Search results for: fractional flow reserve

Authors: H. Falsafi Falsafizadeh, R. Zeinali Zeinali

Abstract:

This paper focuses on determination of total Operating Reserve (OR) level, consisting of spinning and non-spinning reserves, in two small real power systems, in such a way that the system reliability indicator would comply with typical industry standards. For this purpose, the standard used by the North American Electric Reliability Corporation (NERC) – i.e., 1 day outage in 10 years or 0.1 days/year is relied. The simulation of system operation for these systems that was used for the determination of total operating reserve level was performed by industry standard production simulation software in this field, named PLEXOS. In this paper, the operating reserve which meets an annual Loss of Load Expectation (LOLE) of approximately 0.1 days per year is determined in the study year. This reserve is the minimum amount of reserve required in a power system and generally defined as a percentage of the annual peak.

Keywords: frequency control, LOLE, operating reserve, system reliability

Procedia PDF Downloads 320

Authors: Salisu Ibrahim

Abstract:

The paper studies the commutativity associated with fractional order linear time-varying systems (LTVSs), which is an important area of study in control systems engineering. In this paper, we explore the properties of these systems and their ability to commute. We proposed the necessary and sufficient condition for commutativity for fractional order LTVSs. Through a simulation and mathematical analysis, we demonstrate that these systems exhibit commutativity under certain conditions. Our findings have implications for the design and control of fractional order systems in practical applications, science, and engineering. An example is given to show the effectiveness of the proposed method which is been computed by Mathematica and validated by the use of MATLAB (Simulink).

Keywords: fractional differential equation, physical systems, equivalent circuit, analog control

Procedia PDF Downloads 88

Authors: Salisu Ibrahim

Abstract:

The paper studies the commutativity associated with fractional order linear time-varying systems (LTVSs), which is an important area of study in control systems engineering. In this paper, we explore the properties of these systems and their ability to commute. We proposed the necessary and sufficient condition for commutativity for fractional order LTVSs. Through a simulation and mathematical analysis, we demonstrate that these systems exhibit commutativity under certain conditions. Our findings have implications for the design and control of fractional order systems in practical applications, science, and engineering. An example is given to show the effectiveness of the proposed method which is been computed by Mathematica and validated by the use of Matlab (Simulink).

Keywords: fractional differential equation, physical systems, equivalent circuit, and analog control

Procedia PDF Downloads 49

Authors: Muhammad Swilam

Abstract:

Solutions for the wrong division problem of the Extended Multi-Modulus Divider (EMMD) that occurs during modulus extension (i.e. switching the modulus value between two different ranges of division ratios), in open loop fractional dividers and fractional multiplying delay locked loop, is proposed. A detailed study for the MMD with Sigma-Delta is also presented. Moreover, extensive simulations for the divider are presented to ensure and verify its functionality and compared with the conventional dividers.

Keywords: extended multi-modulus divider (EMMD), fractional multiplying delay locked loop, open loop fractional divider, sigma delta modulator

Procedia PDF Downloads 458

Authors: Kazim G. Atman, Hüseyin Sirin

Abstract:

In theoretical physics, nonlocal phenomena has always been subject of debate. However, in the conventional mathematical approach where the developments of the physical systems are investigated by using the standard mathematical tools, nonlocal effects are not taken into account. In order to investigate the nonlocality in quantum mechanics and fractal property of space, fractional derivative operators are employed in this study. In this manner, fractional creation and annihilation operators are introduced and Einstein coefficients are taken into account as an application of concomitant formalism in quantum field theory. Therefore, each energy mode of photons are considered as fractional quantized harmonic oscillator hereby Einstein coefficients are obtained. Nevertheless, wave function and energy eigenvalues of fractional quantum mechanical harmonic oscillator are obtained via the fractional derivative order α which is a measure of the influence of nonlocal effects. In the case α = 1, where space becomes homogeneous and continuous, standard physical conclusions are recovered.

Keywords: Einstein’s Coefficients, Fractional Calculus, Fractional Quantum Mechanics, Nonlocal Theories

Procedia PDF Downloads 148

Authors: Yasser F. Hassan

Abstract:

In this paper, a proposed model of cellular automata is studied by means of fractional integral function. A cellular automaton is a decentralized computing model providing an excellent platform for performing complex computation with the help of only local information. The paper discusses how using fractional integral function for representing cellular automata memory or state. The architecture of computing and learning model will be given and the results of calibrating of approach are also given.

