Fractional Calculus for Continuum Mechanics - anisotropic non-locality

In this paper the generalisation of previous author's formulation of fractional continuum mechanics to the case of anisotropic non-locality is presented. The considerations include the review of competitive formulations available in literature. The overall concept bases on the fractional deformation gradient which is non-local, as a consequence of fractional derivative definition. The main advantage of the proposed formulation is its analogical structure to the general framework of classical continuum mechanics. In this sense, it allows, to give similar physical and geometrical meaning of introduced objects

Fourth-Order Compact Difference Schemes for the Riemann-Liouville and Riesz Derivatives

We propose two new compact difference schemes for numerical approximation of the Riemann-Liouville and Riesz derivatives, respectively. It is shown that these formulas have fourth-order convergence order by means of the Fourier transform method. Finally, some numerical examples are implemented to testify the efficiency of the numerical schemes and confirm the convergence orders

Fractional Order Operator for Symmetric Analysis of Cancer Model on Stem Cells with Chemotherapy

Cancer is dangerous and one of the major diseases affecting normal human life. In this paper, a fractional-order cancer model with stem cells and chemotherapy is analyzed to check the effects of infection in individuals. The model is investigated by the Sumudu transform and a very effective numerical method. The positivity of solutions with the ABC operator of the proposed technique is verified. Fixed point theory is used to derive the existence and uniqueness of the solutions for the fractional order cancer system. Our derived solutions analyze the actual behavior and effect of cancer disease in the human body using different fractional values. Modern mathematical control with the fractional operator has many applications including the complex and crucial study of systems with symmetry. Symmetry analysis is a powerful tool that enables the user to construct numerical solutions of a given fractional differential equation in a fairly systematic way. Such an analysis will provide a better understanding to control the of cancer disease in the human body.This research was funded by Basque Government Grants: IT1555-22 and KK-2022/00090 and MCIN/AEI 269.10.13039/501100011033: Grant PID2021-1235430B-C21 and Grant PID2021-1235430B-C22

Vibration analysis of micro-damaged plates with Riesz-Caputo fractional derivative

Bu çalışmada Riesz Caputo kesirli türev tanımı yardımıyla, nonlokal çekirdekler tanımlamadan, mikrogenleşme teorisi ile modellenen mikro hasarlı plakların nonlokal titreşim analizi yapılmıştır. Dört ucu ankastre-“clamped” (CCCC) mikro hasarlı plağın frekans spektrumu ve mod şekilleri kesirli türev mertebesinin ve birim uyum katsayısının farklı değerleri için elde edilmiştir. 3-boyutlu titreşim analizi Ritz enerji yöntemi ile gerçekleştirilmiştir. Çalışmanın bilimsel literatüre temel katkısı, kesirli türev kavramıyla modellenen nonlokal titireşim analizinin klasik teoriye göre daha uygun bir model olduğunun ve deneysel sonuçlarla daha iyi örtüştüğünün gösterilmesidir.In this study, with the help of Riesz Caputo fractional derivative definition, non-local analysis of micro-damaged plates are investigated by micro-elongation theory without defining the nonlocal kernels. The frequency spectrum and mode shapes of the microelongated plate with four clamped edges for different values of the fractional continua order and the material length scale parameter are carried out. 3-dimensional vibration analysis are done using the Ritz energy method. The main contribution of the study to the scientific literature is the demonstration that the nonlocal vibration analysis modeled with the concept of fractional derivative is a more suitable model than the classical theory and it fits better with the experimental results

About Robust Stability of Caputo Linear Fractional Dynamic Systems with Time Delays through Fixed Point Theory

This paper investigates the global stability and the global asymptotic stability independent of the sizes of the delays of linear time-varying Caputo fractional dynamic systems of real fractional order possessing internal point delays. The investigation is performed via fixed point theory in a complete metric space by defining appropriate nonexpansive or contractive self-mappings from initial conditions to points of the state-trajectory solution. The existence of a unique fixed point leading to a globally asymptotically stable equilibrium point is investigated, in particular, under easily testable sufficiency-type stability conditions. The study is performed for both the uncontrolled case and the controlled case under a wide class of state feedback laws.Ministerio de Educación (DPI2009-07197) y Gobierno Vasco (IT378-10 SAIOTEK S-PE09UN12