Research Article
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Year 2019, Volume: 68 Issue: 1, 1061 - 1072, 01.02.2019
https://doi.org/10.31801/cfsuasmas.501582

Abstract

References

  • Machado, J. T., Kiryakova, V. and Mainardi, F., A poster about the old history of fractional calculus, Fract. Calc. Appl. Anal. 13 (2010), no. 4, 447--454.
  • Machado, J. T., Kiryakova, V. and Mainardi, F., Recent history of fractional calculus, Commun. Nonlinear Sci. Numer. Simul. 16 (2011), no. 3, 1140--1153.
  • Agarwal, R. P., Baleanu, Du., Nieto, J. J., Torres, D. F. M. and Zhou, Y., A survey on fuzzy fractional differential and optimal control nonlocal evolution equations, J. Comput. Appl. Math. 339 (2018), 3--29.
  • Ortigueira, M. D., Fractional calculus for scientists and engineers, Lecture Notes in Electrical Engineering, 84, Springer, Dordrecht, 2011.
  • Podlubny, I., Fractional differential equations, Mathematics in Science and Engineering, 198, Academic Press, Inc., San Diego, CA, 1999.
  • Baleanu, D., Diethelm, K., Scalas, E. and Trujillo, J. J., Fractional calculus, Series on Complexity, Nonlinearity and Chaos, 5, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2017.
  • Miller, K. S. and Ross, B., An introduction to the fractional calculus and fractional differential equations, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1993.
  • Samko, S. G., Kilbas, A. A. and Marichev, O. I., Fractional integrals and derivatives, translated from the 1987 Russian original, Gordon and Breach Science Publishers, Yverdon, 1993.
  • Baleanu, D., Golmankhaneh, A. K., Golmankhaneh, A. K. and Nigmatullin, R. R., Newtonian law with memory, Nonlinear Dynam. 60 (2010), no. 1--2, 81--86.
  • Caputo, M., Linear models of dissipation whose Q is almost frequency independent. II, Geophys. J. R. Astr. Soc. 13 (1967), no. 5, 529--539.
  • Caputo, M. and Mainardi, F., Linear models of dissipation in anelastic solids, Riv. Nuovo Cimento (Ser. II) 1 (1971) 161--198.
  • Hilfer, R., Applications of fractional calculus in physics, World Scientific Publishing Co., Inc., River Edge, NJ, 2000.
  • Kilbas, A. A., Srivastava, H. M. and Trujillo, J. J., Theory and applications of fractional differential equations, North-Holland Mathematics Studies, 204, Elsevier Science B.V., Amsterdam, 2006.
  • Mainardi, F., Fractional calculus: some basic problems in continuum and statistical mechanics, in Fractals and fractional calculus in continuum mechanics (Udine, 1996), 291--348, CISM Courses and Lect., 378, Springer, Vienna, 1997.
  • Khalil, R., Al Horani, M., Yousef. A. and Sababheh, M., A new definition of fractional derivative, J. Comput. Appl. Math. 264 (2014), 65--70.
  • Abdeljawad, T., On conformable fractional calculus, J. Comput. Appl. Math. 279 (2015), 57--66. chung : Chung, W. S. Fractional Newton mechanics with conformable fractional derivative, J. Comput. Appl. Math. 290 (2015), 150--158.
  • Ünal, E., Gökdogan, A. and Çelik, E., Solutions of sequential conformable fractional differential equations around an ordinary point and conformable fractional Hermite differential equation, British J. Appl. Science & Tech. 10 (2015), 1--11.
  • Benkhettou, N., Hassani, S. and Torres, D. F. M., A conformable fractional calculus on arbitrary time scales, J. King Saud Univ. Sci. 28 (2016), no. 1, 93--98.
  • Eslami, M. and Rezazadeh, H., The first integral method for Wu-Zhang system with conformable time-fractional derivative, Calcolo 53 (2016), no. 3, 475--485.
  • Lazo, M. J. and Torres, D. F. M., Variational calculus with conformable fractional derivatives, IEEE/CAA J. Autom. Sin. 4 (2017), no. 2, 340--352.
  • Bayour, B., Hammoudi, A. and Torres, D. F. M., A truly conformable calculus on time scales, Glob. Stoch. Anal. 5 (2018), no. 1, 1--14.
  • Feng, Q. and Meng, F., Oscillation results for a fractional order dynamic equation on time scales with conformable fractional derivative, Adv. Difference Equ. 2018, 2018:193.
  • Gholami, Y. and Ghanbari, K., New class of conformable derivatives and applications to differential impulsive systems, SeMA J. 75 (2018), no. 2, 305--333.
  • Kareem, A. M., Conformable fractional derivatives and it is applications for solving fractional differential equations, IOSR J. Math. 13 (2017), 81--87.
  • Ünal, E. and Gökdogan, A., Solution of conformable fractional ordinary differential equations via differential transform method, Optik -- Int. J. Light and Elect. Optics, 128 (2017), 264--273.
  • Rochdi, K., Solution of some conformable fractional differential equations, Int. J. Pure Appl. Math. 103 (2015), no. 4, 667--673.
  • Bartosz, K., Janiczko, T., Szafraniec, P. and Shillor, M., Dynamic thermoviscoelastic thermistor problem with contact and nonmonotone friction, Appl. Anal. 97 (2018), no. 8, 1432--1453.
  • Hrynkiv, V. and Turchaninova, A., Analytical solution of a one-dimensional thermistor problem with Robin boundary condition, Involve 12 (2019), no. 1, 79--88.
  • Mbehou, M., The theta-Galerkin finite element method for coupled systems resulting from microsensor thermistor problems, Math. Methods Appl. Sci. 41 (2018), no. 4, 1480--1491.
  • Sidi Ammi, M. R. and Torres, D. F. M., Existence and uniqueness of a positive solution to generalized nonlocal thermistor problems with fractional-order derivatives, Differ. Equ. Appl. 4 (2012), no. 2, 267--276.
  • Sidi Ammi, M. R., Jamiai, I. and Torres, D. F. M., Global existence of solutions for a fractional Caputo nonlocal thermistor problem, Adv. Difference Equ. 2017 (2017), no. 363, 14 pp.
  • Sidi Ammi, M. R. and Torres, D. F. M., Existence and uniqueness results for a fractional Riemann-Liouville nonlocal thermistor problem on arbitrary time scales, J. King Saud Univ. Sci. 30 (2018), no. 3, 381--385.
  • Vivek, D., Kanagarajan, K., Sivasundaram, S., Dynamics and stability results for Hilfer fractional type thermistor problem, Fractal Fract. 1 (2017), 5, 14 pp.
  • Bayour, B. and Torres, D. F. M., Existence of solution to a local fractional nonlinear differential equation, J. Comput. Appl. Math. 312 (2017), 127--133.
  • Granas, A. and Dugundji, J, Fixed point theory, Springer Monographs in Mathematics, Springer, New York, 2003.
  • Li, C. and Sarwar, S., Existence and continuation of solutions for Caputo type fractional differential equations, Electron. J. Differential Equations 2016 (2016), Paper No. 207, 14 pp.

