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January  2021, 26(1): 603-632. doi: 10.3934/dcdsb.2020260

Equilibrium validation in models for pattern formation based on Sobolev embeddings

Evelyn Sander and  Thomas Wanner ,

Department of Mathematical Sciences, George Mason University, Fairfax, VA 22030, USA

* Corresponding author: Thomas Wanner

Received  March 2020 Revised  August 2020 Published  January 2021 Early access  August 2020

In the study of equilibrium solutions for partial differential equations there are so many equilibria that one cannot hope to find them all. Therefore one usually concentrates on finding individual branches of equilibrium solutions. On the one hand, a rigorous theoretical understanding of these branches is ideal but not generally tractable. On the other hand, numerical bifurcation searches are useful but not guaranteed to give an accurate structure, in that they could miss a portion of a branch or find a spurious branch where none exists. In a series of recent papers, we have aimed for a third option. Namely, we have developed a method of computer-assisted proofs to prove both existence and isolation of branches of equilibrium solutions. In the current paper, we extend these techniques to the Ohta-Kawasaki model for the dynamics of diblock copolymers in dimensions one, two, and three, by giving a detailed description of the analytical underpinnings of the method. Although the paper concentrates on applying the method to the Ohta-Kawasaki model, the functional analytic approach and techniques can be generalized to other parabolic partial differential equations.

Keywords: Parabolic partial differential equation, pattern formation in equilibria, bifurcation diagram, saddle-node bifurcations, computer-assisted proof, constructive implicit function theorem, Ohta-Kawasaki model.
Mathematics Subject Classification: Primary: 35B40, 35B41, 35K55, 37M20, 65G20; Secondary: 65G30, 65N35, 74N99.
Citation: Evelyn Sander, Thomas Wanner. Equilibrium validation in models for pattern formation based on Sobolev embeddings. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 603-632. doi: 10.3934/dcdsb.2020260
References:
[1] R. A. Adams and J. J. F. Fournier, Sobolev Spaces, 2nd edition, Elsevier/Academic Press, Amsterdam, 2003.   Google Scholar
[2]

G. Arioli and H. Koch, Computer-assisted methods for the study of stationary solutions in dissipative systems, applied to the Kuramoto-Sivashinski equation, Archive for Rational Mechanics and Analysis, 197 (2010), 1033-1051.  doi: 10.1007/s00205-010-0309-7.  Google Scholar

[3]

S. Cai and Y. Watanabe, A computer-assisted method for the diblock copolymer model, Zeitschrift für Angewandte Mathematik und Mechanik, 99 (2019), e201800125, 14pp. doi: 10.1002/zamm.201800125.  Google Scholar

[4]

L. Chierchia, KAM lectures, in Dynamical Systems. Part I, Scuola Normale Superiore, Pisa, Italy, 2003, 1–55. Google Scholar

[5]

R. Choksi, M. Maras and J. F. Williams, 2D phase diagram for minimizers of a Cahn-Hilliard functional with long-range interactions, SIAM Journal on Applied Dynamical Systems, 10 (2011), 1344-1362.  doi: 10.1137/100784497.  Google Scholar

[6]

R. Choksi, M. A. Peletier and J. F. Williams, On the phase diagram for microphase separation of diblock copolymers: An approach via a nonlocal Cahn-Hilliard functional, SIAM Journal on Applied Mathematics, 69 (2009), 1712-1738.  doi: 10.1137/080728809.  Google Scholar

[7]

R. Choksi and X. Ren, On the derivation of a density functional theory for microphase separation of diblock copolymers, Journal of Statistical Physics, 113 (2003), 151-176.  doi: 10.1023/A:1025722804873.  Google Scholar

[8]

R. Choksi and X. Ren, Diblock copolymer/homopolymer blends: Derivation of a density functional theory, Physica D, 203 (2005), 100-119.  doi: 10.1016/j.physd.2005.03.006.  Google Scholar

[9]

J. Cyranka and T. Wanner, Computer-assisted proof of heteroclinic connections in the one-dimensional Ohta-Kawasaki model, SIAM Journal on Applied Dynamical Systems, 17 (2018), 694-731.  doi: 10.1137/17M111938X.  Google Scholar

[10]

S. Day, J.-P. Lessard and K. Mischaikow, Validated continuation for equilibria of PDEs, SIAM Journal on Numerical Analysis, 45 (2007), 1398-1424.  doi: 10.1137/050645968.  Google Scholar

