Research Article
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A spectral technique for solving two-dimensional fractional integral equations with weakly singular kernel

Year 2018, Volume: 47 Issue: 3, 553 - 566, 01.06.2018

Ali H. Bhrawy , Mohamed A. Abdelkawy , Dumitru Baleanu , Ahmed Z.m. Amin

https://izlik.org/JA74KB25AF

Abstract

This paper adapts a new numerical technique for solving two-dimensional fractional integral equations with weakly singular. Using the spectral collocation method, the fractional operators of Legendre and Chebyshev polynomials, and Gauss-quadrature formula, we achieve a reduction of given problems into those of a system of algebraic equations. We apply the reported numerical method to solve several numerical examples in order to test the accuracy and validity. Thus, the novel algorithm is more responsible for solving two-dimensional fractional integral equations with weakly singular.

Keywords

Two-dimensional fractional integral equations with weakly singular Spectral collocation method Gauss quadrature Shifted Legendre polynomials Shifted Chebyshev polynomial

References

  • L.E. Kosarev, Applications of integral equations of the rst kind in experiment physics., Comput. Phys. Commun., 20 (1980) 69-75.
  • H. Brunner, 1896-1996: One hundred years of Volterra integral equation of the rst kind, Appl. Numer. Math., 24 (1997) 83-93.
  • A. M. Wazwaz, Linear and Nonlinear Integral Equations Methods and Applications, Springer, Heidelberg, Dordrecht, London, New York, 2011.
  • A. Wazwaz, A First Course in Integral Equations, World Scientic Publishing, Singapore, 1997.
  • A. Wazwaz, A reliable modication of Adomian decomposition method, Appl. Math. Comput., 102 (1999) 77-86.
  • A. D. Polyanin and A.V. Manzhirov, Handbook of Integral Equations, CRC Press, Boca Raton, 1998.
  • R. Metzler and J. Klafter, The restaurant at the end of the random walk: Recent developments in the description of anomalous transport by fractional dynamics, J. Phys. A, 37 (2004) 161-208.
  • R. T. Baillie, Long memory processes and fractional integration in econometrics, J. Econometrics, 73 (1996) 5-59.
  • R. L. Magin, Fractional Calculus in Bioengineering, Begell House Publishers, 2006.
  • D. A. Benson, S.W. Wheatcraft and M.M. Meerschaert, Application of a fractional advection-dispersion equation, Water Resour. Res., 36 (2000) 1403-1412.
  • S. Abbas and M. Benchohra, Fractional order integral equations of two independent variables, Appl. Math. Comput., 227 (2014) 755-761.
  • S. Abbas and M. Benchohra, Fractional Order Riemann-Liouville Integral Equations with Multiple Time Delays, Applied Mathematics, 12 (2012) 79-87.
  • R. Hilfer, Applications of Fractional Calculus in Physics, World Scientic, Singapore, 2000.
  • Y.L. Jiang and X.L Ding, Waveform relaxation methods for fractional dierential equations with the Caputo derivatives, J. Comput. Appl. Math., 238 (2013) 51-67.
  • A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and Applications of Fractional Dierential Equations, Elsevier, 2006.
