SPECIAL ISSUE PAPER
Well-posedness and iterative formula for fractional oscillator equations with delays
Denghao Pang,
Wei Jiang,
Song Liu,
Azmat Ullah Khan Niazi,
Corresponding Author
Denghao Pang
School of Mathematical Sciences, Anhui University, Hefei, China
Correspondence
Denghao Pang, School of Mathematical Sciences, Anhui University, Hefei 230601, China.
Email: [email protected]
Communicated by: D. Zeidan
Search for more papers by this authorWei Jiang
School of Mathematical Sciences, Anhui University, Hefei, China
Search for more papers by this authorSong Liu
School of Mathematical Sciences, Anhui University, Hefei, China
Search for more papers by this authorAzmat Ullah Khan Niazi
School of Mathematical Sciences, Anhui University, Hefei, China
Search for more papers by this authorDenghao Pang,
Wei Jiang,
Song Liu,
Azmat Ullah Khan Niazi,
Corresponding Author
Denghao Pang
School of Mathematical Sciences, Anhui University, Hefei, China
Correspondence
Denghao Pang, School of Mathematical Sciences, Anhui University, Hefei 230601, China.
Email: [email protected]
Communicated by: D. Zeidan
Search for more papers by this authorWei Jiang
School of Mathematical Sciences, Anhui University, Hefei, China
Search for more papers by this authorSong Liu
School of Mathematical Sciences, Anhui University, Hefei, China
Search for more papers by this authorAzmat Ullah Khan Niazi
School of Mathematical Sciences, Anhui University, Hefei, China
Search for more papers by this authorAbstract
This paper investigates the well-posedness and iterative formula for fractional oscillator equations with two different fractional orders and time delays. By the mathematical induction, a new generalized Grönwall's inequality in terms of a multivariate Mittag-Leffler function is derived with an iteration argument. Subsequently, using the inequality, a generalized Mittag-Leffler estimation of the solution is proposed. Furthermore, the explicit solution and iterative formula, as well as the unique existence, are presented on the basis of the method of steps. Finally, an example is given to illustrate the validity of our main results.
REFERENCES
- 1Podlubny I. Fractional Differential Equations. San Diego, America: Academic Press; 1999.
- 2Kilbas AA, Srivastava HM, Trujillo JJ. Theory and Applications of Fractional Differential Equations. Amsterdam, The Netherlands: Elsevier; 2006.
10.1016/S0304-0208(06)80001-0 Google Scholar
- 3Zhou Y. Fractional Evolution Equations and Inclusions: Analysis and Control. Amsterdam, The Netherlands: Academic Press; 2016.
- 4Hilfer R. Experimental evidence for fractional time evolution in glass forming materials. Chem Phys. 2002; 284: 399-408.
- 5Sun LM, Chen L. Free vibrations of a taut cable with a general viscoelastic damper modeled by fractional derivatives. J Sound Vib. 2015; 335(20): 19-33.
- 6Scott Blair GW. Analytical and integrative aspects of the stress-strain-time problem. J Sci Instrum. 1944; 21: 80-84.
10.1088/0950-7671/21/5/302 Google Scholar
- 7Bagley RL. On the equivalence of the Riemann-Liouville and the Caputo fractional order derivatives in modeling of linear viscoelastic materials. Fractional Calculus Appl Anal. 2007; 10(2): 123-126.
- 8Sun HG, Zhang Y, Baleanu D, Chen W, Chen Y. A new collection of real world applications of fractional calculus in science and engineering. Commun Nonlinear Sci Numer Simul. 2018; 64: 213-231.
- 9Aguilar JFG, Martínez HY, Jiménez RFE, Zaragoza CMA, Reyes JR. Analytical and numerical solutions of electrical circuits described by fractional derivatives. Appl Math Modell. 2016; 40: 9079-9094.
- 10Machado JT, Mata ME. Pseudo phase plane and fractional calculus modeling of western global economic downturn. Commun Nonlinear Sci Numer Simul. 2015; 22: 396-406.
- 11Ionescu C, Lopes A, Copot D, Machado JAT, Bates JHT. The role of fractional calculus in modeling biological phenomena: a review. Commun Nonlinear Sci Numer Simul. 2017; 51: 141-159.
- 12Metzler R, Klafter J. The random walk's guide to anomalous diffusion: a fractional dynamics approach. Phys Rep. 2000; 339: 1-77.
- 13Sandev T, Tomovski Ž. Langevin equation for a free particle driven by power law type of noises. Phys Lett A. 2014; 378: 1-9.
- 14Sandev T, Metzler R, Chechkin A. From continuous time random walks to the generalized diffusion equation. Fractional Calculus Appl Anal. 2018; 21(1): 10-28.
- 15Hilfer R. Applications of Fractional Calculus in Physics. Singapore: World Scientific; 2000.
