Abstract
A novel method is presented for explicitly solving inhomogeneous initial-boundary-value problems (IBVPs) on the half-line for a well-known coupled system of evolution partial differential equations. The so-called double-diffusion model, which is based on a simple, yet general, inhomogeneous diffusion configuration, describes accurately several important physical and mechanical processes and thus emerges in miscellaneous applications, ranging from materials science, heat-mass transport and solid–fluid dynamics, to petroleum and chemical engineering. For instance, it appears in nanotechnology and its inhomogeneous version has recently appeared in the area of lithium-ion rechargeable batteries. Our approach is based on the extension of the unified transform (also called the Fokas method), so that it can be applied to systems of coupled equations. First, we derive formally effective solution representations and then justify a posteriori their validity rigorously. This includes the reconstruction of the prescribed initial and boundary conditions, which requires careful analysis of the various integral terms appearing in the formulae, proving that they converge in a strictly defined sense. The novel solution formulae are also utilized to rigorously deduce the solution’s regularity properties near the boundaries of the spatiotemporal domain. In particular, we prove uniform convergence of the solution to the data, its rapid decay at infinity as well as its smoothness up to (and beyond) the boundary axes, provided certain data compatibility conditions at the quarter-plane corner are satisfied. As a sample of important applications of our analysis and investigation of the boundary behavior of the solution and its derivatives, we both prove a novel uniqueness theorem and construct a ‘non-uniqueness counterexample’. These supplement the preceding ‘constructive existence’ result, within the framework of well-posedness. Moreover, one of the advantages of the unified transform is that it yields representations which are defined on contours in the complex Fourier \(\lambda \)-plane, which exhibit exponential decay for large values of \(\lambda \). This important characteristic of the solutions is expected to allow for an efficient numerical evaluation; this is envisaged in future numerical-analytic investigations. The new formulae and the findings reported herein are also expected to find utility in the study of questions pertaining to well-posedness for nonlinear counterparts too. In addition, our rigorous approach can be extended to IBVPs for other significant models of mathematical physics and potentially also to higher-dimensional and variable-coefficient cases.
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References
Konstantinidis, D.A., Eleftheriadis, I.E., Aifantis, E.C.: On the experimental validation of the double diffusivity model. Scr. Mater. 38, 573–580 (1998)
Konstantinidis, D.A., Eleftheriadis, I.E., Aifantis, E.C.: Application of double diffusivity model to superconductors. J. Mater. Process. Technol. 108, 185–187 (2001)
Konstantinidis, D.A., Aifantis, E.C.: Further experimental evidence of the double diffusivity model. Scr. Mater. 40, 1235–1241 (1999)
Aifantis, E.C., Hill, J.M.: On the theory of diffusion in media with double diffusivity I—basic mathematical results. Q. J. Mech. Appl. Math. 33, 1–21 (1980)
Hill, J.M., Aifantis, E.C.: On the theory of diffusion in media with double diffusivity II—boundary value problems. Q. J. Mech. Appl. Math. 33, 23–41 (1980)
Aifantis, E.C.: A new interpretation of diffusion in high diffusivity paths—a continuum approach. Acta Metall. 27, 683–691 (1979)
Heaviside, O.: On the extra current. Philos. Mag. Ser. 5, 135 (1876)
Barenblatt, G.I., Zheltov, Y.P., Kochina, I.N.: Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks. J. Appl. Math. Mech. 24, 1286–1303 (1960)
Cahn, J.W., Hilliard, J.E.: Free energy of a nonuniform system I. Interfacial free energy. J. Chem. Phys. 28, 258–267 (1958)
Rubenstein, L.I.: On the problem of the process of propagation of heat in heterogeneousmedia. Izv. Akad. Nauk SSSR Ser. Geogr. 1, 12–45 (1948)
Ter Haar, D. (ed.): Collected papers of L.D. Landau. Pergamon, London (1965)
