New Numerical Approach for Solving Abel’s Integral Equations

Şenel, Ayşe Anapalı; Öztürk, Yalçın; Gülsu, Mustafa

doi:10.2478/fcds-2021-0017

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.

References

[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 Back to article
[2] Brenke W. C., An application of Abel’s integral equation, American Mathematics Monthly, 2, 29,1922, 58-60.10.1080/00029890.1922.11986102
Search in Google Scholar Back to article
[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 Back to article
[4] Cimatti G., Application of the Abel integral equation to an inverse problem in thermoelectricity, Europen Journal of Applied Mathematics, 20, 2009, 519–529.10.1017/S095679250999009X
Search in Google Scholar Back to article
[5] Cremers C.J., Birkebak R.C., Application of the Abel Integral Equation to Spectrographic Data, Applied Optics, 5, 1996, 1057-1064.10.1364/AO.5.00105720049010
Search in Google Scholar Back to article
[6] Diethelm K., The Analysis of Fractional Differential Equations, Springer-Verlag, Berlin Heidelberg, 2010.10.1007/978-3-642-14574-2
Search in Google Scholar Back to article
[7] Ganji D.D., The application of He’s homotopy perturbation method to nonlinear equations arising in heat transfer, Physics Letter A, 355, 2006, 337-34.10.1016/j.physleta.2006.02.056
Search in Google Scholar Back to article
[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, 1-11.10.1186/s13662-020-02831-6739693332834818
Search in Google Scholar Back to article
[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, 109929.10.1016/j.chaos.2020.109929783453533519103
Search in Google Scholar Back to article
[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, 210-243.10.1002/num.22526
Search in Google Scholar Back to article
[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, 109696.10.1016/j.chaos.2020.109696
Search in Google Scholar Back to article
[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, 478.10.3390/sym12030478
Search in Google Scholar Back to article
[13] Gorenflo R., Vessella S., Abel Integral Equations: Analysis and Applications, Lecture Notes in Mathematics 1461, Springer-Verlag, Berlin, 1991.10.1007/BFb0084665
Search in Google Scholar Back to article
[14] He J.H., Homotopy perturbation technique, Computer Methods in Applied Mechanics and Engineering, 178, 1999, 257-262.10.1016/S0045-7825(99)00018-3
Search in Google Scholar Back to article
[15] Huang L, Huang Y., Fang-Li X., Approximate solution of Abel integral equation, Compters Mathematics with Applications, 56, 2008, 1748-1757.10.1016/j.camwa.2008.04.003
Search in Google Scholar Back to article
[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, 171–188.10.2478/amns.2020.1.00016
Search in Google Scholar Back to article
[17] Kumar S., Sloan I.H., A new collocation-type method for Hammerstein integral equations, Journal of Mathematics and Computer Science, 48, 1987, 123-129.10.1090/S0025-5718-1987-0878692-4
Search in Google Scholar Back to article
[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 164-168.10.1016/j.jelechem.2008.04.001
Search in Google Scholar Back to article
[19] Munkhammar J. D., Fractional calculus and the Taylor–Riemann series, Undergrad Mathematics Journal, 6, 1, 2005, 6.
Search in Google Scholar Back to article
[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, 664-676.10.1016/j.camwa.2008.10.085
Search in Google Scholar Back to article
[21] Podlubny I., Fractional differential equations. New York: Academic Press, 1999.
Search in Google Scholar Back to article
[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, 504–515.10.1016/j.amc.2017.08.048
Search in Google Scholar Back to article
[23] Vanani S. K., Solevmani F., Tau approximate solution of weakly singular Volterra integral equations, Mathematical and Computer Modelling., 57, 2013, 3-4.10.1016/j.mcm.2012.07.004
Search in Google Scholar Back to article
[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, 4136-4155.10.1002/mma.6179
Search in Google Scholar Back to article
[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, 109661.10.1016/j.chaos.2020.109661
Search in Google Scholar Back to article
[26] Yousefi S.A., Numerical solution of Abel’s integral equation by using Legendre wavelets, Applied Mathematics and Computation, 175, 2006 574-580.10.1016/j.amc.2005.07.032
Search in Google Scholar Back to article
[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, 105791.10.1016/j.sab.2020.105791
Search in Google Scholar Back to article
[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, 6753-6764.10.3390/e17106753
Search in Google Scholar Back to article

New Numerical Approach for Solving Abel’s Integral Equations

Abstract

Paradigm

My account