  • H. Wang and N. Du, Fast alternating-direction nite dierence methods for threedimensional space-fractional diusion equations, J. Comput. Phys., 258 (2014) 305-318.
  • J. Liu and G. Hou, Numerical solutions of the space- and time-fractional coupled Burgers equation by generalized dierential transform method, Appl. Math. Comput., 217 (2011) 7001-7008.
  • L.Wang, Y. Ma and Z. Meng, Haar wavelet method for solving fractional partial dierential equations numerically, Appl. Math. Comput., 227 (2014) 66-76
  • R.K. Pandey, O.P. Singh and V.K. Baranwal, An analytic algorithm for the space-time fractional advection-dispersion equation, Comput. Phys. Commun. 182 (2011) 1134-1144.
  • I. Area, J. D. Djida, J. Losada and J.J. Nieto, On Fractional Orthonormal Polynomials of a Discrete Variable, Discrete Dynamics in Nature and Society 2015 (2015), Article ID 141325, 7 pages
  • I. Podlubny, Fractional Dierential Equations, Academic Press, San Diego, (1999).
  • M.A. Abdelkawy, M.A. Zaky, A.H. Bhrawy and D. Baleanu, Numerical simulation of time variable fractional order mobile-immobile advection-dispersion model, Rom. Rep. Phys., 67 (3) 2015.
  • A.H. Bhrawy and M.A. Zaky, Numerical simulation for two-dimensional variable-order fractional nonlinear cable equation, Nonlinear Dyn., 80(1) (2015) 101-116
  • A.H. Bhrawy, Engy A. Ahmed and D. Baleanu, An ecient collocation technique for solving generalized Fokker-Planck type equations with variable coecients, P. Rom. Acad. A, 15 (2014) 322-330
  • W. Shaoa and X. Wu, An eective Chebyshev tau meshless domain decomposition method based on the integration-dierentiation for solving fourth order equations, J. Comput. Phys., 231(2012) 7695-7714
  • J. Chena, Y. Huanga, H. Ronga, T. Wub and T. Zen, A multiscale Galerkin method for second-order boundary value problems of Fredholm integro-dierential equation, J. Comput. Appl. Math., 290 (2015) 633-640
  • Y. Chen and J. Zhou, Error estimates of spectral Legendre-Galerkin methods for the fourthorder equation in one dimension, Appl. Math. Comput., 268 (2015) 1217-1226
  • G. Kitzler and J. Schöberl, A high order space-momentum discontinuous Galerkin method for the Boltzmann equation, 70 (2015)1539-1554
  • A.H. Bhrawy, An ecient Jacobi pseudospectral approximation for nonlinear complex generalized Zakharov system, Appl. Math. Comput., 247 (2014) 30-46.
  • E.H. Doha, A.H. Bhrawy, M.A. Abdelkawy and R.A.V. Gorder, Jacobi-Gauss-Lobatto collocation method for the numerical solution of 1+1 nonlinear Schrödinger equations, J. Comput. Phys., 261 (2014)244-255.
  • E.H. Doha, A.H. Bhrawy and S.S. Ezz-Eldien, A new Jacobi operational matrix: An application for solving fractional dierential equations, Appl. Math. Model., 36 (2012) 4931-4943.
  • A.H. Bhrawy, E.H. Doha, D. Baleanu and S.S. Ezz-Eldien, A spectral tau algorithm based on Jacobi operational matrix for numerical solution of time fractional diusion-wave equations, J. Comput. Phys.,293 (2015) 142-156.
  • E.H. Doha, On the construction of recurrence relations for the expansion and connection coecients in series of Jacobi polynomials, J. Phys. A Math. Gen. 37 (2004) 657-675.
