Fractional Relaxation Research Articles

Describing NMR relaxation by effective phase diffusion equation

This paper proposes an effective diffusion equation method to analyze nuclear magnetic resonance (NMR) relaxation. NMR relaxation is a spin system recovery process, where the evolution of the spin system is affected by the random field due to Hamiltonians, such as dipolar couplings. The evolution of magnetization can be treated as a random walk in phase space described either by a normal or fractional phase diffusion equation. Based on these phase diffusion equations, the NMR relaxation rates and equations can be obtained, exemplified in the analysis of relaxations affected by an arbitrary random field, and by dipolar coupling for both like and unlike spins. The obtained theoretical results are consistent with the reported results in the literature. Additionally, the anomalous relaxation expression obtained from the Mittag-Leffler function based time correlation function can successfully fit the previously reported 13C T1 NMR experimental data of polyisobutylene (PIB) in the blend of PIB and head-to-head poly(propylene) (hhPP). Furthermore, the proposed phase diffusion approach provides an intuitive way to interpret NMR relaxation, particularly for the fractional NMR relaxation, which is still a challenge to explain by the available theoretical methods. The paper provides additional insights into NMR and magnetic resonance imaging (MRI) relaxation experiments.

Fractional order thermo-viscoelastic theory of biological tissue with dual phase lag heat conduction model

Viscoelasticity is an inherent property of the biological tissue and is often used as a diagnosis parameter in characterizing cancer. Recently, it has been found that fractional calculus agrees well with experimental results when describing the viscoelastic materials. In this study, the fractional linear thermo-viscoelastic theory is developed with considering the fractional relaxation effect of dual phase lag heat conduction model and viscoelastic constitutive relation simultaneously. The transient coupling thermal and mechanical behavior of biological tissue are investigated. The governing equations are solved by Laplace transformation. The effects of fractional parameter on the distributions of temperature, displacement and stress are obtained and illustrated graphically.

Black swans, extreme risks, and the e-pile model of self-organized criticality

A new model of self-organized criticality (SOC) is described which differs from ordinary sand-pile SOC models in that it uses electrically charged particles of different kinds to propagate activities and to generate the critical state. The model, arguably called the e-pile model (as an alternative to sand-pile, with “e-” standing for “electric charge”), is motivated by the problem of dielectric relaxation in self-asembling random lattices with disorder under the action of electrostatic forces, but in principle it may be applied to different SOC processes with the random injection scheme and charitable lattice-redistribution rule. We show that the critical state is that of self-evolving random percolation clusters at the edge of percolation and is also different from known “self-organized” versions of the percolation problem based on the directed percolation. A set of critical exponents is obtained based on the random walks, using the Kramers-Kronig relation and the formalism of frequency-dependent complex conductivity. The relaxation of a supercritical system to SOC is shown to obey the Mittag-Leffler pattern and fractional relaxation equation, with a broad distribution of durations of relaxation events. A peculiar feature of the e-pile system is that it is characterized by an intrinsic bias disregarding a vast majority of incipient relaxation events (by killing them “in their egg”), while it also favors other events that may, then, grow into extremely large sizes. We use this observation to explain the phenomenon of the “black swan,” by which one means a family of rare, large events, whose sudden occurrence is very difficult to predict, despite the important impact this type of event may have over the entire system. The event-size distribution of the black swans is found to be a power law, with the drop-off exponent being sensitive to the complexity features of the underlying percolation clusters. On the basis of this observation, we show that the black swans are less probable in more complex environments, and we use this argument to explain why driven, dissipative systems would develop complex structures as a result of dynamical evolution. The e-pile model reveals the upper critical dimension dc=6 for which a crossover to mean-field SOC is found. In six and higher dimensions, the probability density to observe a black swan of a given size behaves as the inverse cube of the size. Below dc the decay is always slower than the inverse-cube and is flatter at lower dimensions. The implications of the mean-field law in country risk assessment, business, and finance are discussed. Finally, we show that the occurrence of the power-law tails is limited to the rate of the driving, and that for too high a rate a different branch of extreme events may result, with the defining features enabling to associate this branch with the phenomenon of dragon kings.

Assessment of Neuroprotective Effects of Low-Intensity Transcranial Ultrasound Stimulation in a Parkinson's Disease Rat Model by Fractional Anisotropy and Relaxation Time T2∗ Value.