Lappas, K.I.T., Konstantinidis, A.A., Aifantis, E.C.: Modelling triple diffusion of 63Ni in UFG Cu–Zr ingots. Scr. Mater. 201, 113980 (2021)
Aifantis, K.E., Hackney, S.A., Kumar, V.R. (eds.): High Energy Density Lithium Batteries: Materials, Engineering, Applications. Wiley, Weinheim (2010). (Translated in Chinese: China Machine Press, ISBN: 9787111371786 (2011))
Aifantis, E.C.: Internal length gradient (ILG) material mechanics across scales & disciplines. Adv. Appl. Mech. 49, 1–110 (2016)
Aris, R.: On the permeability of membranes with parallel, but interconnected, pathways. Math. Biosci. 77, 5 (1985)
Fokas, A.S.: Lax Pairs: a novel type of separability (invited paper for the special volume of the 25th anniversary of Inverse Problems). Inverse Probl. 25, 1–44 (2009)
Fokas, A.S.: A unified transform method for solving linear and certain nonlinear PDEs. Proc. R. Soc. Lond. Ser. A 453, 1411–1443 (1997)
Fokas, A.S.: On the integrability of linear and nonlinear PDEs. J. Math. Phys. 41, 4188–4237 (2000)
Fokas, A.S., Kaxiras, E.: Modern Mathematical Methods for Computational Sciences and Engineering. World Scientific, Singapore (2022)
Fokas, A.S., Pelloni, B.: The solution of certain IBV problems for the linearized KdV equation. Proc. R. Soc. Lond. A 454, 645–657 (1998)
Fokas, A.S., Pelloni, B.: A method of solving moving boundary value problems for linear evolution equations. Phys. Rev. Lett. 84, 4785–4789 (2000)
Fokas, A.S.: A new transform method for evolution PDEs. IMA J. Appl. Math. 67, 1–32 (2002)
Fokas, A.S., Papageorgiou, D.T.: Absolute and convective instability for evolution PDEs on the half-line. Stud. Appl. Math. 114, 95–114 (2005)
Fokas, A.S., Wang, Z.: Generalised Dirichlet to Neumann maps for linear dispersive equations on half-Line. Math. Proc. Camb. Philos. Soc. 164, 297–324 (2018)
Fokas, A.S., Pelloni, B.: Two-point boundary value problems for linear evolution equations. Math. Proc. Camb. Philos. Soc. 131, 521–543 (2001)
Fokas, A.S., Pelloni, B.: A transform method for evolution PDEs on the interval. IMA J. Appl. Math. 75, 564–587 (2005)
De Lillo, S., Fokas, A.S.: The Dirichlet to Neumann map for the heat equation on a moving boundary. Inverse Probl. 23, 1699–1710 (2007)
Fokas, A.S., Pelloni, B.: Generalized Dirichlet-to-Neumann map in time-dependent domains. Stud. Appl. Math. 129, 51–90 (2012)
Fernandez, A., Baleanu, D., Fokas, A.S.: Solving PDEs of fractional order using the Unified Transform Method. Appl. Math. Comput. 339, 738–749 (2018)
Mantzavinos, D., Fokas, A.S.: The Unified Method for the heat equation: I. Non-separable boundary conditions and non-local constraints in one dimension. Eur. J. Appl. Math. 24, 857–886 (2013)
Batal, A., Fokas, A.S., Ozsari, T.: Fokas method for linear boundary value problems involving mixed spatial derivatives. Proc. R. Soc. A 476, 20200076 (2020)
Miller, P.D., Smith, D.A.: The diffusion equation with nonlocal data. J. Math. Anal. Appl. 466, 1119–1143 (2018)
Deconinck, B., Pelloni, B., Sheils, N.E.: Non-steady-state heat conduction in composite walls. Proc. R. Soc. A: Math. Phys. Eng. Sci. 470, 20130605 (2014)
Sheils, N.E., Deconinck, B.: Interface problems for dispersive equations. Stud. Appl. Math. 134, 253–275 (2015)
Sheils, N.E., Deconinck, B.: Heat conduction on the ring: Interface problems with periodic boundary conditions. Appl. Math. Lett. 37, 107–111 (2014)
Sheils, N.E., Deconinck, B.: Initial-to-interface maps for the heat equation on composite domains. Stud. Appl. Math. 137, 140–154 (2016)
Sheils, N.E., Deconinck, B.: The time-dependent Schrödinger equation with piecewise constant potentials. Eur. J. Appl. Math. 31, 57–83 (2020)
Deconinck, B., Sheils, N.E., Smith, D.A.: The linear KdV equation with an interface. Commun. Math. Phys. 347, 489–509 (2016)
Bona, J.L., Fokas, A.S.: Initial-boundary-value problems for linear and integrable nonlinear dispersive partial differential equations. Nonlinearity 21, T195–T203 (2008)
Fokas, A.S., van der Weele, M.C.: The Unified Transform for time-periodic boundary conditions. Stud. Appl. Math. 147, 1339–1368 (2021)
Fokas, A.S., Pelloni, B., Smith, D.A.: Time-periodic linear boundary value problems on a finite interval. Q. Appl. Math. 80 (2022)