10.1142/3779 Google Scholar
- 16Chaudhary R, Pandey DN. Monotone iterative technique for neutral fractional differential equation with infinite delay. Math Methods Appl Sci. 2016; 39: 4642-4653.
- 17Li MM, Debbouche A, Wang JR. Relative controllability in fractional differential equations with pure delay. Math Methods Appl Sci. 2017; 41(18): 1-9.
- 18Zhang H, Ye RY, Cao JD, Alsaedie A. Delay-independent stability of Riemann-Liouville fractional neutral-type delayed neural networks. Neural Process Lett. 2018; 47(3): 427-442.
- 19Wang XW, Wang JR, Shen D, Zhou Y. Convergence analysis for iterative learning control of conformable fractional differential equations. Math Methods Appl Sci. 2018; 41(17): 1-14.
- 20Zhang YJ, Liu S, Yang R, Tan YY, Li XY. Global synchronization of fractional coupled networks with discrete and distributed delays. Physica A. 2019; 514: 830-837.
- 21Stankovic B, Atanackovic TM. Dynamics of a rod made of generalized Kelvin–Voigt visco-elastic material. J Math Anal Appl. 2002; 268: 550-563.
- 22Naber M. Linear fractionally damped oscillator. Int J Differ Equ. 2010; 2010: 1-12.
- 23Gómez-Aguilar JF, Yépez-Martínez H, Calderón-Ramón C, Cruz-Orduña I, Escobar-Jiménez RF, Olivares-Peregrino VH. Modeling of a mass-spring-damper system by fractional derivatives with and without a singular kernel. Entropy. 2015; 17(9): 6289-6303.
- 24Kempfle S. Causality criteria for solutions of linear fractional differential equations. Fractional Calculus Appl Anal. 1998; 1(4): 351-364.
- 25Balachandran K, Govindaraj V, Rivero M, Trujillo JJ. Controllability of fractional damped dynamical systems. Appl Math Comput. 2015; 257: 66-73.
- 26He B, Zhou H, Kou C. The controllability of fractional damped dynamical systems with control delay. Commun Nonlinear Sci Numer Simul. 2016; 32: 190-198.
- 27Yuan J, Zhang YA, Liu JM, Shi B, Gai MJ, Yang SJ. Mechanical energy and equivalent differential equations of motion for single-degree-of-freedom fractional oscillators. J Sound Vib. 2017; 397: 192-203.
- 28Zhou XF, Jiang W, Hu LG. Controllability of a fractional linear time-invariant neutral dynamical system. Appl Math Lett. 2013; 26: 418-424.
- 29Zhou XF, Yang FL, Jiang W. Analytic study on linear neutral fractional differential equations. Appl Math Comput. 2015; 257: 295-307.
- 30Ding XL, Nieto JJ. Controllability and optimality of linear time-invariant neutral control systems with different fractional orders. Acta Mathematica Scientia. 2015; 35B(5): 1003-1013.
- 31Zhou Y, Jiao F, Li J. Existence and uniqueness for fractional neutral differential equations with infinite delay. Nonlinear Anal. 2009; 71: 3249-3256.
- 32Fec̆kan M, Zhou Y, Wang JR. On the concept and existence of solution for impulsive fractional differential equations. Commun Nonlinear Sci Numer Simul. 2012; 17: 3050-3060.
- 33Wang JR. Approximate mild solutions of fractional stochastic evolution equations in Hilbert spaces. Appl Math Comput. 2015; 256: 315-323.
- 34Zhou Y, Shen XH, Zhang L. Cauchy problem for fractional evolution equations with Caputo derivative. Eur Phys J-Spec Top. 2013; 222: 1747-1764.
- 35Luchko Y, Gorenflo R. An operational method for solving fractional differential equations with the Caputo derivatives. Acta Mathematica Vietnamica. 1999; 24(2): 207-233.
- 36Sandev T, Chechkin A, Kantz H, Metzler R. Diffusion and Fokker-Planck-Smoluchowski equations with generalized memory kernel. Fractional Calculus Appl Anal. 2015; 18(4): 1006-1038.
- 37Sandev T, Chechkin AV, Korabel N, Kantz H, Sokolov IM, Metzler R. Distributed-order diffusion equations and multifractality: models and solutions. Phys Rev E. 2015; 92(42117): 1-19.
- 38Ye HP, Gao J, Ding Y. A generalized Gronwall inequality and its application to a fractional differential equation. J Math Anal Appl. 2007; 328(2): 1075-1081.
- 39Zhang ZF, Wei ZZ. A generalized Gronwall inequality and its application to fractional neutral evolution inclusions. J Inequalities Appl. 2016; 45: 1-18.
- 40Wang JR, Shah K, Ali A. Existence and Hyers-Ulam stability of fractional nonlinear impulsive switched coupled evolution equations. Mathe Methods Appl Sci. 2018; 41: 2392-2402.