  • A.H. Bhrawy, M.A. Zaky and D. Baleanu, New numerical approximations for space-time fractional Burgers' equations via a Legendre spectral-collocation method, Rom. Rep. Phys., 67(2) (2015).
  • A.H. Bhrawy, M. A. Abdelkawy, J. Tenreiro Machado and A. Z. M. Amin, Legender-Gauss- Lobatto collocation method for solving multi-dimensional Fredholm integral equations., Comput. Math. Appl., Doi: 10.1016/j.camwa.2016.04.011
  • M.R. Eslahchi, M. Dehghan and M. Parvizi, Application of the collocation method for solving nonlinear fractional integro-dierential equations,J. Comput. Appl. Math., 257 (2014) 105-128.
  • J. Shen, T. Tang and L. L. Wang, SPECTRAL METHODS Algorithms, Analyses and Applications, (Springer; 2011 edition August 31, 2011).
  • C. Canuto, M. Y. Hussaini, A. Quarteroni and T.A. Zang, Spectral Methods: fundamentals in single domains, (Springer-Verlag, Berlin, Heidelberg, 2006).
  • M. A. Abdelkawy, S. S. Ezz-Eldien and A. Z. M.Amin, A Jacobi Spectral Collocation Scheme for Solving Abel's Integral Equations, Progr. Fract. Dier. Appl.1, 3 (2015) 187- 200.
  • A. H. Bhrawy, A.A. Al-Zahrani, Y.A. Alhamed and D. Baleanu, A new generalized Laguerre-Gauss collocation scheme for numerical solution of generalized fractional Pantograph equations, Rom. J. Phys., 59 (2014) 646-657
  • S. R. Lau and R. H. Price, Sparse spectral-tau method for the three-dimensional helically reduced wave equation on two-center domains, J. Comput. Phys., 231(2012) 7695-7714
  • F. Ghoreishi and S. Yazdani, An extension of the spectral Tau method for numerical solution of multi-order fractional dierential equations with convergence analysis, Comput. Math. Appl., 61 (2011) 30-43
  • E. H. Doha and A. H. Bhrawy, An ecient direct solver for multidimensional elliptic Robin boundary value problems using a Legendre spectral-Galerkin method, Comput. Math. Appl., 64 (2012) 558-571
  • Y. Luke, The Special Functions and Their Approximations, vol. 2, Academic Press, New York, (1969).
  • E. H. Doha, A. H. Bhrawy and R. M. Hafez, A Jacobi-Jacobi dual-Petrov-Galerkin method for third- and fth-order dierential equations, Math. Comput. Model, 53 (2011) 1820- 1832.
  • I. Area, D.K. Dimitrov, E. Godoy, V.G. Paschoa, Approximate Calculation of Sums I: Bounds for the Zeros of Gram Polynomials, SIAM J. Numer. Anal., 54(4), (2014)2210- 2227.
  • I. Area, D.K. Dimitrov, E. Godoy and V.G. Paschoa, Approximate Calculation of Sums II: Gaussian Type Quadrature, SIAM J. Numer. Anal., 52(4), (2016)1867-1886.
  • E.H. Doha and A.H. Bhrawy, An ecient direct solver for multidimensional elliptic Robin boundary value problems using a Legendre spectral-Galerkin method, Computers and Mathematics with Applications 64 (2012) 558-571.
  • E. Godoy, A. Ronveaux, A. Zarzo and I. Area, Minimal recurrence relations for connection coecients between classical orthogonal polynomials: Continuous case, Journal of Computational and Applied Mathematics 84 (1997) 257-275.
  • A. Ronveaux, A. Zarzo, I. Area and E. Godoy, Transverse limits in the Askey tableau, Journal of Computational and Applied Mathematics 99 (1998) 327-335.