Fokas, A.S.: Boundary-value problems for linear PDEs with variable coefficients. Proc. R. Soc. Lond. A 460, 1131–1151 (2004)
Farkas, M., Deconinck, B.: Solving the heat equation with variable thermal conductivity. Appl. Math. Lett. 135, 108395 (2023)
Chatziafratis, A.: Rigorous Analysis of the Fokas Method for Linear Evolution PDEs on the Half-space, Thesis (in Greek), Advisors: N.D. Alikakos, G. Barbatis, I.G. Stratis, National and Kapodistrian University of Athens (2019). https://pergamos.lib.uoa.gr/uoa/dl/object/2877222
Chatziafratis, A., Kamvissis, S., Stratis, I.G.: Boundary behavior of the solution to the linear KdV equation on the half-line. Stud. Appl. Math. 150, 339–379 (2023)
Chatziafratis, A., Mantzavinos, D.: Boundary behavior for the heat equation on the half-line. Math. Methods Appl. Sci. 45, 7364–7393 (2022). https://doi.org/10.48550/arXiv.2401.08331
Fokas, A.S., Pelloni, B.: Boundary value problems for Boussinesq type systems. Math. Phys. Anal. Geom. 8, 59–96 (2005)
Himonas, A.A., Mantzavinos, D.: On the initial-boundary value problem for the linearized Boussinesq equation. Stud. Appl. Math. 134, 62–100 (2014)
Mantzavinos, D., Mitsotakis, D.: Extended water wave systems of Boussinesq equations on a finite interval: theory and numerical analysis. J. Math. Pures Appl. 169, 109–137 (2022)
Deconinck, B., Guo, Q., Shlizerman, E., Vasan, V.: Fokas’s unified transform method for linear systems. Q. Appl. Math. 76, 463–488 (2018)
Pelloni, B., Smith, D.A.: Spectral theory of some non-selfadjoint linear differential operators. Proc. R. Soc. A 469, 20130019 (2013)
Fokas, A.S., Smith, D.A.: Evolution PDEs and augmented eigenfunctions. Finite interval. Adv. Differ. Equ. 21, 735–766 (2016)
Pelloni, B., Smith, D.A.: Evolution PDEs and augmented eigenfunctions. Half line. J. Spectr. Theory 6, 185–213 (2016)
Smith, D.A.: The unified transform method for linear initial-boundary value problems: a spectral interpretation. In: Fokas, A.S., Pelloni, B. (eds.) Unified Transform Method for Boundary Value Problems: Applications and Advances. SIAM, Philadelphia (2015)
Aitzhan, S., Bhandari, S., Smith, D.A.: Fokas diagonalization of piecewise constant coefficient linear differential operators on finite intervals and networks. Acta Appl. Math. 177, 1–69 (2022)
Chatziafratis, A., Grafakos, L., Kamvissis, S.: Long-range instabilities for linear evolution PDE on semi-bounded domains via the Fokas method. Dyn PDE (to appear)
Chatziafratis, A., Grafakos, L., Kamvissis, S., Stratis, I.G.: Instabilities for linear evolution PDE on the half-line via the Fokas method. In: Bountis, T., et al. (eds.) Proceedings of Conference on “Dynamical Systems and Complexity” (Crete, 2022), in Vol. on “Chaos, Fractals and Complexity”. Springer, Berlin (2023). https://link.springer.com/chapter/10.1007/978-3-031-37404-3_20
Chatziafratis, A., Ozawa, T., Tian, S.-F.: Rigorous analysis of the unified transform method and long-range instability for the inhomogeneous time-dependent Schrödinger equation on the quarter-plane. Math. Ann. (2023). https://doi.org/10.1007/s00208-023-02698-4
Chatziafratis, A., Bona, J.L., Chen, H., Kamvissis, S.: New phenomena for the BBM equation in the quarter-plane. Preprint (2023)
Chatziafratis, A., Ozawa, T.: Asymptotic analysis for a Sobolev-type evolution equation in the quarter-plane. Preprint (2023)
Chatziafratis, A., Kamvissis, S.: A note on uniqueness for linear evolution PDEs posed on the quarter-plane. Preprint (2023) https://doi.org/10.48550/arXiv.2401.08531
Himonas, A.A., Yan, F.: The modified KdV system on the half-line. J. Dyn. Differ. Equ. 333, 55–102 (2023)
Alexandrou Himonas, A., Mantzavinos, D.: Well-posedness of the nonlinear Schrödinger equation on the half-plane. Nonlinearity 33, 5567–5609 (2020)
Alexandrou Himonas, A., Mantzavinos, D., Yan, F.: The Korteweg-de Vries equation on an interval. J. Math. Phys. 60, 1–26 (2019)
Alexandrou Himonas, A., Mantzavinos, D., Yan, F.: The nonlinear Schrödinger equation on the half-line with Neumann boundary conditions. Appl. Numer. Math. 141, 2–18 (2019)
Fokas, A.S., Alexandrou Himonas, A., Mantzavinos, D.: The nonlinear Schrödinger equation on the half-line. Trans. Am. Math. Soc. 369, 681–709 (2017)