[11]

J. P. Desi, H. Edrees, J. Price, E. Sander and T. Wanner, The dynamics of nucleation in stochastic Cahn-Morral systems, SIAM Journal on Applied Dynamical Systems, 10 (2011), 707-743.  doi: 10.1137/100801378.  Google Scholar

[12]

A. Dhooge, W. Govaerts and Y. A. Kuznetsov, MATCONT: A MATLAB package for numerical bifurcation analysis of ODEs, Association for Computing Machinery. Transactions on Mathematical Software, 29 (2003), 141-164.  doi: 10.1145/779359.779362.  Google Scholar

[13]

E. Doedel, AUTO: A program for the automatic bifurcation analysis of autonomous systems, in Proceedings of the Tenth Manitoba Conference on Numerical Mathematics and Computing, Vol. I (Winnipeg, Man., 1980), 30 (1981), 265–284.  Google Scholar

[14]

M. Gameiro, J.-P. Lessard and K. Mischaikow, Validated continuation over large parameter ranges for equilibria of PDEs, Mathematics and Computers in Simulation, 79 (2008), 1368-1382.  doi: 10.1016/j.matcom.2008.03.014.  Google Scholar

[15]

Z. G. Huseynov and A. M. Shykhammedov, On bases of sines and cosines in Sobolev spaces, Applied Mathematics Letters, 25 (2012), 275-278.  doi: 10.1016/j.aml.2011.08.026.  Google Scholar

[16]

I. Johnson, E. Sander and T. Wanner, Branch interactions and long-term dynamics for the diblock copolymer model in one dimension, Discrete and Continuous Dynamical Systems. Series A, 33 (2013), 3671-3705.  doi: 10.3934/dcds.2013.33.3671.  Google Scholar

[17]

T. Kinoshita, Y. Watanabe and M. T. Nakao, An alternative approach to norm bound computation for inverses of linear operators in Hilbert spaces, Journal of Differential Equations, 266 (2019), 5431-5447.  doi: 10.1016/j.jde.2018.10.027.  Google Scholar

[18]

J.-P. Lessard, E. Sander and T. Wanner, Rigorous continuation of bifurcation points in the diblock copolymer equation, Journal of Computational Dynamics, 4 (2017), 71-118.  doi: 10.3934/jcd.2017003.  Google Scholar

[19]

S. Maier-Paape, U. Miller, K. Mischaikow and T. Wanner, Rigorous numerics for the Cahn-Hilliard equation on the unit square, Revista Matematica Complutense, 21 (2008), 351-426.  doi: 10.5209/rev_REMA.2008.v21.n2.16380.  Google Scholar

[20]

S. Maier-Paape, K. Mischaikow and T. Wanner, Structure of the attractor of the Cahn-Hilliard equation on a square, International Journal of Bifurcation and Chaos, 17 (2007), 1221-1263.  doi: 10.1142/S0218127407017781.  Google Scholar

[21]

S. Maier-Paape and T. Wanner, Spinodal decomposition for the Cahn-Hilliard equation in higher dimensions: Nonlinear dynamics, Archive for Rational Mechanics and Analysis, 151 (2000), 187-219.  doi: 10.1007/s002050050196.  Google Scholar

[22]

T. R. Muradov and V. F. Salmanov, On the basis property of trigonometric systems with linear phase in a weighted Sobolev space, Dokl. Math., 90 (2014), 611-612.  doi: 10.1134/s1064562414060301.  Google Scholar

[23]

M. T. Nakao, M. Plum and Y. Watanabe, Numerical Verification Methods and Computer-Assisted Proofs for Partial Differential Equations, Springer-Verlag, Berlin, 2019. doi: 10.1007/978-981-13-7669-6.  Google Scholar

[24]

T. Ohta and K. Kawasaki, Equilibrium morphology of block copolymer melts, Macromolecules, 19 (1986), 2621-2632.  doi: 10.1021/ma00164a028.  Google Scholar

[25]

M. Plum, Existence and enclosure results for continua of solutions of parameter-dependent nonlinear boundary value problems, Journal of Computational and Applied Mathematics, 60 (1995), 187-200.  doi: 10.1016/0377-0427(94)00091-E.  Google Scholar

[26]

