1. bookVolume 46 (2021): Issue 3 (September 2021)
Journal Details
License
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Journal
First Published
24 Oct 2012
Publication timeframe
4 times per year
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English
access type Open Access

New Numerical Approach for Solving Abel’s Integral Equations

Published Online: 17 Sep 2021
Page range: 255 - 271
Received: 14 Jun 2020
Accepted: 22 Mar 2021
Journal Details
License
Format
Journal
First Published
24 Oct 2012
Publication timeframe
4 times per year
Languages
English
Abstract

In this article, we present an efficient method for solving Abel’s integral equations. This important equation is consisting of an integral equation that is modeling many problems in literature. Our proposed method is based on first taking the truncated Taylor expansions of the solution function and fractional derivatives, then substituting their matrix forms into the equation. The main character behind this technique’s approach is that it reduces such problems to solving a system of algebraic equations, thus greatly simplifying the problem. Numerical examples are used to illustrate the preciseness and effectiveness of the proposed method. Figures and tables are demonstrated to solutions impress. Also, all numerical examples are solved with the aid of Maple.

Keywords

[1] Avazzadeh Z, Shafiee B., Loghmani G.B., Fractional calculus for solving Abel’s integral equations using Chebyshev polynomials, Applied. Mathematical Science, 5, 45, 2011, 227-2216. Search in Google Scholar

[2] Brenke W. C., An application of Abel’s integral equation, American Mathematics Monthly, 2, 29,1922, 58-60. Search in Google Scholar

[3] Caputo M., Fabrizio M., A new definition of fractional derivative without singular Kernel, Progress in Fractional Differentiation and Applications 1, 2015, 73–85. Search in Google Scholar

[4] Cimatti G., Application of the Abel integral equation to an inverse problem in thermoelectricity, Europen Journal of Applied Mathematics, 20, 2009, 519–529. Search in Google Scholar

[5] Cremers C.J., Birkebak R.C., Application of the Abel Integral Equation to Spectrographic Data, Applied Optics, 5, 1996, 1057-1064. Search in Google Scholar

[6] Diethelm K., The Analysis of Fractional Differential Equations, Springer-Verlag, Berlin Heidelberg, 2010. Search in Google Scholar

[7] Ganji D.D., The application of He’s homotopy perturbation method to nonlinear equations arising in heat transfer, Physics Letter A, 355, 2006, 337-34. Search in Google Scholar

[8] Gao W., Baskonus H.M., Shi L., New investigation of bats-hosts-reservoir-people coronavirus model and application to 2019-nCoV system, Advance in Difference Equation, 391, 2020, 1-11. Search in Google Scholar

[9] Gao W., Veeresha P., Baskonus H.M., Prakasha D.G., Kumar P., A new study of unreported cases of 2019-nCOV epidemic outbreaks, Chaos, Solitons and Fractals, 138, 2020, 109929. Search in Google Scholar

[10] Gao W., Veeresha P., Prakasha D.G., Baskonus H.M., New numerical simulation for fractional Benney–Lin equation arising in falling film problems using two novel techniques, Numerical methods for partial differential equation, 37, 1, 2020, 210-243. Search in Google Scholar

[11] Gao W., Veeresha P., Prakasha D.G., Baskonus H.M., Yel G., New approach for the model describing the deathly disease in pregnant women using Mittag-Leffler function, Chaos, Solitons and Fractals, 134, 2020, 109696. Search in Google Scholar

[12] Gao W., Veeresha P., Prakasha D.G., Baskonus H.M., Yel G., New Numerical Results for the Time-Fractional Phi-Four Equation Using a Novel Analytical Approach, Symmetry, 2020, 12, 478. Search in Google Scholar

[13] Gorenflo R., Vessella S., Abel Integral Equations: Analysis and Applications, Lecture Notes in Mathematics 1461, Springer-Verlag, Berlin, 1991. Search in Google Scholar

[14] He J.H., Homotopy perturbation technique, Computer Methods in Applied Mechanics and Engineering, 178, 1999, 257-262. Search in Google Scholar

[15] Huang L, Huang Y., Fang-Li X., Approximate solution of Abel integral equation, Compters Mathematics with Applications, 56, 2008, 1748-1757. Search in Google Scholar

[16] Ilhan E., Kıymaz O., A generalization of truncated M-fractional derivative and applications to fractional differential equations, Applied Mathematics and Nonlinear Sciences, 5, 1, 2020, 171–188. Search in Google Scholar

[17] Kumar S., Sloan I.H., A new collocation-type method for Hammerstein integral equations, Journal of Mathematics and Computer Science, 48, 1987, 123-129. Search in Google Scholar

[18] Mirčeski V., Tomovski Z., Analytical solutions of integral equations for modeling of reversible electrode processes under voltammetric conditions, Journal of Electroanalytical Chemistry, 619, 620, 2008 164-168. Search in Google Scholar

[19] Munkhammar J. D., Fractional calculus and the Taylor–Riemann series, Undergrad Mathematics Journal, 6, 1, 2005, 6. Search in Google Scholar

[20] Pandey R. K., Singh O. P., Singh V. K., Efficient algorithms to solve singular integral equations of Abel type, Computers Mathematics with Applications, 57, 2009, 664-676. Search in Google Scholar

[21] Podlubny I., Fractional differential equations. New York: Academic Press, 1999. Search in Google Scholar

[22] Singh J., Kumar D., Hammouch Z., Atangana A., A fractional epidemiological model for computer viruses pertaining to a new fractional derivative, Applied Mathematics and Computation, 316, 2018, 504–515. Search in Google Scholar

[23] Vanani S. K., Solevmani F., Tau approximate solution of weakly singular Volterra integral equations, Mathematical and Computer Modelling., 57, 2013, 3-4. Search in Google Scholar

[24] Veeresha P., Prakasha D.G., Baskonus H.M., Yel G., An efficient analytical approach for fractional Lakshmanan-Porsezian-Daniel model, Mathematical methods in applied science, 43, 2020, 4136-4155. Search in Google Scholar

[25] Veeresha P., Baskonus H.M., Prakasha D.G., Gao W., Yel G., Regarding new numerical solution of fractional Schistosomiasis disease arising in biological phenomena, Chaos, Solitons and Fractals, 133, 2020, 109661. Search in Google Scholar

[26] Yousefi S.A., Numerical solution of Abel’s integral equation by using Legendre wavelets, Applied Mathematics and Computation, 175, 2006 574-580. Search in Google Scholar

[27] Wu J., Zhou Y., Hang C., A singularity free and derivative free approach for Abel integral equation in analyzing the laser-induced breakdown spectroscopy, Spectrochimica Acta Part B: Atomic Spectroscopy,167, 2020, 105791. Search in Google Scholar

[28] Zhang Y., Cattani C., Yang X.J., Local Fractional Homotopy Perturbation Method for Solving Non-Homogeneous Heat Conduction Equations in Fractal Domain, Entropy, 17, 2015, 6753-6764. Search in Google Scholar

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