Year 2018, Volume: 47 Issue: 3, 553 - 566, 01.06.2018

Ali H. Bhrawy , Mohamed A. Abdelkawy , Dumitru Baleanu , Ahmed Z.m. Amin

https://izlik.org/JA74KB25AF

Abstract

Keywords

Two-dimensional fractional integral equations with weakly singular Spectral collocation method Gauss quadrature Shifted Legendre polynomials Shifted Chebyshev polynomial

References

  • L.E. Kosarev, Applications of integral equations of the rst kind in experiment physics., Comput. Phys. Commun., 20 (1980) 69-75.
  • H. Brunner, 1896-1996: One hundred years of Volterra integral equation of the rst kind, Appl. Numer. Math., 24 (1997) 83-93.
  • A. M. Wazwaz, Linear and Nonlinear Integral Equations Methods and Applications, Springer, Heidelberg, Dordrecht, London, New York, 2011.
  • A. Wazwaz, A First Course in Integral Equations, World Scientic Publishing, Singapore, 1997.
  • A. Wazwaz, A reliable modication of Adomian decomposition method, Appl. Math. Comput., 102 (1999) 77-86.
  • A. D. Polyanin and A.V. Manzhirov, Handbook of Integral Equations, CRC Press, Boca Raton, 1998.
  • R. Metzler and J. Klafter, The restaurant at the end of the random walk: Recent developments in the description of anomalous transport by fractional dynamics, J. Phys. A, 37 (2004) 161-208.
  • R. T. Baillie, Long memory processes and fractional integration in econometrics, J. Econometrics, 73 (1996) 5-59.
  • R. L. Magin, Fractional Calculus in Bioengineering, Begell House Publishers, 2006.
  • D. A. Benson, S.W. Wheatcraft and M.M. Meerschaert, Application of a fractional advection-dispersion equation, Water Resour. Res., 36 (2000) 1403-1412.
  • S. Abbas and M. Benchohra, Fractional order integral equations of two independent variables, Appl. Math. Comput., 227 (2014) 755-761.
  • S. Abbas and M. Benchohra, Fractional Order Riemann-Liouville Integral Equations with Multiple Time Delays, Applied Mathematics, 12 (2012) 79-87.
  • R. Hilfer, Applications of Fractional Calculus in Physics, World Scientic, Singapore, 2000.
  • Y.L. Jiang and X.L Ding, Waveform relaxation methods for fractional dierential equations with the Caputo derivatives, J. Comput. Appl. Math., 238 (2013) 51-67.
  • A.A. Kilbas, H.M. Srivastava and J.J. Trujillo, Theory and Applications of Fractional Dierential Equations, Elsevier, 2006.
  • H. Wang and N. Du, Fast alternating-direction nite dierence methods for threedimensional space-fractional diusion equations, J. Comput. Phys., 258 (2014) 305-318.
  • J. Liu and G. Hou, Numerical solutions of the space- and time-fractional coupled Burgers equation by generalized dierential transform method, Appl. Math. Comput., 217 (2011) 7001-7008.
  • L.Wang, Y. Ma and Z. Meng, Haar wavelet method for solving fractional partial dierential equations numerically, Appl. Math. Comput., 227 (2014) 66-76
  • R.K. Pandey, O.P. Singh and V.K. Baranwal, An analytic algorithm for the space-time fractional advection-dispersion equation, Comput. Phys. Commun. 182 (2011) 1134-1144.
  • I. Area, J. D. Djida, J. Losada and J.J. Nieto, On Fractional Orthonormal Polynomials of a Discrete Variable, Discrete Dynamics in Nature and Society 2015 (2015), Article ID 141325, 7 pages
  • I. Podlubny, Fractional Dierential Equations, Academic Press, San Diego, (1999).
  • M.A. Abdelkawy, M.A. Zaky, A.H. Bhrawy and D. Baleanu, Numerical simulation of time variable fractional order mobile-immobile advection-dispersion model, Rom. Rep. Phys., 67 (3) 2015.
  • A.H. Bhrawy and M.A. Zaky, Numerical simulation for two-dimensional variable-order fractional nonlinear cable equation, Nonlinear Dyn., 80(1) (2015) 101-116
  • A.H. Bhrawy, Engy A. Ahmed and D. Baleanu, An ecient collocation technique for solving generalized Fokker-Planck type equations with variable coecients, P. Rom. Acad. A, 15 (2014) 322-330
  • W. Shaoa and X. Wu, An eective Chebyshev tau meshless domain decomposition method based on the integration-dierentiation for solving fourth order equations, J. Comput. Phys., 231(2012) 7695-7714
  • J. Chena, Y. Huanga, H. Ronga, T. Wub and T. Zen, A multiscale Galerkin method for second-order boundary value problems of Fredholm integro-dierential equation, J. Comput. Appl. Math., 290 (2015) 633-640
  • Y. Chen and J. Zhou, Error estimates of spectral Legendre-Galerkin methods for the fourthorder equation in one dimension, Appl. Math. Comput., 268 (2015) 1217-1226