M. Plum, Enclosures for two-point boundary value problems near bifurcation points, in Scientific Computing and Validated Numerics (Wuppertal, 1995), vol. 90 of Mathematical Research, Akademie Verlag, Berlin, 1996,265–279.  Google Scholar

[27]

M. Plum, Computer-assisted proofs for semilinear elliptic boundary value problems, Japan Journal of Industrial and Applied Mathematics, 26 (2009), 419-442.  doi: 10.1007/BF03186542.  Google Scholar

[28]

S. M. Rump, INTLAB - INTerval LABoratory, in Developments in Reliable Computing (ed. T. Csendes), Kluwer Academic Publishers, Dordrecht, 1999, 77–104, http://www.ti3.tuhh.de/rump/. Google Scholar

[29]

S. M. Rump, Verification methods: Rigorous results using floating-point arithmetic, Acta Numerica, 19 (2010), 287-449.  doi: 10.1017/S096249291000005X.  Google Scholar

[30]

E. Sander and T. Wanner, Validated saddle-node bifurcations and applications to lattice dynamical systems, SIAM Journal on Applied Dynamical Systems, 15 (2016), 1690-1733.  doi: 10.1137/16M1061011.  Google Scholar

[31] L. N. Trefethen and M. Embree, Spectra and Pseudospectra, Princeton University Press, Princeton, NJ, 2005.   Google Scholar
[32]

J. B. van den Berg and J. F. Williams, Validation of the bifurcation diagram in the 2D Ohta-Kawasaki problem, Nonlinearity, 30 (2017), 1584-1638.  doi: 10.1088/1361-6544/aa60e8.  Google Scholar

[33]

J. B. van den Berg and J. F. Williams, Optimal periodic structures with general space group symmetries in the Ohta-Kawasaki problem, arXiv: 1912.00059. Google Scholar

[34]

J. B. van den Berg and J. F. Williams, Rigorously computing symmetric stationary states of the Ohta-Kawasaki problem in three dimensions, SIAM Journal on Mathematical Analysis, 51 (2019), 131-158.  doi: 10.1137/17M1155624.  Google Scholar

[35]

T. Wanner, Topological analysis of the diblock copolymer equation, in Mathematical Challenges in a New Phase of Materials Science (eds. Y. Nishiura and M. Kotani), vol. 166 of Springer Proceedings in Mathematics & Statistics, Springer-Verlag, 2016, 27–51. doi: 10.1007/978-4-431-56104-0_2.  Google Scholar

[36]

T. Wanner, Computer-assisted equilibrium validation for the diblock copolymer model, Discrete and Continuous Dynamical Systems, Series A, 37 (2017), 1075-1107.  doi: 10.3934/dcds.2017045.  Google Scholar

[37]

T. Wanner, Computer-assisted bifurcation diagram validation and applications in materials science, Proceedings of Symposia in Applied Mathematics, 74 (2018), 123-174.   Google Scholar

[38]

T. Wanner, Validated bounds on embedding constants for Sobolev space Banach algebras, Mathematical Methods in the Applied Sciences, 41 (2018), 9361-9376.  doi: 10.1002/mma.5294.  Google Scholar

[39]

Y. Watanabe, T. Kinoshita and M. T. Nakao, An improved method for verifying the existence and bounds of the inverse of second-order linear elliptic operators mapping to dual space, Japan Journal of Industrial and Applied Mathematics, 36 (2019), 407-420.  doi: 10.1007/s13160-019-00344-8.  Google Scholar

[40]

Y. Watanabe, K. Nagatou, M. Plum and M. T. Nakao, Norm bound computation for inverses of linear operators in Hilbert spaces, Journal of Differential Equations, 260 (2016), 6363-6374.  doi: 10.1016/j.jde.2015.12.041.  Google Scholar

[41]

N. Yamamoto, A numerical verification method for solutions of boundary value problems with local uniqueness by Banach's fixed-point theorem, SIAM Journal on Numerical Analysis, 35 (1998), 2004-2013.  doi: 10.1137/S0036142996304498.  Google Scholar

[42]

N. Yamamoto, M. T. Nakao and Y. Watanabe, A theorem for numerical verification on local uniqueness of solutions to fixed-point equations, Numerical Functional Analysis and Optimization, 32 (2011), 1190-1204.  doi: 10.1080/01630563.2011.594348.  Google Scholar

show all references

References:
[1] R. A. Adams and J. J. F. Fournier, Sobolev Spaces, 2nd edition, Elsevier/Academic Press, Amsterdam, 2003.   Google Scholar
[2]

