A new Willis aneurysm system is proposed, which contains the Atangana-Baleanu(AB) fractional derivative.we obtain the numerical solution of the Atangana–Baleanu fractional Willis aneurysm system(ABWAS) with the AB fr...A new Willis aneurysm system is proposed, which contains the Atangana-Baleanu(AB) fractional derivative.we obtain the numerical solution of the Atangana–Baleanu fractional Willis aneurysm system(ABWAS) with the AB fractional integral and the predictor–corrector scheme.Moreover, we research the chaotic properties of ABWAS with phase diagrams and Poincare sections.The different values of pulse pressure and system order are used to evaluate and compare their effects on ABWAS.The simulations verify that the changes of pulse pressure and system order are the significant reason for ABWAS'states varying from chaotic to steady.In addition, compared with Caputo fractional WAS(FWAS),ABWAS shows less state that is chaotic.Furthermore, the results of bifurcation diagrams of blood flow damping coefficient and reciprocal heart rate show that the blood flow velocity tends to stabilize with the increase of blood flow damping coefficient or reciprocal heart rate, which is consistent with embolization therapy and drug therapy for clinical treatment of cerebral aneurysms.Finally, in view of the fact that ABWAS in chaotic state increases the possibility of rupture of cerebral aneurysms, a reasonable controller is designed to control ABWAS based on the stability theory.Compared with the control results of FWAS by the same method, the results show that the blood flow velocity in the ABWAS system varies in a smaller range.Therefore, the control effect of ABWAS is better and more stable.The new Willis aneurysm system with Atangana–Baleanu fractional derivative provides new information for the further study on treatment and control of brain aneurysms.展开更多
This work provides a new fuzzy variable fractional COVID-19 model and uses a variablefractional operator, namely, the fuzzy variable Atangana–Baleanu fractional derivativesin the Caputo sense. Next, we explore the pr...This work provides a new fuzzy variable fractional COVID-19 model and uses a variablefractional operator, namely, the fuzzy variable Atangana–Baleanu fractional derivativesin the Caputo sense. Next, we explore the proposed fuzzy variable fractional COVID-19 model using the fixed point theory approach and determine the solution’s existenceand uniqueness conditions. We choose an appropriate mapping and with the help ofthe upper/lower solutions method. We prove the existence of a positive solution for theproposed fuzzy variable fractional COVID-19 model and also obtain the result on theexistence of a unique positive solution. Moreover, we discuss the generalized Hyers–Ulam stability and generalized Hyers–Ulam–Rassias stability. Further, we investigate theresults on maximum and minimum solutions for the fuzzy variable fractional COVID-19model.展开更多
This paper deals with the closed-form solutions to the family of Boussinesq-like equations with the effect of spatio-temporal dispersion.The sine-Gordon expansion and the hyperbolic function approaches are efficiently...This paper deals with the closed-form solutions to the family of Boussinesq-like equations with the effect of spatio-temporal dispersion.The sine-Gordon expansion and the hyperbolic function approaches are efficiently applied to the family of Boussinesq-like equations to explore novel solitary,kink,anti-kink,combo,and singular-periodic wave solutions.The attained solutions are expressed by the trigonometric and hyperbolic functions including tan,sec,cot,csc,tanh,sech,coth,csch,and of their combination.In addition,the mentioned two approaches are applied to the aforesaid models in the sense of Atangana conformable derivative or Beta derivative to attain new wave solutions.Three-dimensional and two-dimensional graphs of some of the obtained novel solutions satisfying the corresponding equations of interest are provided to understand the underlying mechanisms of the stated family.For the bright wave solutions in terms of Atangana’s conformable derivative,the amplitudes of the bright wave gradually decrease,but the smoothness increases when the fractional parametersαandβincrease.On the other hand,the periodicities of periodic waves increase.The attained new wave solutions can motivate applied scientists for engineering their ideas to an optimal level and they can be used for the validation of numerical simulation results in the propagation of waves in shallow water and other nonlinear cases.The performed approaches are found to be simple and efficient enough to estimate the solutions attained in the study and can be used to solve various classes of nonlinear partial differential equations arising in mathematical physics and engineering.展开更多
