Solving the Unstable Linear Fredholm Integral Equation of the First Kind by Means of a New Stochastic Algorithm

• Autorzy: Kowalczyk P. , Solarz L. , Terzyk A. P. , Gauden P. A. , Gun'ko V. M.
• Języki publikacji: EN

Abstrakty

EN

A new "adsorption stochastic algorithm" (called ASA) is proposed for solving the unstable linear Fredholm integral equation of the first kind. The developed algorithm was applied for the calculation of the pore size distribution of activated carbons from single adsorption isotherms assuming different forms of the kernel (i.e. Dubinin and Radushkevich (DR) and/or Nguyen and Do (ND)) of a linear Fredholm integral equation of the first kind. The results obtained by ASA are compared with obtained applying, developed by Provencher, the advanced regularization CONTIN algorithm, advanced evolutionary algorithm GABI written by Arabas and modified by Kowalczyk, and simple evolutionary algorithm based on the mutation strategy labeled SASA. Additionally, the ASA results obtained by solving the integral equation with the ND kernel are compared with the results obtained by regularization solution of the integral equation with density functional theory (DFT) local isotherms as a kernel. It is shown that the developed ASA algorithm always provides stable and very similar results to the Tikhonov regularization method. Moreover, the ASA computations obtained for the ND local isotherms as a kernel are very similar to the results obtained by the most sophisticated regularization DFT software.

Słowa kluczowe

EN

reinforcement learning method; Fredholm integral equation of the first kind; optimization methods; genetic algorithms; adsorption; activated carbon

• Wydawca: Wydawnictwo Uniwersytetu Jagiellońskiego
• Czasopismo: Schedae Informaticae
• Rocznik: 2002
• Tom: Vol. 11
• Strony: 75--98
• Opis fizyczny: Bibliogr. 59 poz., rys.

Twórcy

autor

Kowalczyk P.

• Military Institute of Chemistry and Radiometry, Department of Respiratory Protection, gen. Chruściel Avenue 105, 00-910 Warsaw, Poland

autor

Solarz L.

• Institute of Physics, Military Technical Academy, Kaliski St. 2, 00-908 Warsaw, Poland

autor

Terzyk A. P.

• Copernicus University, Department of Chemistry, Physicochemistry of Carbon Materials Research Group, Gagarin St. 7, 87-100 Toruń, Poland

autor

Gauden P. A.

• . Copernicus University, Department of Chemistry, Physicochemistry of Carbon Materials Research Group, Gagarin St. 7, 87-100 Toruń, Poland

autor

Gun'ko V. M.