G. Arioli and H. Koch, Computer-assisted methods for the study of stationary solutions in dissipative systems, applied to the Kuramoto-Sivashinski equation, Archive for Rational Mechanics and Analysis, 197 (2010), 1033-1051.  doi: 10.1007/s00205-010-0309-7.  Google Scholar

[3]

S. Cai and Y. Watanabe, A computer-assisted method for the diblock copolymer model, Zeitschrift für Angewandte Mathematik und Mechanik, 99 (2019), e201800125, 14pp. doi: 10.1002/zamm.201800125.  Google Scholar

[4]

L. Chierchia, KAM lectures, in Dynamical Systems. Part I, Scuola Normale Superiore, Pisa, Italy, 2003, 1–55. Google Scholar

[5]

R. Choksi, M. Maras and J. F. Williams, 2D phase diagram for minimizers of a Cahn-Hilliard functional with long-range interactions, SIAM Journal on Applied Dynamical Systems, 10 (2011), 1344-1362.  doi: 10.1137/100784497.  Google Scholar

[6]

R. Choksi, M. A. Peletier and J. F. Williams, On the phase diagram for microphase separation of diblock copolymers: An approach via a nonlocal Cahn-Hilliard functional, SIAM Journal on Applied Mathematics, 69 (2009), 1712-1738.  doi: 10.1137/080728809.  Google Scholar

[7]

R. Choksi and X. Ren, On the derivation of a density functional theory for microphase separation of diblock copolymers, Journal of Statistical Physics, 113 (2003), 151-176.  doi: 10.1023/A:1025722804873.  Google Scholar

[8]

R. Choksi and X. Ren, Diblock copolymer/homopolymer blends: Derivation of a density functional theory, Physica D, 203 (2005), 100-119.  doi: 10.1016/j.physd.2005.03.006.  Google Scholar

[9]

J. Cyranka and T. Wanner, Computer-assisted proof of heteroclinic connections in the one-dimensional Ohta-Kawasaki model, SIAM Journal on Applied Dynamical Systems, 17 (2018), 694-731.  doi: 10.1137/17M111938X.  Google Scholar

[10]

S. Day, J.-P. Lessard and K. Mischaikow, Validated continuation for equilibria of PDEs, SIAM Journal on Numerical Analysis, 45 (2007), 1398-1424.  doi: 10.1137/050645968.  Google Scholar

[11]

J. P. Desi, H. Edrees, J. Price, E. Sander and T. Wanner, The dynamics of nucleation in stochastic Cahn-Morral systems, SIAM Journal on Applied Dynamical Systems, 10 (2011), 707-743.  doi: 10.1137/100801378.  Google Scholar

[12]

A. Dhooge, W. Govaerts and Y. A. Kuznetsov, MATCONT: A MATLAB package for numerical bifurcation analysis of ODEs, Association for Computing Machinery. Transactions on Mathematical Software, 29 (2003), 141-164.  doi: 10.1145/779359.779362.  Google Scholar

[13]

E. Doedel, AUTO: A program for the automatic bifurcation analysis of autonomous systems, in Proceedings of the Tenth Manitoba Conference on Numerical Mathematics and Computing, Vol. I (Winnipeg, Man., 1980), 30 (1981), 265–284.  Google Scholar

[14]

M. Gameiro, J.-P. Lessard and K. Mischaikow, Validated continuation over large parameter ranges for equilibria of PDEs, Mathematics and Computers in Simulation, 79 (2008), 1368-1382.  doi: 10.1016/j.matcom.2008.03.014.  Google Scholar

[15]

Z. G. Huseynov and A. M. Shykhammedov, On bases of sines and cosines in Sobolev spaces, Applied Mathematics Letters, 25 (2012), 275-278.  doi: 10.1016/j.aml.2011.08.026.  Google Scholar

[16]

I. Johnson, E. Sander and T. Wanner, Branch interactions and long-term dynamics for the diblock copolymer model in one dimension, Discrete and Continuous Dynamical Systems. Series A, 33 (2013), 3671-3705.  doi: 10.3934/dcds.2013.33.3671.  Google Scholar

[17]