Background: Low-intensity transcranial ultrasound (LITUS) may have a therapeutic effect on Parkinson’s disease (PD) patients to some extent. Fractional anisotropy (FA) and relaxation time T2∗ that indicate the integrity of fiber tracts and iron concentrations in brain tissue have been used to evaluate the therapeutic effects of LITUS.Purpose: This study aims to use FA and T2∗ values to evaluate the therapeutic effects of LITUS in a PD rat model.Materials and Methods: Twenty Sprague-Dawley rats were randomly divided into a hemi-PD group (n = 10) and a LITUS group (n = 10). Single-shot spin echo echo-planar imaging and fast low-angle shot T2WI sequences at 3.0 T were used. The FA and T2∗ values on the right side of the substantia nigra (SN) pars compacta were measured to evaluate the therapeutic effect of LITUS in the rats.Results: One week after PD-like signs were induced in the rats, the FA value in the LITUS group was significantly larger compared with the PD group (0.214 ± 0.027 vs. 0.340 ± 0.032, t = 2.864, P = 0.011). At the 5th and 6th weeks, the FA values in the LITUS group were significantly smaller compared with the PD group (5th week: 0.290 ± 0.037 vs. 0.405 ± 0.027, t = 2.385, P = 0.030; 6th week: 0.299 ± 0.021 vs. 0.525 ± 0.028, t = 6.620, P < 0.0001). In the 5th and 6th weeks, the T2∗ values in the injected right SN of the LITUS group were significantly higher compared with the PD group (5th week, 12.169 ± 0.826 in the LITUS group vs. 7.550 ± 0.824 in the PD group; 6th week, 11.749 ± 0.615 in the LITUS group vs. 7.550 ± 0.849 in the PD group).Conclusion: LITUS had neuroprotective effects and can reduce the damage of 6-OHDA-induced neurotoxicity in hemi-PD rats. The combination of FA and T2∗ assessments can potentially serve as a new and effective method to evaluate the therapeutic effects of LITUS.

Myocardial changes detected with modern cardiac magnetic resonance techniques in patients with early rheumatoid arthritis

Abstract Funding Acknowledgements Type of funding sources: Public hospital(s). Main funding source(s): HUS diagnostic imaging center Aims Subclinical myocardial disease is common in patients with rheumatoid arthritis (RA). Impaired cardiac function, myocardial fibrosis and inflammation have previously correlated with RA disease activity. Our aim was to study whether myocardial changes are detectable by cardiac magnetic resonance (CMR) at the time of RA diagnosis. Material and methods: We recruited 21 untreated early RA patients without history of heart disease in Helsinki University Hospital and Lohja Hospital (Finland) between 10/2018 and 2/2020, and nine healthy volunteers. The patients underwent a clinical examination, laboratory tests, and CMR including mapping of extracellular volume fraction (ECV), and T1 and T2 relaxation times. The healthy controls underwent non-contrast CMR. Results The RA patients were older than the controls (median 58.1 years vs. 41.6 years, respectively, table 1.). T1 was slightly higher in RA patients compared with healthy controls in anteroseptal segments (1015 ms vs. 982 ms, P = 0.017) (table 2). No difference in T2 was detected and the ECV values were considered normal. Segmental T1, T2 or ECV showed no significant correlations with age, duration of the symptoms or with RA disease activity (DAS28-CRP score). Conclusions The minor, but statistically significant, elevation of T1 relaxation time in the anteroseptal segments suggests that myocardial changes may occur already in the early phase of RA, the anteroseptal segments being most vulnerable. The elevation of T1 relaxation time can be caused by mild myocardial inflammation or fibrosis. Although no significant correlation with DAS28-CRP was observed, subclinical systemic inflammation may have contributed to the myocardial abnormalities. Table 1. Pre-contrast T1 relaxation time (ms) T2 relaxation time (ms) ECV (%) Mean RA patients Controls P-value RA patients Controls P-value RA patients Global myocardial mean 996 (978-1011) 982 (964-1000) 0.304 48.0 (44.6-49.7) 46.3 (44.1-48.7) 0.295 26.6 (25.7-28.5) Anterior segments 954 (919-1000) 951 (921-998) 0.929 47.7 (46.4-49.8) 47.9 (43.5-50.1) 0.871 26.7 (25.3-28.8) Anteroseptal segments 1015 (987-1041) 982 (947-996) 0.017 48.3 (45.5-49.9) 45.7 (43.8-48.4) 0.150 28.3 (26.8-29.2) Inferoseptal segments 1012 (992-1023) 998 (967-1003) 0.077 48.0 (43.5-49.4) 44.9 (43.5-46.2) 0.533 26.7 (26.0-27.9) Inferior segments 1016 (987-1064) 1003 (992-1025) 0.625 47.5 (45.1-50.3) 46.1 (44.2-48.4) 0.304 27.3 (25.6-29.0) Inferolateral segments 997 (974-1043) 992 (980-1016) &amp;gt;0.999 46.6 (42.6-49.1) 45.1 (42.4-49.1) 0.689 25.8 (25.3-28.2) Anterolateral segments 973 (945-973) 981 (954-1010) 0.563 47.0 (45.1-48.5) 46.6 (43.5-49.6) 0.625 26.4 (24.7-28.0) T1 and T2 relaxation time mapping and ECV results. Abstract Figure. ECV mapping

