The price adjustment equation with different types of conformable derivatives in market equilibrium

Erdal Bas; Bahar Acay; Ramazan Ozarslan; Erdal Bas; Bahar Acay; Ramazan Ozarslan

doi:10.3934/math.2019.3.805

AIMS Mathematics

2019, Volume 4, Issue 3: 805-820. doi: 10.3934/math.2019.3.805

Previous Article Next Article

Research article

The price adjustment equation with different types of conformable derivatives in market equilibrium

Department of Mathematics, Firat University, 23119, Elazig, Turkey

Received: 03 May 2019 Accepted: 04 July 2019 Published: 12 July 2019
MSC : 97M30, 97M10, 97M99

In the current study, price adjustment equation which takes an important place in market equilibrium is presented in consideration of truncated M-derivative including Mittag-Leffler function, beta-derivative and conformable derivative defined in the form of limit for α-differentiable functions. These popular limit-based derivative and integral definitions enable α to vary between (0; 1], whereupon we can observe the intrinsic behavior of the competitive market at different times. The reason of popularity of the underlying definitions is that the natural appearances of their applications. Due to their similarity to classical derivative, it can be taken a good deal of advantages of them in terms of applicability to the diverse governing models. Hence we derive some novel solutions of the market equilibrium models which are big parts of our lives and in order to solve the linear ordinary differential equations in the sense of M-derivative and beta derivative, the solution methods are given. Moreover, we carry out simulation analysis in order to confirm the usefulness of results obtained.
- stability,
- time paths,
- M-derivative,
- beta-derivative,
- market equilibrium
Citation: Erdal Bas, Bahar Acay, Ramazan Ozarslan. The price adjustment equation with different types of conformable derivatives in market equilibrium[J]. AIMS Mathematics, 2019, 4(3): 805-820. doi: 10.3934/math.2019.3.805

Related Papers:

Abstract

In the current study, price adjustment equation which takes an important place in market equilibrium is presented in consideration of truncated M-derivative including Mittag-Leffler function, beta-derivative and conformable derivative defined in the form of limit for α-differentiable functions. These popular limit-based derivative and integral definitions enable α to vary between (0; 1], whereupon we can observe the intrinsic behavior of the competitive market at different times. The reason of popularity of the underlying definitions is that the natural appearances of their applications. Due to their similarity to classical derivative, it can be taken a good deal of advantages of them in terms of applicability to the diverse governing models. Hence we derive some novel solutions of the market equilibrium models which are big parts of our lives and in order to solve the linear ordinary differential equations in the sense of M-derivative and beta derivative, the solution methods are given. Moreover, we carry out simulation analysis in order to confirm the usefulness of results obtained.

References

[1]	A. Takayama, Mathematical economics, Cambridge University Press, 1985.
[2]	A. C. Chiang, Fundamental methods of mathematical economics, 1984.
[3]	R. K. Nagle , E. B. Staff, A. D. Snider, Fundamentals Differential Equations, Pearson, 2008.
[4]	R. Khalil, M. Al Horani, A. Yousef, et al. A new definition of fractional derivative, J. Comput. Appl. Math., 264 (2014), 65-70. doi: 10.1016/j.cam.2014.01.002
[5]	T. Abdeljawad, On conformable fractional calculus, J. Comput. Appl. Math., 279 (2015), 57-66. doi: 10.1016/j.cam.2014.10.016
[6]	T. Abdeljawad, J. Alzabut, F. Jarad, A generalized Lyapunov-type inequality in the frame of conformable derivatives, Adv. Differ. Equ-Ny, 2017 (2017), 321.
[7]	T. Abdeljawad, R. P. Agarwal, J. Alzabut, et al. Lyapunov-type inequalities for mixed non-linear forced differential equations within conformable derivatives, J. Inequal. Appl., 2018 (2018), 143.
[8]	F. M. Alharbi, D. Baleanu, A. Ebaid, Physical properties of the projectile motion using the conformable derivative, Chinese J. Phys., 58 (2019), 18-28. doi: 10.1016/j.cjph.2018.12.010
[9]	A. Atangana, D. Baleanu, A. Alsaedi, New properties of conformable derivative, Open Math., 13 (2015), 889-898.
[10]	M. Ilie, J. Biazar, Z. Ayati, General solution of Bernoulli and Riccati fractional differential equations based on conformable fractional derivative, International Journal of Applied Mathematical Research, 6 (2017), 49-51. doi: 10.14419/ijamr.v6i2.7014
[11]	E. Bas, R. Ozarslan, Real world applications of fractional models by Atangana-Baleanu fractional derivative, Chaos Soliton. Fract., 116 (2018), 121-125. doi: 10.1016/j.chaos.2018.09.019
[12]	E. Bas, R. Ozarslan, D. Baleanu, et al. Comparative simulations for solutions of fractional Sturm-Liouville problems with non-singular operators, Adv. Differ. Equ-Ny, 2018 (2018), 350.
[13]	E. Bas, The Inverse Nodal problem for the fractional diffusion equation, Acta Sci-Technol, 37 (2015), 251-257. doi: 10.4025/actascitechnol.v37i2.17273
[14]	E. Bas, B. Acay, R. Ozarslan, Fractional models with singular and non-singular kernels for energy efficient buildings, Chaos, 29 (2019), 023110.
[15]	A. Yusuf, S. Qureshi, M. Inc, et al. Two-strain epidemic model involving fractional derivative with Mittag-Leffler kernel, Chaos, 28 (2018), 123121.
[16]	S. Qureshi, A. Yusuf, A. A. Shaikh, et al. Fractional modeling of blood ethanol concentration system with real data application, Chaos, 29 (2019), 013143.
[17]	M. Al-Refai, T. Abdeljawad, Fundamental results of conformable Sturm-Liouville eigenvalue problems, Complexity, 2017 (2017), 1-7.
[18]	S. Qureshi, A. Yusuf, Modeling chickenpox disease with fractional derivatives: From caputo to Atangana-Baleanu, Chaos Soliton. Fract., 122 (2019), 111-118. doi: 10.1016/j.chaos.2019.03.020
[19]	S. Qureshi, A. Yusuf, Fractional derivatives applied to MSEIR problems: Comparative study with real world data, The European Physical Journal Plus, 134 (2019), 171.
[20]	S. Sakarya, M. Yavuz, A. D. Karaoglan, et al. Stock market index prediction with neural network during financial crises: a review on Bist-100, Financial Risk and Management Reviews, 1 (2015), 53-67. doi: 10.18488/journal.89/2015.1.2/89.2.53.67
[21]	M. Yavuz, Novel solution methods for initial boundary value problems of fractional order with conformable differentiation, An International Journal of Optimization and Control: Theories & Applications (IJOCTA), 8 (2017), 1-7.
[22]	A. Atangana, D. Baleanu, A. Alsaedi, Analysis of time-fractional Hunter-Saxton equation: a model of neumatic liquid crystal, Open Phys., 14 (2016), 145-149.
[23]	A. Atangana, A novel model for the lassa hemorrhagic fever: deathly disease for pregnant women, Neural Comput. Appl., 26 (2015), 1895-1903. doi: 10.1007/s00521-015-1860-9
[24]	U. N. Katugampola, A new fractional derivative with classical properties, arXiv preprint arXiv:1410.6535, 2014.
[25]	J. V. D. C. Sousa, E. C. de Oliveira, A new truncated M-fractional derivative type unifying some fractional derivative types with classical properties, Int. J. Anal. Appl., 16 (2018), 83-96.

Reader Comments

Your name:*

Email:*
© 2019 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)