  • G. Kitzler and J. Schöberl, A high order space-momentum discontinuous Galerkin method for the Boltzmann equation, 70 (2015)1539-1554
  • A.H. Bhrawy, An ecient Jacobi pseudospectral approximation for nonlinear complex generalized Zakharov system, Appl. Math. Comput., 247 (2014) 30-46.
  • E.H. Doha, A.H. Bhrawy, M.A. Abdelkawy and R.A.V. Gorder, Jacobi-Gauss-Lobatto collocation method for the numerical solution of 1+1 nonlinear Schrödinger equations, J. Comput. Phys., 261 (2014)244-255.
  • E.H. Doha, A.H. Bhrawy and S.S. Ezz-Eldien, A new Jacobi operational matrix: An application for solving fractional dierential equations, Appl. Math. Model., 36 (2012) 4931-4943.
  • A.H. Bhrawy, E.H. Doha, D. Baleanu and S.S. Ezz-Eldien, A spectral tau algorithm based on Jacobi operational matrix for numerical solution of time fractional diusion-wave equations, J. Comput. Phys.,293 (2015) 142-156.
  • E.H. Doha, On the construction of recurrence relations for the expansion and connection coecients in series of Jacobi polynomials, J. Phys. A Math. Gen. 37 (2004) 657-675.
  • A.H. Bhrawy, M.A. Zaky and D. Baleanu, New numerical approximations for space-time fractional Burgers' equations via a Legendre spectral-collocation method, Rom. Rep. Phys., 67(2) (2015).
  • A.H. Bhrawy, M. A. Abdelkawy, J. Tenreiro Machado and A. Z. M. Amin, Legender-Gauss- Lobatto collocation method for solving multi-dimensional Fredholm integral equations., Comput. Math. Appl., Doi: 10.1016/j.camwa.2016.04.011
  • M.R. Eslahchi, M. Dehghan and M. Parvizi, Application of the collocation method for solving nonlinear fractional integro-dierential equations,J. Comput. Appl. Math., 257 (2014) 105-128.
  • J. Shen, T. Tang and L. L. Wang, SPECTRAL METHODS Algorithms, Analyses and Applications, (Springer; 2011 edition August 31, 2011).
  • C. Canuto, M. Y. Hussaini, A. Quarteroni and T.A. Zang, Spectral Methods: fundamentals in single domains, (Springer-Verlag, Berlin, Heidelberg, 2006).
  • M. A. Abdelkawy, S. S. Ezz-Eldien and A. Z. M.Amin, A Jacobi Spectral Collocation Scheme for Solving Abel's Integral Equations, Progr. Fract. Dier. Appl.1, 3 (2015) 187- 200.
  • A. H. Bhrawy, A.A. Al-Zahrani, Y.A. Alhamed and D. Baleanu, A new generalized Laguerre-Gauss collocation scheme for numerical solution of generalized fractional Pantograph equations, Rom. J. Phys., 59 (2014) 646-657
  • S. R. Lau and R. H. Price, Sparse spectral-tau method for the three-dimensional helically reduced wave equation on two-center domains, J. Comput. Phys., 231(2012) 7695-7714
  • F. Ghoreishi and S. Yazdani, An extension of the spectral Tau method for numerical solution of multi-order fractional dierential equations with convergence analysis, Comput. Math. Appl., 61 (2011) 30-43
  • E. H. Doha and A. H. Bhrawy, An ecient direct solver for multidimensional elliptic Robin boundary value problems using a Legendre spectral-Galerkin method, Comput. Math. Appl., 64 (2012) 558-571
  • Y. Luke, The Special Functions and Their Approximations, vol. 2, Academic Press, New York, (1969).
  • E. H. Doha, A. H. Bhrawy and R. M. Hafez, A Jacobi-Jacobi dual-Petrov-Galerkin method for third- and fth-order dierential equations, Math. Comput. Model, 53 (2011) 1820- 1832.
  • I. Area, D.K. Dimitrov, E. Godoy, V.G. Paschoa, Approximate Calculation of Sums I: Bounds for the Zeros of Gram Polynomials, SIAM J. Numer. Anal., 54(4), (2014)2210- 2227.
  • I. Area, D.K. Dimitrov, E. Godoy and V.G. Paschoa, Approximate Calculation of Sums II: Gaussian Type Quadrature, SIAM J. Numer. Anal., 52(4), (2016)1867-1886.
  • E.H. Doha and A.H. Bhrawy, An ecient direct solver for multidimensional elliptic Robin boundary value problems using a Legendre spectral-Galerkin method, Computers and Mathematics with Applications 64 (2012) 558-571.
  • E. Godoy, A. Ronveaux, A. Zarzo and I. Area, Minimal recurrence relations for connection coecients between classical orthogonal polynomials: Continuous case, Journal of Computational and Applied Mathematics 84 (1997) 257-275.
  • A. Ronveaux, A. Zarzo, I. Area and E. Godoy, Transverse limits in the Askey tableau, Journal of Computational and Applied Mathematics 99 (1998) 327-335.
There are 50 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Research Article
Authors

