وجود جواب برای معادلات انتگرالی کسری ‏‎‏تناسبی هادامارد با استفاده از قضیه نقطه ثابت

نوع مقاله : مقاله پژوهشی

نویسنده

عضو هیات علمی گروه ریاضی، دانشگاه آزاد اسلامی، آشتیان، ایران

چکیده

در این مقاله، با استفاده از اندازه نافشردگی و قضیه نقطه ثابت پترشن در فضای باناخ، یک قضیه وجودی برای برخی معادلات انتگرالی کسری تناسبی هادامارد، ارائه شده است. مطالعه این معادلات انتگرالی بسیار حائز اهمیت هستند چرا که دربرگیرنده موارد خاص زیادی از معادلات انتگرالی می‌باشند که در شاخه‌های زیادی از آنالیز غیر خطی و کاربردهای آن ظاهر می‌شوند. تفاوت قضیه نقطه ثابت پترشن با قضایای نقطه ثابت شاودر و نقطه ثابت داربو، دراین است که ما را قادر می‌سازد تا از نشان دادن خواص بسته، محدب و فشردگی عملگرهای مورد بررسی صرف نظر کنیم . در پایان، برای صحت و کارایی نتایج به‌دست آمده، چند مثال ارائه شده است.

کلیدواژه‌ها

موضوعات


عنوان مقاله [English]

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

نویسنده [English]

  • Manochehr Kazemi
Department of Mathematics & amp, lrm, Ashtian Branch & amp;lrm, & amp, lrm, Islamic Azad University& amp, lrm, AshtianIran
چکیده [English]

‎‎‎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.

کلیدواژه‌ها [English]

  • Hadamard proportional fractional integral equations‎
  • ‎Existence of solutions‎
  • ‎Measure of noncompactness‎
  • Fixed point ‎theorems
  • Fractional calculus
[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