Study of composite fractional relaxation differential equation using fractional operators with and without singular kernels and special functions

Our aim in this article is to solve the composite fractional relaxation differential equation by using different definitions of the non-integer order derivative operator D_{t}^{alpha }, more specifically we employ the definitions of Caputo, Caputo–Fabrizio and Atangana–Baleanu of non-integer order derivative operators. We apply the Laplace transform method to solve the problem and express our solutions in terms of Lorenzo and Hartley’s generalised G function. Furthermore, the effects of the parameters involved in the model are graphically highlighted.

Fractional Stress Relaxation Model of Rock Freeze‐Thaw Damage

Freeze‐thaw cycle is a type of fatigue loading, and rock stress relaxation under freeze‐thaw cycles takes into account the influence of the freeze‐thaw cycle damage and deterioration. Rock stress relaxation under freeze‐thaw cycles is one of the paramount issues in tunnel and slope stability research. To accurately describe the mechanical behaviour of stress relaxation of rocks under freeze‐thaw, the software element is constructed based on the theory of fractional calculus to replace the ideal viscous element in the traditional element model. The freeze‐thaw damage degradation of viscosity coefficient is considered. A new three‐element model is established to better reflect the nonlinear stress relaxation behavior of rocks under freeze‐thaw. The freeze‐thaw and stress relaxation of rock are simulated by ABAQUS, the relevant model parameters are determined, and the stress relaxation equation of rock under freeze‐thaw cycle is obtained based on numerical simulation results. The research shows that the test results are consistent with the calculated results, indicating that the constitutive equation can better describe the stress relaxation characteristics of rocks under freeze‐thaw and provide theoretical basis for surrounding rock support in cold region.

Upper and Lower Solution method for Positive solution of generalized Caputo fractional differential equations

In this research paper, the nonlinear fractional relaxation equation involving the generalized Caputo derivative is reduced to an equivalent integral equation via the generalized Laplace transform. Moreover, the upper and lower solutions method combined with some fixed point theorems, and the properties of the Mittag-Leffler function are applied to investigate the existence and uniqueness of positive solutions for the problem at hand. At the end, to illustrate our results, we give an example.

An efficient numerical method for fractional neutron diffusion equation in the presence of different types of reactivities

In this paper, we construct an efficient numerical technique for solving a fractional neutron diffusion equation with step, ramp and sinusoidal reactivity effects combined with one group of delayed neutron precursor concentration equation. This problem describes neutron transport in a nuclear reactor. We use L1 approximation technique for discretization of time derivative and a collocation method based on quintic B-spline (QBS) basis function for discretization of space derivative. Some numerical experiments are performed to show the accuracy and efficiency of the method. The effects of order of fractional derivative, relaxation time and radioactive decay constant on the neutron flux profile are examined. It is shown that the suggested method is of order O(k1+h4) convergence, where k and h represent the step size for time and space, respectively. The numerical result of fractional model is compared with that of classic one. The CPU time is provided to show the computational efficiency of the method.