Existence of solution to a nonlocal conformable fractional thermistor problem

Year 2019, Volume: 68 Issue: 1, 1061 - 1072, 01.02.2019
https://doi.org/10.31801/cfsuasmas.501582

Abstract

We study a nonlocal thermistor problem for fractional derivatives in the conformable sense. Classical Schauder's fixed point theorem is used to derive the existence of a tube solution.

References

  • Machado, J. T., Kiryakova, V. and Mainardi, F., A poster about the old history of fractional calculus, Fract. Calc. Appl. Anal. 13 (2010), no. 4, 447--454.
  • Machado, J. T., Kiryakova, V. and Mainardi, F., Recent history of fractional calculus, Commun. Nonlinear Sci. Numer. Simul. 16 (2011), no. 3, 1140--1153.
  • Agarwal, R. P., Baleanu, Du., Nieto, J. J., Torres, D. F. M. and Zhou, Y., A survey on fuzzy fractional differential and optimal control nonlocal evolution equations, J. Comput. Appl. Math. 339 (2018), 3--29.
  • Ortigueira, M. D., Fractional calculus for scientists and engineers, Lecture Notes in Electrical Engineering, 84, Springer, Dordrecht, 2011.
  • Podlubny, I., Fractional differential equations, Mathematics in Science and Engineering, 198, Academic Press, Inc., San Diego, CA, 1999.
  • Baleanu, D., Diethelm, K., Scalas, E. and Trujillo, J. J., Fractional calculus, Series on Complexity, Nonlinearity and Chaos, 5, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2017.
  • Miller, K. S. and Ross, B., An introduction to the fractional calculus and fractional differential equations, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1993.
  • Samko, S. G., Kilbas, A. A. and Marichev, O. I., Fractional integrals and derivatives, translated from the 1987 Russian original, Gordon and Breach Science Publishers, Yverdon, 1993.
  • Baleanu, D., Golmankhaneh, A. K., Golmankhaneh, A. K. and Nigmatullin, R. R., Newtonian law with memory, Nonlinear Dynam. 60 (2010), no. 1--2, 81--86.
  • Caputo, M., Linear models of dissipation whose Q is almost frequency independent. II, Geophys. J. R. Astr. Soc. 13 (1967), no. 5, 529--539.
  • Caputo, M. and Mainardi, F., Linear models of dissipation in anelastic solids, Riv. Nuovo Cimento (Ser. II) 1 (1971) 161--198.
  • Hilfer, R., Applications of fractional calculus in physics, World Scientific Publishing Co., Inc., River Edge, NJ, 2000.
  • Kilbas, A. A., Srivastava, H. M. and Trujillo, J. J., Theory and applications of fractional differential equations, North-Holland Mathematics Studies, 204, Elsevier Science B.V., Amsterdam, 2006.
  • Mainardi, F., Fractional calculus: some basic problems in continuum and statistical mechanics, in Fractals and fractional calculus in continuum mechanics (Udine, 1996), 291--348, CISM Courses and Lect., 378, Springer, Vienna, 1997.
  • Khalil, R., Al Horani, M., Yousef. A. and Sababheh, M., A new definition of fractional derivative, J. Comput. Appl. Math. 264 (2014), 65--70.
  • Abdeljawad, T., On conformable fractional calculus, J. Comput. Appl. Math. 279 (2015), 57--66. chung : Chung, W. S. Fractional Newton mechanics with conformable fractional derivative, J. Comput. Appl. Math. 290 (2015), 150--158.
  • Ünal, E., Gökdogan, A. and Çelik, E., Solutions of sequential conformable fractional differential equations around an ordinary point and conformable fractional Hermite differential equation, British J. Appl. Science & Tech. 10 (2015), 1--11.
