# Existence of solution for Hadamard proportional fractional integral equations by Fixed point ‎theorem

Document Type : Original Paper

Author

Department of Mathematics & amp, lrm, Ashtian Branch & amp;lrm, & amp, lrm, Islamic Azad University& amp, lrm, AshtianIran

Abstract

‎‎‎In this article‎, ‎using the technique of the measure of non-compactness and the Petryshyn's fixed point theorem in Banach algebra an existence theorem for some Hadamard proportional fractional integral equations is provided.‎ The study of these integral equations are important because they contain lots of particular cases of integral equations that arise in many branches of nonlinear analysis and its applications. ‎Comparing Petryshyn's fixed point theorem to Schauder and‎ ‎Darbo's fixed point theorems‎, ‎that is, it enables us to skip demonstrating closed‎, ‎convex, and compactness‎ ‎properties on the investigated operators‎. ‎‎ Finally, some examples are provided for the accuracy and efficiency of the obtained results.

Highlights

[1] Abbaszadeh, M., & Dehghan, M. 2017. An improved meshless method for solving two­dimensional distributed order time­fractional diffusion­wave equation with error estimate, Numer. Algor., 75, 173­-211. doi:10.1007/s11075­016­0201­0
[2] Aghdam, Y. E., Mesgarani, H., Asadi, Z., & Nguyen, V. T. 2023. Investigation and analysis of the
numerical approach to solve the multi­term time­fractional advection­diffusion model. AIMS Mathematics, 8(12), 29474­29489. doi:10.3934/math.20231509
[3] Ansari, A., & Moradi, M. 2013. Exact solutions to some models of distributed­order time fractional diffusion equations via the Fox H functions, ScienceAsia 39S, 57­66. doi: 10.2306/scienceasia1513­1874.2013.39S.057

[4] Chen, H., Lü, S., & Chen, W. 2016. Finite difference/spectral approximations for the distributed order time fractional reaction­diffusion equation on an unbounded domain, J. Comput. Phys., 315, 84­ -97.doi:10.1016/j.jcp.2016.03.044
[5] Chen, M., & Deng, W. 2014. A second­order numerical method for two­dimensional two­sided
space fractional convection diffusion equation. Applied Mathematical Modelling, 38(13), 3244­3259.doi:10.1016/j.apm.2013.11.043
[6] Chen, M., Deng, W., & Wu, Y. 2013. Superlinearly convergent algorithms for the two­dimensional space­time Caputo­Riesz fractional diffusion equation. Applied Numerical Mathematics, 70, 22­41. doi:10.1016/j.apnum.2013.03.006
[7] Esmaeelzade Aghdam, Y., Mesgarani , H., & Asadi, Z. 2023. Estimate of the fractional advectiondiffusion equation with a time­fractional term based on the shifted Legendre polynomials, ”J. Math. Model.”, 11(4), 731­744. doi: 10.22124/jmm.2023.24479.2191
[8] Gorenflo, R., Luchko, Yu., Stojanović, M. 2013. Fundamental solution of a distributed order timefractional diffusion­wave equation as probability density, Fract. Calc. Appl. Anal., 16, 297­316.
doi:10.2478/s13540­013­0019­6
[9] Hilfer, R. 2000. ”Applications of fractional calculus in physics”. Singapore: World Scientific.
[10] Jafari, H., Aghdam, Y. E., Farnam, B., Nguyen, V. T., & Masetshaba, M. T. 2023. A convergence
analysis of the mobile­ immobile advection­dispersion model of temporal fractional order arising in
watershed catchments and rivers, Fractals, 31(04), 2340068. doi:10.1142/S0218348X23400686
[11] Li, C., & Zeng, F. 2015. Numerical methods for fractional calculus. Boca Raton, FL: CRC Press,
Taylor & Francis Group.
[12] Li, Z., Luchko, Yu, & Yamamoto, M. 2014. Asymptotic estimates of solutions to initial­boundaryvalue problems for distributed order time­fractional diffusion equations, Fract. Calc. Appl. Anal., 17, 1114­1136. doi:10.2478/s13540­014­0217­x
[13] Luchko, Yu. 2011. Boundary value problems for the generalized time­fractional diffusion equation of distributed order, Fract. Calc. Appl. Anal., 12, 409­422. doi: 10.1016/j.jmaa.2010.08.048
[14] Mesgarani, H., Aghdam, Y. E., Khoshkhahtinat, M., & Farnam, B. 2023. Analysis of the numerical scheme of the one­dimensional fractional Rayleigh–Stokes model arising in a heated generalized problem. AIP Advances, 13(8), 085024. doi: 10.1063/5.0156586
[15] Magin, R. L. 2021. ”Fractional calculus in bioengineering”. New York: Begell.
[16] McLean, W., 2010. Regularity of solutions to a time­fractional diffusion equation, ANZIAM J. 52. 123–138. doi:10.1017/S1446181111000617
[17] Meerschaert, M .M., Nane, E., Vellaisamy, P. 2011. Distributed­order fractional diffusions on
bounded domains, J. Math. Anal. Appl., 379, 216­228. doi:10.1016/j.jmaa.2010.12.056
[18] Mesgarani, H., Aghdam, Y. E., & Vafapisheh, M. 2023. A numerical procedure for approximating time fractional nonlinear Burgers–Fisher models and its error analysis. AIP Advances, 13(5), 055313. doi: 10.1063/5.0143690
[19] Miller, K. S., & Ross, B. 1993. ”An introduction to the fractional calculus and fractional differential equations”. New York, NY: Wiley