Abstract 14479: Fontan Patients With Dominant Left Ventricular Morphology Have Severe Ventricular Electromechanical Discoordination Associated With Worse Exercise Capacity

Introduction: Fontan patients with a single left ventricle (SLV) develop late systolic and diastolic dysfunction. The development of SLV dysfunction has been associated with abnormal loading conditions, myocardial ischemia and fibrosis, and compromised ventricular-aortic coupling. However, electromechanical discoordination, a prime determinant of ventricular performance, has not been adequately evaluated in patients with SLV. Hypothesis: Patients with SLV and Fontan circulation will have abnormal electromechanical discoordination compared to controls, and this will be associated with their functional status. Methods: Patients with SLV (n=20) and healthy controls (n=20) underwent MRI-based tissue deformation analysis. Circumferential strain and strain rate data were used to calculate systolic stretch fraction (SSF) and diastolic relaxation fraction (DRF) (Figure 1). SSF is the proportion of myocardial segments undergoing relaxation in systole, while the DRF is the proportion of myocardial segments undergoing contraction in diastole. SSF and DRF were correlated with standard MRI indices and standard exercise tests. Results: Global circumferential strain was decreased in SLV compared to controls (16.4±2.7 vs 19.9±1.7, P&lt;0.001). SLV had increased median SSF when compared to the LV of controls [(0.023 (IQR: 0.006-0.036) vs. 0.007 (IQR: 0.006-0.013), P=0.023] and increased median DRF [0.39 (IQR: 0.30-0.44) vs. 0.30 (IQR: 0.26-0.32), P=0.002]. SSF correlated with the peak VO 2 index (R=-0.56, P=0.027) and DRF correlated with ejection fraction (R=-0.56, P=0.010). Conclusion: Fontan patients with SLV exhibit systolic and diastolic electromechanical discoordination, which is associated with lower peak VO 2 and reduced ejection fraction, respectively. Electromechanical discoordination might contribute to global SLV dysfunction by limiting both effective contractility and preload generation.

Abstract 15057: Electromechanical Discoordination is Present in Patients With Duchenne Muscular Dystrophy Independent of Tissue Fibrosis

Introduction: Progressive ventricular dysfunction is a cardinal symptom in Duchenne Muscular Dystrophy (DMD). Some of the earliest signs of cardiomyopathy in DMD are myocardial fibrotic deposition and LV strain defects. Electromechanical discoordination, as measured by Systolic Stretch Fraction (SSF) and Diastolic Relaxation Fraction (DRF), has been shown to be a sensitive marker of ventricular dysfunction. The presence of this discoordination in relation to fibrotic deposition in DMD has yet to be elucidated. Hypothesis: Patients with DMD will have abnormal SSF and DRF on cardiac MRI (CMR) tissue tracking analysis and associated with fibrotic deposition. Methods: Patients with DMD (n=31)(mean age: 14 ± 4 yrs) and controls (n=20) (mean age: 15 ± 3 yrs) underwent CMR for volumetric and functional analysis as well as Gadolinium (Gd) enhancement to evaluate the presence of fibrosis. Circumferential strain and strain rate indices from each segment were used to calculate electromechanical discoordination. Strain rate data was used to calculate SSF and DRF. Results: Patients with DMD showed increased median LV SSF compared to controls [0.027 (IQR: 0.015-0.041) vs 0.007 (IQR:0.005-0.013), P = 0.002] as well as increased median LV DRF [0.371 (IQR: 0.310-0.473) vs 0.300 (IQR: 0.264-0.325), P &lt; 0.001] (Figure). When comparing Gd(+) (n=14) vs Gd(-) (n=17) DMD patients, there was no difference between groups in either SSF [0.027 (IQR: 0.016-0.042) vs 0.026 (IQR: 0.008-0.040), P= 0.929] or DRF [0.371 (IQR: 0.309-0.537) vs 0.379 (IQR: 0.322-0.464), P= 0.931]. The SSF was associated with ESVi (R=0.71, P&lt;0.001), EDVi (R=0.65, P&lt; 0.001) and inversely associated with EF (R=-0.63, P&lt;0.001). Conclusion: Patients with DMD showed increased levels of LV electromechanical discoordination independent of qualitative presence of fibrosis noted by Gd enhancement. This allows speculation that changes in electromechanical discoordination may precede visible fibrotic change in DMD.