  • Benkhettou, N., Hassani, S. and Torres, D. F. M., A conformable fractional calculus on arbitrary time scales, J. King Saud Univ. Sci. 28 (2016), no. 1, 93--98.
  • Eslami, M. and Rezazadeh, H., The first integral method for Wu-Zhang system with conformable time-fractional derivative, Calcolo 53 (2016), no. 3, 475--485.
  • Lazo, M. J. and Torres, D. F. M., Variational calculus with conformable fractional derivatives, IEEE/CAA J. Autom. Sin. 4 (2017), no. 2, 340--352.
  • Bayour, B., Hammoudi, A. and Torres, D. F. M., A truly conformable calculus on time scales, Glob. Stoch. Anal. 5 (2018), no. 1, 1--14.
  • Feng, Q. and Meng, F., Oscillation results for a fractional order dynamic equation on time scales with conformable fractional derivative, Adv. Difference Equ. 2018, 2018:193.
  • Gholami, Y. and Ghanbari, K., New class of conformable derivatives and applications to differential impulsive systems, SeMA J. 75 (2018), no. 2, 305--333.
  • Kareem, A. M., Conformable fractional derivatives and it is applications for solving fractional differential equations, IOSR J. Math. 13 (2017), 81--87.
  • Ünal, E. and Gökdogan, A., Solution of conformable fractional ordinary differential equations via differential transform method, Optik -- Int. J. Light and Elect. Optics, 128 (2017), 264--273.
  • Rochdi, K., Solution of some conformable fractional differential equations, Int. J. Pure Appl. Math. 103 (2015), no. 4, 667--673.
  • Bartosz, K., Janiczko, T., Szafraniec, P. and Shillor, M., Dynamic thermoviscoelastic thermistor problem with contact and nonmonotone friction, Appl. Anal. 97 (2018), no. 8, 1432--1453.
  • Hrynkiv, V. and Turchaninova, A., Analytical solution of a one-dimensional thermistor problem with Robin boundary condition, Involve 12 (2019), no. 1, 79--88.
  • Mbehou, M., The theta-Galerkin finite element method for coupled systems resulting from microsensor thermistor problems, Math. Methods Appl. Sci. 41 (2018), no. 4, 1480--1491.
  • Sidi Ammi, M. R. and Torres, D. F. M., Existence and uniqueness of a positive solution to generalized nonlocal thermistor problems with fractional-order derivatives, Differ. Equ. Appl. 4 (2012), no. 2, 267--276.
  • Sidi Ammi, M. R., Jamiai, I. and Torres, D. F. M., Global existence of solutions for a fractional Caputo nonlocal thermistor problem, Adv. Difference Equ. 2017 (2017), no. 363, 14 pp.
  • Sidi Ammi, M. R. and Torres, D. F. M., Existence and uniqueness results for a fractional Riemann-Liouville nonlocal thermistor problem on arbitrary time scales, J. King Saud Univ. Sci. 30 (2018), no. 3, 381--385.
  • Vivek, D., Kanagarajan, K., Sivasundaram, S., Dynamics and stability results for Hilfer fractional type thermistor problem, Fractal Fract. 1 (2017), 5, 14 pp.
  • Bayour, B. and Torres, D. F. M., Existence of solution to a local fractional nonlinear differential equation, J. Comput. Appl. Math. 312 (2017), 127--133.
  • Granas, A. and Dugundji, J, Fixed point theory, Springer Monographs in Mathematics, Springer, New York, 2003.
  • Li, C. and Sarwar, S., Existence and continuation of solutions for Caputo type fractional differential equations, Electron. J. Differential Equations 2016 (2016), Paper No. 207, 14 pp.
There are 36 citations in total.