[20] Ren, J., & Chen, H. 2019. A numerical method for distributed order time fractional diffusion equation with weakly singular solutions, Appl. Math. Lett., 96, 159–165. doi:10.1016/j.aml.2019.04.030
[21] Stynes, M. ÓRiordan, E. & Gracia, J. L. 2017. Error analysis of a finite difference method on
graded meshes for a time­fractional diffusion equation, SIAM J. Numer. Anal., 55, 1057–1079.
doi:10.1137/16M1082329
[22] Wei, L., Liu, L., & Sun, H. 2018. Stability and convergence of a local discontinuous Galerkin method for the fractional diffusion equation with distributed order, J. Appl. Math. Comput., 59, 323­341.doi:10.1007/s12190­018­1182­z
[23] Ye, H. Liu, F., & Anh, V. 2015. Compact difference scheme for distributed­order timefractional diffusion­wave equation on bounded domains, J. Comput. Phys., 98, 652–660.
doi:10.1016/j.jcp.2015.06.025
[24] Zhang, Y., Sun, Z., & Liao, H. 2014. Finite difference methods for the time fractional
diffusion equation on non­uniform meshes. Journal of Computational Physics, 265, 195­210.
doi:10.1016/j.jcp.2014.02.008

Keywords

Main Subjects

#### References

[1] Abbaszadeh, M., & Dehghan, M. 2017. An improved meshless method for solving two­dimensional distributed order time­fractional diffusion­wave equation with error estimate, Numer. Algor., 75, 173­-211. doi:10.1007/s11075­016­0201­0
[2] Aghdam, Y. E., Mesgarani, H., Asadi, Z., & Nguyen, V. T. 2023. Investigation and analysis of the
numerical approach to solve the multi­term time­fractional advection­diffusion model. AIMS Mathematics, 8(12), 29474­29489. doi:10.3934/math.20231509
[3] Ansari, A., & Moradi, M. 2013. Exact solutions to some models of distributed­order time fractional diffusion equations via the Fox H functions, ScienceAsia 39S, 57­66. doi: 10.2306/scienceasia1513­1874.2013.39S.057
[4] Chen, H., Lü, S., & Chen, W. 2016. Finite difference/spectral approximations for the distributed order time fractional reaction­diffusion equation on an unbounded domain, J. Comput. Phys., 315, 84­ -97.doi:10.1016/j.jcp.2016.03.044
[5] Chen, M., & Deng, W. 2014. A second­order numerical method for two­dimensional two­sided
space fractional convection diffusion equation. Applied Mathematical Modelling, 38(13), 3244­3259.doi:10.1016/j.apm.2013.11.043
[6] Chen, M., Deng, W., & Wu, Y. 2013. Superlinearly convergent algorithms for the two­dimensional space­time Caputo­Riesz fractional diffusion equation. Applied Numerical Mathematics, 70, 22­41. doi:10.1016/j.apnum.2013.03.006
[7] Esmaeelzade Aghdam, Y., Mesgarani , H., & Asadi, Z. 2023. Estimate of the fractional advectiondiffusion equation with a time­fractional term based on the shifted Legendre polynomials, ”J. Math. Model.”, 11(4), 731­744. doi: 10.22124/jmm.2023.24479.2191
[8] Gorenflo, R., Luchko, Yu., Stojanović, M. 2013. Fundamental solution of a distributed order timefractional diffusion­wave equation as probability density, Fract. Calc. Appl. Anal., 16, 297­316.
doi:10.2478/s13540­013­0019­6
[9] Hilfer, R. 2000. ”Applications of fractional calculus in physics”. Singapore: World Scientific.