Abstract 12877: Magnesium Deficiency Causes Reversible Metabolic Diastolic and Systolic Cardiomyopathies

Introduction: Low circulating magnesium (Mg) level is associated with increased cardiovascular mortality, and conversely, dietary Mg intake is associated with a decreased risk of developing heart failure. Hypothesis: We investigated whether Mg deficiency alone could cause cardiomyopathy. Methods: C57BL/6J mice were fed with a low-Mg diet (low-Mg, 15-30 mg/kg Mg) or a normal diet (nl-Mg, 600 mg/kg Mg) for 6 weeks. To test reversibility, half of the low-Mg mice were fed then with normal diet for another 6 weeks (low→nl-Mg group). Results: Mg deficiency increased mortality, especially in female mice. After 6 weeks of low-Mg diet, surviving mice showed significantly decreased serum Mg (0.9±0.1 vs. 2.8±0.1 mg/dL for nl-Mg) and a reciprocal increase in serum Ca, K, and Na. Low-Mg mice exhibited a decreased cardiac ejection fraction (EF%, 39.8±1.9% vs. 52.0±1.7% of nl-Mg) and impaired relaxation (E/e’: 21.1±1.1 vs. 15.4±0.4 of nl-Mg) in echocardiography. At the cellular level, ATP, Ca transient amplitude, sarcoplasmic reticulum (SR) Ca load, resting sarcomere length, and sarcomere shortening were all decreased significantly in low-Mg hearts and cardiomyocytes. These changes were accompanied by evidence of mitochondrial dysfunction with significantly increased mitochondrial ROS production and mitochondrial membrane depolarization. Mg repletion normalized electrolytes, contraction, relaxation, and cellular changes. The SR Ca pump (SERCA) was decreased, the SR Ca channel RyR2 oxidized, and cardiac myosin binding protein C S-glutathionylated in low-Mg mouse hearts. These changes were normalized with Mg repletion. In vivo , mitoTEMPO treatment during low Mg diet improved the cardiac relaxation and increased cellular ATP levels without improving contraction. Conclusions: Mg deficiency caused a reversible diastolic and systolic cardiomyopathy associated with mitochondrial dysfunction. This cardiomyopathy may explain the relationship of hypomagnesemia and worsening heart failure. Mg intake could reverse these changes, reinforcing the known correlation of increased Mg intake and reduced heart failure symptoms and mortality. In deficiency states, Mg supplementation may represent a novel treatment for systolic and diastolic heart failure.

Unsteady fractional stress relaxation time effect model

The study of rock stress relaxation characteristics is an important aspect of the research of its long-term mechanical characteristics. It is also the key to ensure the long-term safety and stability of the surrounding rock of the high-ground-stress roadway. Using the MTS815.02 triaxial test system, the stress relaxation test of the sandstone of Fuxin Hengda Coal Mine with a confining pressure of 30 MPa is carried out. A new unsteady fractional stress relaxation aging constitutive model is established by constructing the relationship between damage variables, time, and model parameters and considering that the fractional order is affected by the time effect. The correctness and rationality of the model are verified by experimental data. Test results show that as the relaxation damage develops during relaxation, the relaxation has evident nonlinear characteristics. The model curve is highly consistent with the experimental curve, which indicates that the unsteady model is suitable for reflecting the deformation of rock stress relaxation. It can better describe the whole process of stress relaxation change. This study can provide a scientific basis for the long-term stability analysis of Fuxin Hengda high in situ stress sandstone roadway.

Optimal Budget-Feasible Mechanisms for Additive Valuations

In this article, we show a tight approximation guarantee for budget-feasible mechanisms with an additive buyer. We propose a new simple randomized mechanism with approximation ratio of 2, improving the previous best known result of 3. Our bound is tight with respect to either the optimal offline benchmark or its fractional relaxation. We also present a simple deterministic mechanism with the tight approximation guarantee of 3 against the fractional optimum, improving the best known result of (2+ √ 2) for the weaker integral benchmark.

Thermoelastic guided wave in fractional order functionally graded plates: An analytical integration Legendre polynomial approach

In this paper, an analytical integration Legendre polynomial approach (AILPA) is proposed to investigate the guided thermoelastic wave in functionally graded material (FGM) plates in the context of the fractional order Lord-Shulman (LS) thermoelastic theory. Coupled wave equations and fractional order heat conduction equation are solved by the presented approach, which proposes the analytical integral instead of numerical integration in the available conventional Legendre polynomial approach (CLPA). Comparison of the CPU time between two approaches indicates the higher efficiency of the presented approach. Furthermore, a new treatment of the adiabatic boundary condition for the Legendre polynomial is developed, other than the CLPA can only deal with the isothermal boundary condition. Finally, the phase velocity dispersion curves, attenuation curves, the displacement and temperature distributions for functionally graded plates with different fractional orders are analysed. Both the fractional order and relaxation time have weak influence on the elastic mode velocity, but they have considerable influence on the elastic mode attenuation.