• Institute of Surface Chemistry, 17 General Naumov Street, 03164 Kiev, Ukraine

Bibliografia

• [1] Phillips D.L., A technique for the numerical solution of certain integral equations of the first kind, J. Assoc. Comp. Mach., 9. 1962, pp. 85-97.
• [2] Tikhonov A.N., Arsenin V.Y.; Solutions of Ill-posted Problems, Wiley, New York, 1977.
• [3] Morozov, V.A.; Methods for Solving Incorrectly Posed Problems. Springer, Berlin, 1984.
• [4] Press W.H., Teukolsky S.A., Vettcrling W.T., Flannery B.P.; Numerical Recipes in Fortran, Cambridge University Press, Cambridge. 1992.
• [5] Oliver L.P.; Modeling pysical adsorption on porous and nonporous solids using density functional theory. Journal of Porous Materials, 2, 1995, pp. 9-17. [5] Oliver L.P.; Modeling pysical adsorption on porous and nonporous solids using density functional theory. Journal of Porous Materials, 2, 1995, pp. 9-17
• [6] v. Szombathely M., Brauer M., Jaroniec M.; The solution of adsorption integral equations by means of the regularization method, J. Comput. Chem., 13, 1992, pp. 17-32.
• [7] Cherkashinin G.Y., Ismakaev M.K., Bubnov A.V., Drozdov V.A.; The technique of micropore size distribution reconstruction on the basis of the Dubinin theory of volume filling. Inverse Probl., 16, 2000, pp. 1421-1440.
• [8] Puziy A.M., Matynia T., Gawdzik B., Poddubnaya 0.1.; Use of CONTIN for calculation of adsorption energy distribution, Langmuir, 15, 1999, pp. 6016-6025.
• [9] Gun'ko V.M., Leboda R., Skubiszewska-Zięba J., Turov V.V.. Kowalczyk P.; Structure of silica gel Si-60 and pyrocarbon/silica gel adsorbents thermally and liydrothermally treated, Langmuir, 17, 2001, pp. 3148-3161.
• [10] Gun'ko V.M., Leboda R., Marciniak M., Grzegorczyk W., Skubiszewska-Zięba J., Malygin A.A., Maikov A.A.; CVD-titania/silica gel carbonized due to pyrolysis of cyclohexene, Langmuir, 16, 2000, pp. 3227-3243.
• |11] Puziy A.M., Poddubnaya O.I., Ritter J.A., Ebner A.D., Holland Ch.E.; Elucidation of the ion binding mechanism in heterogeneous carbon-composite adsorbents, Carbon. 39, 2001, pp. 2313-2324.
• [12] Leboda R., Gun’ko V.M., Tomaszewski W., Trznadel B.J.; Relationship between structural characteristics of activated carbons and their concentrating efficiency with respect to nitroorganics, J. Colloid Interface Sci., 238, 2001, pp. 489-500.
• [13] Leboda R., Turov V.V., Tomaszewski W., Skubiszewska-Zięba J., Gun’ko V.M.; Effect of adsorption of nitroaromatic compounds on the characteristics of bound water layers in aqueous suspensions of activated carbons, Carbon, 40, 2002, pp. 389-396.
• [14] Gun’ko V.M., Leboda R., Turov V.V., Charmas B., Skubiszewska-Zięba J.; Structural and energetic heterogeneities of hybrid carbon-mineral adsorbents, Apll. Surf. Sci.. 191. 2002, pp. 286-299.
• [15] Gun’ko V.M.. Leboda R., Turov V.V., Villiiras F., Skubiszewska-Zi^ba J., Chodorowski S., Marciniak M.; Structural and energetic nonuniformities of pyrocarbon-mineral adsorbents, J. Colloid Interface Sci.. 238, 2001, pp. 340-356.
• [16] Gun'ko V.M., Do, D.D.; Characterisation of pore structure of carbon adsorbents using regularization procedure, Colloids Surf. A., 193, 2001, pp. 71-83.
• [17] Kowalczyk P., Szmigielski R., Terzyk A.P.. Gauden P.A., Ziętek S.. Palijczuk D.; The evaluation of the structural helerogenity of activated carbons on the basis of Polanyi - Dubinin theory of adsorption. Bull. WIChiR. 31, 2001, pp. 91-112.
• [18] Provencher S.W.; CONTIN - Users Manual (Version 2), Max-Planck-lnstitut fllr Biophysikalische Chemie, Gottingen, 1984.
• [19] Kowalczyk P.. Terzyk A.P., Gauden P.A., Gun'ko V.M.; Evaluation of the structural and energetic heterogeneity of microporous solids by means of novel adsorption methods and numerical algorithms, J. Colloid Interface Sci. (submitted).
• [20| Kowalczyk P., Terzyk A.P., Gauden P.A.; The application of a CONTIN package for the evaluation of micropore size distribution functions, Langmuir (in press).
• [21] Fortuna Z., Macukow B., Wąsowski J.; Numerical Methods, WNT. Warszawa, 1995 (in Polish).
• [22] Jankowska J., Jankowski M.; Review of Numerical Methods and Algorithms, WNT. Warszawa.
• [23] Fogel D.B. (ed.); Evolutionary Computation: The Fossil Record, IEEE Press, Piscataway, 1998.
• [24] Rudziński W„ Everett D.H.; Adsorption of Gases on Heterogeneous Surfaces, Academic Press, New York, 1992.
• [25] Rudziński W., Steele W.A. Zgrablich G. (Eds.); Equilibria and Dynamics of Gas Adsorption on Heterogeneous Solid Surfaces, Elsevier, Amsterdam, 1997.
• [26] Wojsz R.; Characteristics of the Structural and Energetic Heterogeneity of Microporous Carbon Adsorbents Regarding the Adsorption of Polar Substances, UMK, Toruń, 1989 (in Polish).
• [27] Do D.D.; Adsorption Analysis: Equilibria and Kinetics, Imperial College Press. London, 1998.
• [28] Jankowska H.. Świątkowski A., Choma J.; Active Carbon. WNT. Warszawa. 1985 (in Polish).
• [29] Gauden P.A.; Theoretical Description of the Structural and Energetic Heterogeneity of Carbonaceous Materials (Thesis), UMK. Toruń, 2001 (in Polish).