Ali H. Bhrawy

Mohamed A. Abdelkawy

Dumitru Baleanu

Ahmed Z.m. Amin This is me

Publication Date June 1, 2018
IZ https://izlik.org/JA74KB25AF
Published in Issue Year 2018 Volume: 47 Issue: 3

Cite

APA Bhrawy, A. H., Abdelkawy, M. A., Baleanu, D., & Amin, A. Z. (2018). A spectral technique for solving two-dimensional fractional integral equations with weakly singular kernel. Hacettepe Journal of Mathematics and Statistics, 47(3), 553-566. https://izlik.org/JA74KB25AF
AMA 1.Bhrawy AH, Abdelkawy MA, Baleanu D, Amin AZ. A spectral technique for solving two-dimensional fractional integral equations with weakly singular kernel. Hacettepe Journal of Mathematics and Statistics. 2018;47(3):553-566. https://izlik.org/JA74KB25AF
Chicago Bhrawy, Ali H., Mohamed A. Abdelkawy, Dumitru Baleanu, and Ahmed Z.m. Amin. 2018. “A Spectral Technique for Solving Two-Dimensional Fractional Integral Equations With Weakly Singular Kernel”. Hacettepe Journal of Mathematics and Statistics 47 (3): 553-66. https://izlik.org/JA74KB25AF.
EndNote Bhrawy AH, Abdelkawy MA, Baleanu D, Amin AZ (June 1, 2018) A spectral technique for solving two-dimensional fractional integral equations with weakly singular kernel. Hacettepe Journal of Mathematics and Statistics 47 3 553–566.
IEEE [1]A. H. Bhrawy, M. A. Abdelkawy, D. Baleanu, and A. Z. Amin, “A spectral technique for solving two-dimensional fractional integral equations with weakly singular kernel”, Hacettepe Journal of Mathematics and Statistics, vol. 47, no. 3, pp. 553–566, June 2018, [Online]. Available: https://izlik.org/JA74KB25AF
ISNAD Bhrawy, Ali H. - Abdelkawy, Mohamed A. - Baleanu, Dumitru - Amin, Ahmed Z.m. “A Spectral Technique for Solving Two-Dimensional Fractional Integral Equations With Weakly Singular Kernel”. Hacettepe Journal of Mathematics and Statistics 47/3 (June 1, 2018): 553-566. https://izlik.org/JA74KB25AF.
JAMA 1.Bhrawy AH, Abdelkawy MA, Baleanu D, Amin AZ. A spectral technique for solving two-dimensional fractional integral equations with weakly singular kernel. Hacettepe Journal of Mathematics and Statistics. 2018;47:553–566.
MLA Bhrawy, Ali H., et al. “A Spectral Technique for Solving Two-Dimensional Fractional Integral Equations With Weakly Singular Kernel”. Hacettepe Journal of Mathematics and Statistics, vol. 47, no. 3, June 2018, pp. 553-66, https://izlik.org/JA74KB25AF.
Vancouver 1.Bhrawy AH, Abdelkawy MA, Baleanu D, Amin AZ. A spectral technique for solving two-dimensional fractional integral equations with weakly singular kernel. Hacettepe Journal of Mathematics and Statistics [Internet]. 2018 June 1;47(3):553-66. Available from: https://izlik.org/JA74KB25AF

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