Kalimeris, K., Özsarı, T.: An elementary proof of the lack of null controllability for the heat equation on the half line. Appl. Math. Lett. 104, 106241 (2020)
Kalimeris, K., Özsarı, T., Dikaios, N.: Numerical computation of Neumann controls for the heat equation on a finite interval. IEEE Trans. Autom. Control 69, 1–13 (2024)
Chatziafratis, A., Fokas, A.S., Aifantis, E.C.: On Barenblatt’s pseudoparabolic equation with forcing on the half-line via the Fokas method. Z Angew Math Mech (2024). https://doi.org/10.1002/zamm.202300614
Chatziafratis, A., Aifantis, E.C., Fokas, A.S., Miranville, A.: Variations on heat equation: fourth-order diffusion and Cahn–Hilliard models on the half-line revisited. Preprint (2023)
Flyer, N., Fokas, A.S.: A hybrid analytical-numerical method for solving evolution partial differential equations. I. The half-line. Proc. R. Soc. A. 464, 1823–1849 (2008)
Sifalakis, A.G., Fokas, A.S., Fulton, S.R., Saridakis, Y.G.: The generalized Dirichlet to Neumann map for linear elliptic PDEs and its numerical implementation. Comput. Appl. Math. 219, 9–34 (2008)
Fokas, A.S., Spence, E.A.: Novel analytical and numerical methods for elliptic boundary value problems. In: Engquist, B., Fokas, A., Hairer, E., Iserles, A. (eds.) Highly Oscillatory Problems, London Mathematical Society Lecture Note Series, vol. 366. Cambridge University Press, Cambridge (2009)
Smitheman, S.A., Spence, E.A., Fokas, A.S.: A spectral collocation method for the Laplace and modified Helmholtz equations in a convex polygon. IMA J. Numer. Anal. 30, 1184–1205 (2010)
Fornberg, B., Flyer, N.: A numerical implementation of Fokas boundary integral approach: Laplace’s equation on a polygonal domain. Proc. R. Soc. Lond. A 467, 2983–3003 (2011)
Hashemzadeh, P., Fokas, A.S., Smitheman, S.A.: A numerical technique for linear elliptic partial differential equations in polygonal domains. Proc. R. Soc. Lond. A 471, 1–13 (2015)
Ashton, A.C.L., Crooks, K.M.: Numerical analysis of Fokas’ unified method for linear elliptic PDEs. Appl. Numer. Math. 104, 120–132 (2016)
Kesici, E., Pelloni, B., Pryer, T., Smith, D.A.: A numerical implementation of the unified Fokas transform for evolution problems on a finite interval. Eur. J. Appl. Math. 29(3), 543–567 (2018)
de Barros, F.P.J., Colbrook, M.J., Fokas, A.S.: A hybrid analytical-numerical method for solving advection-dispersion problems on a half-line. Int. J. Heat Mass Trans. 139, 482–491 (2019)
Acknowledgements
Tony Carbery was partially supported by a Leverhulme Fellowship. Andreas Chatziafratis’ research was partially funded, at different stages of this project, through a Grant awarded by the Academy of Athens, a Scholarship from the Onassis Foundation and a Fellowship from the Leventis Foundation; he wishes to acknowledge the Riemann International School of Mathematics (RISM) as well as the Johann Radon Institute for Computational and Applied Mathematics (RICAM) of the Austrian Academy of Sciences, where parts of this work were completed, for travel grants and for kind hospitality, and gratefully also thanks Professors: N.D. Alikakos, C. Dafermos, B. Deconinck, T. Hatziafratis, A.A. Himonas, S. Kamvissis, A. Katsevich, T. Ozawa, I.G. Stratis and N. Stylianopoulos for providing inspiration, encouragement and academic support altogether. We thank the anonymous referees for useful suggestions and stimulating comments which led to an improved version of the paper.
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Leverhulme Trust (for A. Carbery) and Academy of Athens (for A. Chatziafratis).
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Chatziafratis, A., Aifantis, E.C., Carbery, A. et al. Integral representations for the double-diffusivity system on the half-line. Z. Angew. Math. Phys. 75, 54 (2024). https://doi.org/10.1007/s00033-023-02174-8
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DOI: https://doi.org/10.1007/s00033-023-02174-8
Keywords
- Double-diffusion model
- Fokas unified-transform method
- Nonhomogeneous initial-boundary-value problems on the quarter-plane
- Closed-form solutions
- Rigorous analysis
- Integral representations
- Well-posedness
- Cahn–Hilliard model
- Barenblatt’s pseudoparabolic equation
- Mechanics of solids
- Nanotechnology