourth-Order Locally One-Dimensional Schemes for The 3D Elastic Wave Equation. Journal of Computational and Applied Mathematics. 2019; 348: 246-260. DOI: https://doi.org/10.1016/j.cam.2018.08.056.
Arioli G and Gazzola F. On a Nonlinear Nonlocal Hyperbolic System Modeling Suspension Bridges. Milan Journal of Mathematics. 2015; 83: 211-236. DOI: https://doi.org/10.1007/s00032-015-0239-9.
Bennour A, Khodja FA, and Teniou D. Exact and Approximate Controllability of Coupled One-Dimensional Hyperbolic Equations. Evolution Equations and Control Theory. 2017; 6(4): 487-516. DOI: https://doi.org/10.3934/eect.2017025.
Hui Y, Kang HJ, Law SS, and Hua XG. Effect of Cut-Off Order of Nonlinear Stiffness on The Dynamics of A Sectional Suspension Bridge Model. Engineering Structures. 2019; 185: 377-391. DOI: https://doi.org/10.1016/j.engstruct.2019.01.129.
Fan N, Zhao LF, Xie XB, and Yao ZX. A Discontinuous-Grid Finite-Difference Scheme for Frequency-Domain 2D Scalar Wave Modeling. Geophysics. 2018; 83(4): T235-T244. DOI: https://doi.org/10.1190/geo2017-0535.1.
Strichartz RS. A Guide to Distribution Theory and Fourier Transforms. USA: World Scientific; 2003.
Strichartz RS. Differential Equations on Fractals: A Tutorial. New Jersey: Princeton University Press; 2006.
Shen J, Tang T, and Wang LL. Spectral Methods: Algorithms, Analysis and Applications. Springer Series in Computational Mathematics, Vol. 41. New York: Springer; 2011. DOI: http://dx.doi.org/10.1007/978-3-540-71041-7.
Korkmaz A. Explicit Exact Solutions to Some One-Dimensional Conformable Time Fractional Equations. Waves in Random and Complex Media. 2019; 29(1): 124-137. DOI: https://doi.org/10.1080/17455030.2017.1416702.
Xie J, Xu X, Li A, and Zhu Q. Experimental Validation of Frequency-Domain Finite-Difference Model of Active Pipe-Embedded Building Envelope in Time Domain by Using Fourier Series Analysis. Energy and Buildings. 2015; 99: 177-188. DOI: https://doi.org/10.1016/j.enbuild.2015.04.043.
Don WS, Gao Z, Li P, and Wen X. Hybrid Compact-WENO Finite Difference Scheme with Conjugate Fourier Shock Detection Algorithm for Hyperbolic Conservation Laws. SIAM Journal on Scientific Computing. 2016; 38(2): A691-A711. DOI: https://doi.org/10.1137/15M1021520.
Ji S. Periodic Solutions for One Dimensional Wave Equation with Bounded Nonlinearity. Journal of Differential Equations. 2018; 264(9): 5527-5540. DOI: https://doi.org/10.1016/j.jde.2018.02.001.
Ji S, Gao Y, and Zhu W. Existence and Multiplicity of Periodic Solutions for DirichletNeumann Boundary Value Problem of A Variable Coefficient Wave Equation. Advanced Nonlinear Studies. 2016; 16(4): 765-773. DOI: https://doi.org/10.1515/ans-2015-5058.
Chambers DH, Chandrasekaran H, and Walston SE. Fourier Method for Calculating Fission Chain Neutron Multiplicity Distributions. Nuclear Science and Engineering. 2016; 184(2): 244-253. DOI: https://doi.org/10.13182/NSE15-109.
Zhang TW and Han SF. Multiplicity of Almost Periodic Oscillations in Delayed Harvesting Predator-Prey Model with Modified Leslie-Gower Holling-Type II Schemes. ScienceAsia. 2019; 45(5): 494-501. DOI: https://doi.org/10.2306/scienceasia1513-1874.2019.45.494.
Ambrosio V, Bartolo R, and Bisci GM. A Multiplicity Result for A Non-Local Parametric Problem with Periodic Boundary Conditions. Arkiv för Matematik. 2020; 58(1): 1-18. DOI: https://dx.doi.org/10.4310/ARKIV.2020.v58.n1.a1.
Kuznetsov DF. Expansion of Iterated Stratonovich Stochastic Integrals of Multiplicity 3 Based on Generalized Multiple Fourier Series Converging in The Mean: General Case of Series Summation. Preprint arXiv:1801.01564.2018; 2018. Available from: https://arxiv.org/abs/1801.01564.
Blömker D, Wacker P, and Wanner T. Probabilistic Estimates of the Maximum Norm of Random Neumann Fourier Series. Communications in Nonlinear Science and Numerical Simulation. 2017; 47: 348-369. DOI: https://doi.org/10.1016/j.cnsns.2016.11.023.
Gourevitch D, Gustafsson H, Kleinschmidt A, Persson D, and Sahi S. Fourier Coefficients and Small Automorphic Representations; 2018. Available from: http://hdl.handle.net/21.11116/0000-0002-82B6-D.