基金Project supported by the State Key Program of the National Natural Science of China(Grant No.91324201)the Fundamental Research Funds for the Central Universities of China+1 种基金the Self–determined and Innovative Research Funds of WUT,China(Grant No.2018IB017)the Natural Science Foundation of Hubei Province of China(Grant No.2014CFB865)
文摘A new Willis aneurysm system is proposed, which contains the Atangana-Baleanu(AB) fractional derivative.we obtain the numerical solution of the Atangana–Baleanu fractional Willis aneurysm system(ABWAS) with the AB fractional integral and the predictor–corrector scheme.Moreover, we research the chaotic properties of ABWAS with phase diagrams and Poincare sections.The different values of pulse pressure and system order are used to evaluate and compare their effects on ABWAS.The simulations verify that the changes of pulse pressure and system order are the significant reason for ABWAS'states varying from chaotic to steady.In addition, compared with Caputo fractional WAS(FWAS),ABWAS shows less state that is chaotic.Furthermore, the results of bifurcation diagrams of blood flow damping coefficient and reciprocal heart rate show that the blood flow velocity tends to stabilize with the increase of blood flow damping coefficient or reciprocal heart rate, which is consistent with embolization therapy and drug therapy for clinical treatment of cerebral aneurysms.Finally, in view of the fact that ABWAS in chaotic state increases the possibility of rupture of cerebral aneurysms, a reasonable controller is designed to control ABWAS based on the stability theory.Compared with the control results of FWAS by the same method, the results show that the blood flow velocity in the ABWAS system varies in a smaller range.Therefore, the control effect of ABWAS is better and more stable.The new Willis aneurysm system with Atangana–Baleanu fractional derivative provides new information for the further study on treatment and control of brain aneurysms.
文摘This work provides a new fuzzy variable fractional COVID-19 model and uses a variablefractional operator, namely, the fuzzy variable Atangana–Baleanu fractional derivativesin the Caputo sense. Next, we explore the proposed fuzzy variable fractional COVID-19 model using the fixed point theory approach and determine the solution’s existenceand uniqueness conditions. We choose an appropriate mapping and with the help ofthe upper/lower solutions method. We prove the existence of a positive solution for theproposed fuzzy variable fractional COVID-19 model and also obtain the result on theexistence of a unique positive solution. Moreover, we discuss the generalized Hyers–Ulam stability and generalized Hyers–Ulam–Rassias stability. Further, we investigate theresults on maximum and minimum solutions for the fuzzy variable fractional COVID-19model.
文摘This paper deals with the closed-form solutions to the family of Boussinesq-like equations with the effect of spatio-temporal dispersion.The sine-Gordon expansion and the hyperbolic function approaches are efficiently applied to the family of Boussinesq-like equations to explore novel solitary,kink,anti-kink,combo,and singular-periodic wave solutions.The attained solutions are expressed by the trigonometric and hyperbolic functions including tan,sec,cot,csc,tanh,sech,coth,csch,and of their combination.In addition,the mentioned two approaches are applied to the aforesaid models in the sense of Atangana conformable derivative or Beta derivative to attain new wave solutions.Three-dimensional and two-dimensional graphs of some of the obtained novel solutions satisfying the corresponding equations of interest are provided to understand the underlying mechanisms of the stated family.For the bright wave solutions in terms of Atangana’s conformable derivative,the amplitudes of the bright wave gradually decrease,but the smoothness increases when the fractional parametersαandβincrease.On the other hand,the periodicities of periodic waves increase.The attained new wave solutions can motivate applied scientists for engineering their ideas to an optimal level and they can be used for the validation of numerical simulation results in the propagation of waves in shallow water and other nonlinear cases.The performed approaches are found to be simple and efficient enough to estimate the solutions attained in the study and can be used to solve various classes of nonlinear partial differential equations arising in mathematical physics and engineering.