Details

Primary Language English
Journal Section Review Articles
Authors

Moulay Rchid Sidi Ammi This is me 0000-0002-4488-9070

Delfim F. M. Torres 0000-0001-8641-2505

Publication Date February 1, 2019
Submission Date June 20, 2018
Acceptance Date August 26, 2018
Published in Issue Year 2019 Volume: 68 Issue: 1

Cite

APA Ammi, M. R. S., & Torres, D. F. M. (2019). Existence of solution to a nonlocal conformable fractional thermistor problem. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 68(1), 1061-1072. https://doi.org/10.31801/cfsuasmas.501582
AMA Ammi MRS, Torres DFM. Existence of solution to a nonlocal conformable fractional thermistor problem. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. February 2019;68(1):1061-1072. doi:10.31801/cfsuasmas.501582
Chicago Ammi, Moulay Rchid Sidi, and Delfim F. M. Torres. “Existence of Solution to a Nonlocal Conformable Fractional Thermistor Problem”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 68, no. 1 (February 2019): 1061-72. https://doi.org/10.31801/cfsuasmas.501582.
EndNote Ammi MRS, Torres DFM (February 1, 2019) Existence of solution to a nonlocal conformable fractional thermistor problem. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 68 1 1061–1072.
IEEE M. R. S. Ammi and D. F. M. Torres, “Existence of solution to a nonlocal conformable fractional thermistor problem”, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., vol. 68, no. 1, pp. 1061–1072, 2019, doi: 10.31801/cfsuasmas.501582.
ISNAD Ammi, Moulay Rchid Sidi - Torres, Delfim F. M. “Existence of Solution to a Nonlocal Conformable Fractional Thermistor Problem”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 68/1 (February 2019), 1061-1072. https://doi.org/10.31801/cfsuasmas.501582.
JAMA Ammi MRS, Torres DFM. Existence of solution to a nonlocal conformable fractional thermistor problem. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2019;68:1061–1072.
MLA Ammi, Moulay Rchid Sidi and Delfim F. M. Torres. “Existence of Solution to a Nonlocal Conformable Fractional Thermistor Problem”. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, vol. 68, no. 1, 2019, pp. 1061-72, doi:10.31801/cfsuasmas.501582.
Vancouver Ammi MRS, Torres DFM. Existence of solution to a nonlocal conformable fractional thermistor problem. Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 2019;68(1):1061-72.

Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics.

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