Liu ZJ, Adamu MY, Enoch S, and Ji-Huan H. Hybridization of Homotopy Perturbation Method and Laplace Transformation for the Partial Differential Equations. Thermal Science. 2017; 21(4): 1843-1846. DOI: https://doi.org/10.2298/TSCI160715078L.
Andrews U, Bonik G, Chen JP, Martin RW, and Teplyaev A. Wave Equation on One-Dimensional Fractals with Spectral Decimation and the Complex Dynamics of Polynomials. Journal of Fourier Analysis and Applications. 2017; 23: 994-1027. DOI: https://doi.org/10.1007/s00041-016-9494-6.
Dobrokhotov SY and Nazaikinskii VE. Characteristics with Singularities and the Boundary Values of the Asymptotic Solution of the Cauchy Problem for A Degenerate Wave Equation. Mathematical Notes. 2016; 100: 695-713. DOI: https://doi.org/10.1134/S0001434616110067.
Bilbao S and Hamilton B. Higher-Order Accurate Two-Step Finite Difference Schemes for the Many-Dimensional Wave Equation. Journal of Computational Physics. 2018; 367: 134-165. DOI: https://doi.org/10.1016/j.jcp.2018.04.012.
Murty AVSN, Srinivas MN, and Sastry DRVSRK. New Scheme for One Dimensional Wave Equation. International Journal of Civil Engineering and Technology (IJCIET). 2017; 8(8): 1028-1031. Available from: http://www.iaeme.com/ijciet/IJCIET_Paper.asp?sno=8807.
Szyszka B. A Nine-Point Finite Difference Scheme for One-Dimensional Wave Equation. AIP Conference Proceedings. 2017; 1863(1): 560078. DOI: https://doi.org/10.1063/1.4992761.
Agom EU, Ogunfiditimi FO, and Assi PN. Numerical Application of Adomian Decomposition Method to One Dimensional Wave Equations. Journal of Mathematical and Computational Science. 2017; 7(3): 554-563. Available from: http://scik.org/index.php/jmcs/article/view/3036.
Guo W and Guo BZ. Performance Output Tracking for A Wave Equation Subject to Unmatched General Boundary Harmonic Disturbance. Automatica. 2016; 68: 194-202. DOI: https://doi.org/10.1016/j.automatica.2016.01.041.
Guo W, Zhou HC, and Krstic M. Adaptive Error Feedback Regulation Problem for 1D Wave Equation. International Journal of Robust and Nonlinear Control. 2018; 28(15): 4309-4329. DOI: https://doi.org/10.1002/rnc.4234.
Leukauf A, Schirrer A, and Talic E. Fourier Galerkin Approach to Wave Equation with Absorbing Boundary Conditions. World Academy of Science, Engineering and Technology. 2017; 11(8): 380-385. DOI: https://doi.org/10.5281/zenodo.1131844.
Lissy P and Rovenţa I. Optimal Filtration for the Approximation of Boundary Controls for the One-Dimensional Wave Equation Using A Finite-Difference Method. Mathematics of Computation. 2019; 88: 273-291. DOI: https://doi.org/10.1090/mcom/3345.
An Y. Uniform Dispersion Reduction Schemes for the One-Dimensional Wave Equation in Isotropic Media. Journal of Computational Physics. 2017; 341: 13-21. DOI: https://doi.org/10.1016/j.jcp.2017.04.015.
Gerdt VP, Robertz D, and Blinkov YA. Strong Consistency and Thomas Decomposition of Finite Difference Approximations to Systems of Partial Differential Equations. Preprint arXiv:2009.01731.2020; 2020. Available from: https://arxiv.org/abs/2009.01731.
Taghizadeh N and Noori SRM. Reduced Differential Transform Method for Solving Parabolic-Like and Hyperbolic-Like Equations. SeMA Journal. 2017; 74(4): 559-567. DOI: https://doi.org/10.1007/s40324-016-0101-1.
Jufriansah A, Pramudya Y, Hermanto A, and Khusnani A. Surface Wave Topography using the 4 Point FDM Simulator. Science and Technology Indonesia. 2020; 5(4): 117-120. DOI: https://doi.org/10.26554/sti.2020.5.4.117-120.
Peng B, Luo CH, Sinha N, Tai CC, Xie X, and Xie H. Fourier Series Analysis for Novel Spatiotemporal Pulse Waves: Normal, Taut, and Slippery Pulse Images. Evidence-Based Complementary and Alternative Medicine. 2019; 2019: 5734018. DOI: https://doi.org/10.1155/2019/5734018.
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2020-12-31
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Jufriansah, A., Khusnani, A., Hermanto, A., Toifur, M. and Prasetyo, E. (2020) “The Existence of Fourier Coefficients and Periodic Multiplicity Based on Initial Values and One-Dimensional Wave Limits Requirements”, Jurnal Penelitian Fisika dan Aplikasinya (JPFA), 10(2), pp. 146–157. doi: 10.26740/jpfa.v10n2.p146-157.
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