• [30] Pfeifer P., Avnir D.; Chemistry in noninteger dimensions between two and three. I. Fractal theory of heterogeneous surfaces, J. Chem. Phys.. 79, 1983. pp. 3558-3565.
• [31] Pfeifer P., Avnir D.; Chemistry in noninteger dimensions between two and three. I. Fractal theory of heterogeneous surfaces (erratum), J. Chem. Phys., 80, 1984, pp. 4573.
• [32] Terzyk A.P., Wojsz R., Rychlicki G., Gauden P.A.; Fractal dimension of microporous carbon on the basis of Polany ¡-Dubinin theory of adsorption. Dubinin-Radushkevich adsorption isotherm equation, Colloids Surf. A, 119, 1996, pp. 175-181.
• [33] Terzyk A. P.. Wojsz R., Rychlicki G., Gauden P. A; Fractal dimension of microporous carbon on the basis of the Polanyi-Dubinin theory of adsorption. Part 2: Dubinin-Astakhov adsorption isotherm equation, Colloids Surf. A, 126, 1997, pp. 67-73.
• [34] Terzyk A.P., Wojsz R., Rychlicki G., Gauden P. A.: Fractal dimension of microporous carbon on the basis of the Polanyi-Dubinin theory of adsorption. Part 3: Adsorption and adsorption thermodynamics in the micropores of fractal carbons. Colloids Surf. A, 136, 1998, pp. 245-261.
• [35] Terzyk A.P., Gauden P.A., Rychlicki G., Wojsz R; Fractal dimension of microporous carbon on the basis of Polanyi-Dubinin theory of adsorption. Part IV. The comparative analysis of two alternative solutions of the overall adsorption isotherm equation for microporous fractal solids, Colloids Surf. A, 152, 1999. pp. 293-313.
• [36] Gauden P.A., Rychlicki G., Terzyk A.P.. Wojsz R.; Thermodynamics of adsorption on microporous fractal solids, J. Therm. Anal., 54. 1998, pp. 351-361.
• [37] Wojsz R., Terzyk A.P.; Fractal dimension of microporous carbon on the basis of first solution of a Laplace transform using an incomplete gamma function, Computers. Chem., 20. 1996, pp. 427-430.
• [38] Wojsz R., Terzyk A.P.; The structural parameters of microporous solid, including fractal dimension, on the basis of the potential theory of adsorption - the general solution, Computers. Chem., 21. 1997. pp. 83-87.
• [39] Terzyk A.P., Gauden P.A., Rychlicki G., Wojsz R.; Comments on "An isotherm equation for adsorption on fractal surfaces of heterogeneous porous materials”, Langmuir, 15, 1999, pp. 285-288.
• [40] Leboda R., Marciniak M., Gun’ko V.M., Grzegorczyk W., Malygin A.A., Maikov A.A.; Structure of carbonized mesoporous silica gel/CVD-titania, Colloids Surf. A, 167, 2000, pp. 275-285.
• [41] Leboda R., Gun’ko V.M., Marciniak M., Malygin A.A.. Malkin A.A., Grzegorczyk W., Trznadel B.J., Pakhlov E.M., Voronin E.F.; Structure of chemical vapor deposition litania/silica gel, J. Colloid Interface Sei., 218, 1999. pp. 23-29.
• [42] Leboda R., Turov V.V., Charmas B., Skubiszewska-Zięba J., Gun'ko V.M.; Surface properties of mesoporous carbon-silica gel adsorbents, J. Colloid Interface Sci., 223, 2000, pp. 112-125.
• [43] Terzyk A.P., Gauden P.A., Zawadzki J., Rychlicki G., Wiśniewski M., Kowalczyk P.; Toward the characterization of microporosity of carbonaceous films, J. Colloid Interface Sei., 243, 2001. pp. 183-192.
• [44] Stoeckli H.F., Kraehenbuehl F., Ballerini L., De Bemardini S.; Recent Development in the Dubinin Equation. Carbon, 27, 1989, pp. 125-128.
• [45] Stoeckli H.F.; Carbon, 36. 1998, pp. 363-368.
• [46] Vishnyakov, A., Russo R.. Neimark A.V., Ravikovitch P.I.; Unified approach to pore size characterization of microporous carbonaceous materials from Ni, Ar, and C02 adsorption isotherms. Langmuir, 16. 2000. pp. 2311-2320.
• [47] Nguyen C., Do D.D.; Simple optimization approach for the characterization of pore size distribution, Langmuir, 16,2000, pp. 1319-1322.
• [48] Nguyen C., Do D.D.; New method for the characterization of porous materials, Langmuir, 15, 1999, pp. 3608-3615.
• [49] Nguyen C., Do D.D.; The Dubinin-Radushkevich equation and the underlying microscopic adsorption description, Carbon, 39, 2001, pp. 1327-1336.
• [50] Do D.D.. Nguyen C., Do H.D.; Cluiracterization of micro-mesoporous carbon media. Colloids Surf. A. 187-188. 2001, pp. 51-71.
• [51] Do D.D., Do H.D.; Effects of adsorbate-adsorbate interaction in the description of adsorption isotherm of hydrocarbons in micro-mesoporous carbonaceous materials, Appl. Surf. Sei.. 7821, 2002. pp. 1-17.
• [52] Do D.D., Do H.D.; Characterization of micro-mesoporous carbonaceous materials, calculations of adsorption isotherm of hydrocarbons, Langmuir. 18, 2002, pp. 93-99.
• [53] Michalewicz Z., Fogel D.; How to Solve It: Modern Heuristics, Springer-Verlag, Berlin, 2000.
• [54] Michalewicz Z.; Genetic Algorithms + Data Structures = Evolution Programs, Springer- Verlag, Berlin, 1996. 
• [55] Schwefel H.P.; Evolution and Optimum Seeking, John Wiley, Chichester, 1995.
• [56] Golberg, D.E.; Genetic Algorithms in Search, Optimization and Machine Learning, WNT. Warszawa, 1995 (in Polish).
• [57] Arabas J.; Lecture Notes on Evolutionary Computation, WNT, Warszawa, 2001 (in Polish).
• [58] Findeisen W., Szymanowski J., Wierzbicki A.; The Theory and the Computational Methods of Optymalisation. PWN, Warszawa, 1980 (in Polish).
• [59] Back T., Fogel D., Michalewicz Z. (Eds.); Handbook of Evolutionary Computation, Oxford University Press, New York, 1996.