T. Kinoshita, Y. Watanabe and M. T. Nakao, An alternative approach to norm bound computation for inverses of linear operators in Hilbert spaces, Journal of Differential Equations, 266 (2019), 5431-5447.  doi: 10.1016/j.jde.2018.10.027.  Google Scholar

[18]

J.-P. Lessard, E. Sander and T. Wanner, Rigorous continuation of bifurcation points in the diblock copolymer equation, Journal of Computational Dynamics, 4 (2017), 71-118.  doi: 10.3934/jcd.2017003.  Google Scholar

[19]

S. Maier-Paape, U. Miller, K. Mischaikow and T. Wanner, Rigorous numerics for the Cahn-Hilliard equation on the unit square, Revista Matematica Complutense, 21 (2008), 351-426.  doi: 10.5209/rev_REMA.2008.v21.n2.16380.  Google Scholar

[20]

S. Maier-Paape, K. Mischaikow and T. Wanner, Structure of the attractor of the Cahn-Hilliard equation on a square, International Journal of Bifurcation and Chaos, 17 (2007), 1221-1263.  doi: 10.1142/S0218127407017781.  Google Scholar

[21]

S. Maier-Paape and T. Wanner, Spinodal decomposition for the Cahn-Hilliard equation in higher dimensions: Nonlinear dynamics, Archive for Rational Mechanics and Analysis, 151 (2000), 187-219.  doi: 10.1007/s002050050196.  Google Scholar

[22]

T. R. Muradov and V. F. Salmanov, On the basis property of trigonometric systems with linear phase in a weighted Sobolev space, Dokl. Math., 90 (2014), 611-612.  doi: 10.1134/s1064562414060301.  Google Scholar

[23]

M. T. Nakao, M. Plum and Y. Watanabe, Numerical Verification Methods and Computer-Assisted Proofs for Partial Differential Equations, Springer-Verlag, Berlin, 2019. doi: 10.1007/978-981-13-7669-6.  Google Scholar

[24]

T. Ohta and K. Kawasaki, Equilibrium morphology of block copolymer melts, Macromolecules, 19 (1986), 2621-2632.  doi: 10.1021/ma00164a028.  Google Scholar

[25]

M. Plum, Existence and enclosure results for continua of solutions of parameter-dependent nonlinear boundary value problems, Journal of Computational and Applied Mathematics, 60 (1995), 187-200.  doi: 10.1016/0377-0427(94)00091-E.  Google Scholar

[26]

M. Plum, Enclosures for two-point boundary value problems near bifurcation points, in Scientific Computing and Validated Numerics (Wuppertal, 1995), vol. 90 of Mathematical Research, Akademie Verlag, Berlin, 1996,265–279.  Google Scholar

[27]

M. Plum, Computer-assisted proofs for semilinear elliptic boundary value problems, Japan Journal of Industrial and Applied Mathematics, 26 (2009), 419-442.  doi: 10.1007/BF03186542.  Google Scholar

[28]

S. M. Rump, INTLAB - INTerval LABoratory, in Developments in Reliable Computing (ed. T. Csendes), Kluwer Academic Publishers, Dordrecht, 1999, 77–104, http://www.ti3.tuhh.de/rump/. Google Scholar

[29]

S. M. Rump, Verification methods: Rigorous results using floating-point arithmetic, Acta Numerica, 19 (2010), 287-449.  doi: 10.1017/S096249291000005X.  Google Scholar

[30]

E. Sander and T. Wanner, Validated saddle-node bifurcations and applications to lattice dynamical systems, SIAM Journal on Applied Dynamical Systems, 15 (2016), 1690-1733.  doi: 10.1137/16M1061011.  Google Scholar

[31] L. N. Trefethen and M. Embree, Spectra and Pseudospectra, Princeton University Press, Princeton, NJ, 2005.   Google Scholar
[32]

J. B. van den Berg and J. F. Williams, Validation of the bifurcation diagram in the 2D Ohta-Kawasaki problem, Nonlinearity, 30 (2017), 1584-1638.  doi: 10.1088/1361-6544/aa60e8.  Google Scholar

[33]

J. B. van den Berg and J. F. Williams, Optimal periodic structures with general space group symmetries in the Ohta-Kawasaki problem, arXiv: 1912.00059. Google Scholar

[34]

J. B. van den Berg and J. F. Williams, Rigorously computing symmetric stationary states of the Ohta-Kawasaki problem in three dimensions, SIAM Journal on Mathematical Analysis, 51 (2019), 131-158.  doi: 10.1137/17M1155624.  Google Scholar