[10] Jafari, H., Aghdam, Y. E., Farnam, B., Nguyen, V. T., & Masetshaba, M. T. 2023. A convergence
analysis of the mobile­ immobile advection­dispersion model of temporal fractional order arising in
watershed catchments and rivers, Fractals, 31(04), 2340068. doi:10.1142/S0218348X23400686
[11] Li, C., & Zeng, F. 2015. Numerical methods for fractional calculus. Boca Raton, FL: CRC Press,
Taylor & Francis Group.
[12] Li, Z., Luchko, Yu, & Yamamoto, M. 2014. Asymptotic estimates of solutions to initial­boundaryvalue problems for distributed order time­fractional diffusion equations, Fract. Calc. Appl. Anal., 17, 1114­1136. doi:10.2478/s13540­014­0217­x
[13] Luchko, Yu. 2011. Boundary value problems for the generalized time­fractional diffusion equation of distributed order, Fract. Calc. Appl. Anal., 12, 409­422. doi: 10.1016/j.jmaa.2010.08.048
[14] Mesgarani, H., Aghdam, Y. E., Khoshkhahtinat, M., & Farnam, B. 2023. Analysis of the numerical scheme of the one­dimensional fractional Rayleigh–Stokes model arising in a heated generalized problem. AIP Advances, 13(8), 085024. doi: 10.1063/5.0156586
[15] Magin, R. L. 2021. ”Fractional calculus in bioengineering”. New York: Begell.
[16] McLean, W., 2010. Regularity of solutions to a time­fractional diffusion equation, ANZIAM J. 52. 123–138. doi:10.1017/S1446181111000617
[17] Meerschaert, M .M., Nane, E., Vellaisamy, P. 2011. Distributed­order fractional diffusions on
bounded domains, J. Math. Anal. Appl., 379, 216­228. doi:10.1016/j.jmaa.2010.12.056
[18] Mesgarani, H., Aghdam, Y. E., & Vafapisheh, M. 2023. A numerical procedure for approximating time fractional nonlinear Burgers–Fisher models and its error analysis. AIP Advances, 13(5), 055313. doi: 10.1063/5.0143690
[19] Miller, K. S., & Ross, B. 1993. ”An introduction to the fractional calculus and fractional differential equations”. New York, NY: Wiley
[20] Ren, J., & Chen, H. 2019. A numerical method for distributed order time fractional diffusion equation with weakly singular solutions, Appl. Math. Lett., 96, 159–165. doi:10.1016/j.aml.2019.04.030
[21] Stynes, M. ÓRiordan, E. & Gracia, J. L. 2017. Error analysis of a finite difference method on
graded meshes for a time­fractional diffusion equation, SIAM J. Numer. Anal., 55, 1057–1079.
doi:10.1137/16M1082329
[22] Wei, L., Liu, L., & Sun, H. 2018. Stability and convergence of a local discontinuous Galerkin method for the fractional diffusion equation with distributed order, J. Appl. Math. Comput., 59, 323­341.doi:10.1007/s12190­018­1182­z
[23] Ye, H. Liu, F., & Anh, V. 2015. Compact difference scheme for distributed­order timefractional diffusion­wave equation on bounded domains, J. Comput. Phys., 98, 652–660.
doi:10.1016/j.jcp.2015.06.025
[24] Zhang, Y., Sun, Z., & Liao, H. 2014. Finite difference methods for the time fractional
diffusion equation on non­uniform meshes. Journal of Computational Physics, 265, 195­210.
doi:10.1016/j.jcp.2014.02.008

### History

• Receive Date: 21 October 2023
• Revise Date: 07 April 2024
• Accept Date: 12 May 2024