Fractional-Order Thermoelastic Wave Assessment in a Two-Dimensional Fiber-Reinforced Anisotropic Material

The present work is aimed at studying the effect of fractional order and thermal relaxation time on an unbounded fiber-reinforced medium. In the context of generalized thermoelasticity theory, the fractional time derivative and the thermal relaxation times are employed to study the thermophysical quantities. The techniques of Fourier and Laplace transformations are used to present the problem exact solutions in the transformed domain by the eigenvalue approach. The inversions of the Fourier-Laplace transforms hold analytical and numerically. The numerical outcomes for the fiber-reinforced material are presented and graphically depicted. A comparison of the results for different theories under the fractional time derivative is presented. The properties of the fiber-reinforced material with the fractional derivative act to reduce the magnitudes of the variables considered, which can be significant in some practical applications and can be easily considered and accurately evaluated.

On Complete Monotonicity of Solution to the Fractional Relaxation Equation with the nth Level Fractional Derivative

In this paper, we first deduce the explicit formulas for the projector of the nth level fractional derivative and for its Laplace transform. Afterwards, the fractional relaxation equation with the nth level fractional derivative is discussed. It turns out that, under some conditions, the solutions to the initial-value problems for this equation are completely monotone functions that can be represented in form of the linear combinations of the Mittag–Leffler functions with some power law weights. Special attention is given to the case of the relaxation equation with the second level derivative.

The double fractional Cattaneo model on anomalous transport of compounds in spiny dendrites structure

The double fractional Cattaneo model is first introduced to simulate the anomalous transport of inert compounds in spiny dendrites comb structure. Formulated governing equations with relaxation parameter are solved by Laplace transform and Mellin transform methods. The particle distribution and mean square displacement (MSD) are represented by Fox H-function and Mittag-Leffler function, respectively. We find that MSD has many forms, which can be used to describe complex diffusion in spiny dendrites problems. Considering the application in spinous dendrites, qualitative simulations are made and the physical significance of fractional parameters and delay parameters are obtained. Fractional parameter β represents the density of dendritic spines and relaxation parameter τ reflects the velocity of reaching spines. Besides, the effects of the fractional parameters and relaxation parameter on dynamic characteristics are graphically analyzed and discussed.

Assortment Optimization over Dense Universe is Easy

Assortment optimization is an important problem in revenue management arising in industries such as online advertising, retailing and airline ticketing. We study assortment optimization under an arbitrary mixture of multi-nomial logit (MNL) models, when the universe of products is dense. In general, it is NP-hard to approximate assortment optimization under mixture of MNLs model within any reasonable factor. However, if the universe of products is dense, we show that assortment optimization has an optimal solution that is well approximated by a nested-by-revenue policy. We argue this by considering an alternate, continuous-space model where product features follow a density on a continuous space, and offer set can be any open subset on the feature space. In this model, we show that there exists a nested-by-revenue policy that is optimal. This characterization holds regardless of the mixture distribution (including discrete and continuous) and the dimension of feature space. Through this characterization and linking the continuous-space to the standard discrete-item model via a fractional relaxation, we also show that, in the finite and discrete item case, nested-by-revenue performs nearly optimally. In particular, we derive an optimality gap, explicitly dependent on the model parameters, that shrinks to zero under suitable scaling as the number of items grows.

Characterization of the Photothermal Interaction of a Semiconducting Solid Sphere Due to the Fractional Deformation, Relaxation Time, and Various Reference Temperature under L-S Theory

In the present paper, we constructed a new mathematical model of a thermoelastic semiconducting solid sphere based on the L-S model to study the photothermal interaction. Among this work, we considered the fractional-order strain, which has been introduced by Youssef. The bounding surface of the sphere has been thermally loaded by a thermal shock. Laplace transform has been applied, and its inversions have been obtained numerically. For silicon (Si) material, the numerical results of the temperature increment, carrier density increment, strain, displacement, stress, and stress-strain energy have been represented in figures with various values of the fractional-order parameter, relaxation time parameter, and reference temperature. The relaxation time parameter and reference temperature have significant effects on all the studied functions. The fractional-order parameter has limited effects on the temperature increment and carrier density increment, while it has significant effects on the strain, displacement, stress, and stress-strain energy.