[35]

T. Wanner, Topological analysis of the diblock copolymer equation, in Mathematical Challenges in a New Phase of Materials Science (eds. Y. Nishiura and M. Kotani), vol. 166 of Springer Proceedings in Mathematics & Statistics, Springer-Verlag, 2016, 27–51. doi: 10.1007/978-4-431-56104-0_2.  Google Scholar

[36]

T. Wanner, Computer-assisted equilibrium validation for the diblock copolymer model, Discrete and Continuous Dynamical Systems, Series A, 37 (2017), 1075-1107.  doi: 10.3934/dcds.2017045.  Google Scholar

[37]

T. Wanner, Computer-assisted bifurcation diagram validation and applications in materials science, Proceedings of Symposia in Applied Mathematics, 74 (2018), 123-174.   Google Scholar

[38]

T. Wanner, Validated bounds on embedding constants for Sobolev space Banach algebras, Mathematical Methods in the Applied Sciences, 41 (2018), 9361-9376.  doi: 10.1002/mma.5294.  Google Scholar

[39]

Y. Watanabe, T. Kinoshita and M. T. Nakao, An improved method for verifying the existence and bounds of the inverse of second-order linear elliptic operators mapping to dual space, Japan Journal of Industrial and Applied Mathematics, 36 (2019), 407-420.  doi: 10.1007/s13160-019-00344-8.  Google Scholar

[40]

Y. Watanabe, K. Nagatou, M. Plum and M. T. Nakao, Norm bound computation for inverses of linear operators in Hilbert spaces, Journal of Differential Equations, 260 (2016), 6363-6374.  doi: 10.1016/j.jde.2015.12.041.  Google Scholar

[41]

N. Yamamoto, A numerical verification method for solutions of boundary value problems with local uniqueness by Banach's fixed-point theorem, SIAM Journal on Numerical Analysis, 35 (1998), 2004-2013.  doi: 10.1137/S0036142996304498.  Google Scholar

[42]

N. Yamamoto, M. T. Nakao and Y. Watanabe, A theorem for numerical verification on local uniqueness of solutions to fixed-point equations, Numerical Functional Analysis and Optimization, 32 (2011), 1190-1204.  doi: 10.1080/01630563.2011.594348.  Google Scholar

Figure 1.  Ten sample validated one-dimensional equilibrium solutions. For all solutions we choose $ \lambda = 150 $ and $ \sigma = 6 $. Three of the solutions have total mass $ \mu = 0 $, three are for mass $ \mu = 0.1 $, three for $ \mu = 0.3 $, and finally one for $ \mu = 0.5 $
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Figure 2.  There is a tradeoff between high-dimensional calculations and optimal results. The top left figure shows how the bound of $ K $ varies with the dimension of the truncated approximation matrix used to calculate $ K_N $. These calculations are for dimension one, but a similar effect occurs in higher dimensions as well. The top right figure shows the corresponding estimate for $ \delta_x $, and the bottom panel shows the estimate for $ \delta_\alpha $, where $ \alpha $ is each of the three parameters. The size of the validated interval grows larger as the truncation dimension grows, but with diminishing returns on the computational investment
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Figure 3.  Six of the seventeen validated two-dimensional equilibrium solutions. For all seventeen solutions we use $ \sigma = 6 $. Five of these solutions are for $ \lambda = 75 $ and $ \mu = 0 $ (top left). The rest of them use $ \lambda = 150 $ and $ \mu = 0 $ (top middle and top right), $ \mu = 0.1 $ (bottom left), $ \mu = 0.3 $ (bottom middle), and $ \mu = 0.5 $ (bottom right)
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Figure 4.  A three-dimensional validated solution for the parameter values $ \lambda = 75 $, $ \sigma = 6 $, and $ \mu = 0 $
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Table 1.  These values are rigorous upper bounds for the embedding constants in (11)
Dimension $ d $ $ 1 $ $ 2 $ $ 3 $
Sobolev Embedding Constant $ C_m $ $ 1.010947 $ $ 1.030255 $ $ 1.081202 $
Sobolev Embedding Constant $ \overline{C}_m $ $ 0.149072 $ $ 0.248740 $ $ 0.411972 $
Banach Algebra Constant $ C_b $ $ 1.471443 $ $ 1.488231 $ $ 1.554916 $
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Dimension $ d $ $ 1 $ $ 2 $ $ 3 $
Sobolev Embedding Constant $ C_m $ $ 1.010947 $ $ 1.030255 $ $ 1.081202 $
Sobolev Embedding Constant $ \overline{C}_m $ $ 0.149072 $ $ 0.248740 $ $ 0.411972 $
Banach Algebra Constant $ C_b $ $ 1.471443 $ $ 1.488231 $ $ 1.554916 $
Table 2.  A sample of the one-dimensional solution validation parameters for three typical solutions. In each case, we use $ \sigma = 6 $ and $ \lambda = 150 $. If we had chosen a larger value of $ N $, we could significantly improve the results
$ \mu $ $ K $ $ N $ $ P $ $ \delta_\alpha $ $ \delta_x $
$ 0 $ 6.2575 89 $ \lambda $ 0.0016 0.0056
$ \sigma $ 2.9259e-04 0.0056
$ \mu $ 2.8705e-06 0.0044
$ 0.1 $ 6.4590 104 $ \lambda $ 0.0011 0.0050
$ \sigma $ 2.5369e-04 0.0050
$ \mu $ 2.5579e-06 0.0041
$ 0.5 $ 3.1030 74 $ \lambda $ 0.0052 0.0107
$ \sigma $ 0.0011 0.0106
$ \mu $ 1.2871e-05 0.0092
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$ \mu $ $ K $ $ N $ $ P $ $ \delta_\alpha $ $ \delta_x $
$ 0 $ 6.2575 89 $ \lambda $ 0.0016 0.0056
$ \sigma $ 2.9259e-04 0.0056
$ \mu $ 2.8705e-06 0.0044
$ 0.1 $ 6.4590 104 $ \lambda $ 0.0011 0.0050
$ \sigma $ 2.5369e-04 0.0050
$ \mu $ 2.5579e-06 0.0041
$ 0.5 $ 3.1030 74 $ \lambda $ 0.0052 0.0107
$ \sigma $ 0.0011 0.0106
$ \mu $ 1.2871e-05 0.0092
Table 3.  A sample of the two-dimensional validation parameters for a couple of typical solutions. In all cases, we use $ \sigma = 6 $. Again as in the previous table, we could improve results by choosing a larger value of $ N $, but in this case since $ N $ is only the linear dimension, the dimension of the calculation varies with $ N^2 $
$ (\lambda,\mu) $ $ K $ $ N $ $ P $ $ \delta_\alpha $ $ \delta_x $
$ (75,0) $ 21.1303 28 $ \lambda $ 1.6124e-04 0.0020
$ \sigma $ 6.1338e-05 0.0020
$ \mu $ 5.9914e-07 0.0016
$ (150,0.1) $ 30.1656 72 $ \lambda $ 1.1833e-05 4.7710e-04
$ \sigma $ 5.1514e-06 4.7858e-04
$ \mu $ 4.4558e-08 4.2316e-04
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$ (\lambda,\mu) $ $ K $ $ N $ $ P $ $ \delta_\alpha $ $ \delta_x $
$ (75,0) $ 21.1303 28 $ \lambda $ 1.6124e-04 0.0020
$ \sigma $ 6.1338e-05 0.0020
$ \mu $ 5.9914e-07 0.0016
$ (150,0.1) $ 30.1656 72 $ \lambda $ 1.1833e-05 4.7710e-04
$ \sigma $ 5.1514e-06 4.7858e-04
$ \mu $ 4.4558e-08 4.2316e-04
Table 4.  Validation parameters for a three-dimensional sample solution
$ (\lambda,\sigma,\mu) $ $ K $ $ N $ $ P $ $ \delta_\alpha $ $ \delta_x $
$ (75,6,0) $ 22.6527 22 $ \lambda $ 0.1143e-04 0.5917e-03
$ \sigma $ 0.1707e-04 0.5955e-03
$ \mu $ 0.0010e-04 0.4901e-03
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$ (\lambda,\sigma,\mu) $ $ K $ $ N $ $ P $ $ \delta_\alpha $ $ \delta_x $
$ (75,6,0) $ 22.6527 22 $ \lambda $ 0.1143e-04 0.5917e-03
$ \sigma $ 0.1707e-04 0.5955e-03
$ \mu $ 0.0010e-04 0.4901e-03
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