Keywords: fractional integral, cellular automata, memory, learning

Procedia PDF Downloads 383

Authors: Jorge Olivares, Fernando Maass, Pablo Martin

Abstract:

Fractional derivatives have been considered recently as a way to solve different problems in Engineering. In this way, second kind modified Bessel functions are considered here. The order α fractional differential equations of second kind Bessel functions, Kᵥ(x), are studied with simple initial conditions. The Laplace transform and Caputo definition of fractional derivatives are considered. Solutions have been found for ν=1/3, 1/2, 2/3, -1/3, -1/2 and (-2/3). In these cases, the solutions are the sum of two hypergeometric functions. The α fractional derivatives have been for α=1/3, 1/2 and 2/3, and the above values of ν. No convergence has been found for the integer values of ν Furthermore when α has been considered as a rational found m/p, no general solution has been found. Clearly, this case is more difficult to treat than those of first kind Bessel Function.

Keywords: Caputo, modified Bessel functions, hypergeometric, linear fractional differential equations, transform Laplace

Procedia PDF Downloads 308

Authors: A. Guezane-Lakoud, S. Bensebaa

Abstract:

In this paper, we study a boundary value problem of nonlinear fractional differential equation. Existence and positivity results of solutions are obtained.

Keywords: positive solution, fractional caputo derivative, Banach contraction principle, Avery and Peterson fixed point theorem

Procedia PDF Downloads 385

Authors: Karunambigai Mevani Govindasamy, Sathishkumar Ayyappan

Abstract:

In this paper, we formulate the algorithm to find out the dominating function parameters of Intuitionistic Fuzzy Graphs(IFG). The methodology we adopted here is converting any physical problem into an IFG, and that has been transformed into Intuitionistic Fuzzy Matrix. Using Linear Program Solver software (LiPS), we found the defined parameters for the given IFG. We obtained these parameters for a path and cycle IFG. This study can be extended to other varieties of IFG. In particular, we obtain the definition of edge dominating function, minimal edge dominating function, fractional edge domination number (γ_if^') and upper fractional edge domination number (Γ_if^') of an intuitionistic fuzzy graph. Also, we formulated an algorithm which is appropriate to work on LiPS to find fractional edge domination number and upper fractional edge domination number of an IFG.

Keywords: fractional edge domination number, intuitionistic fuzzy cycle, intuitionistic fuzzy graph, intuitionistic fuzzy path

Procedia PDF Downloads 142

Authors: Muhammad Danish Khan, Imran Naeem, Mudassar Imran

Abstract:

In this article, we study time-space Fractional Advection-Dispersion (FADE) equation and time-space Fractional Whitham-Broer-Kaup (FWBK) equation that have a significant role in hydrology. We introduce suitable transformations to convert fractional order derivatives to integer order derivatives and as a result these equations transform into Partial Differential Equations (PDEs). Then the Lie symmetries and corresponding optimal systems of the resulting PDEs are derived. The symmetry reductions and exact independent solutions based on optimal system are investigated which constitute the exact solutions of original fractional differential equations.

Keywords: modified Riemann-Liouville fractional derivative, lie-symmetries, optimal system, invariant solutions

Procedia PDF Downloads 407

Authors: Arrate Barrenechea Garro

Abstract:

Background: Bilingualism has been shown to enhance cognitive reserve and potentially delay the onset of dementia symptoms. This study investigates the impact of bilingualism on cognitive reserve and the age of diagnosis in Parkinson's Disease (PD). Methodology: The study involves 16 non-demented monolingual PD patients and 12 non-demented bilingual PD patients, matched for age, sex, and years of education. All participants are native Spanish speakers, with Spanish as their first language (L1). Cognitive performance is assessed through a neuropsychological examination covering all cognitive domains. Cognitive reserve is measured using the Cognitive Reserve Index Questionnaire (CRIq), while language proficiency is evaluated using the Bilingual Language Profile (BLP). The age at diagnosis is recorded for both monolingual and bilingual patients. Results: Bilingual PD patients demonstrate higher scores on the CRIq compared to monolingual PD patients, with significant differences between the groups. Furthermore, there is a positive correlation between cognitive reserve (CRIq) and the utilization of the second language (L2) as indicated by the BLP. Bilingual PD patients are diagnosed, on average, three years later than monolingual PD patients. Conclusion: Bilingual PD patients exhibit higher levels of cognitive reserve compared to monolingual PD patients, as indicated by the CRIq scores. The utilization of the second language (L2) is positively correlated with cognitive reserve. Bilingual PD patients are diagnosed with PD, on average, three years later than monolingual PD patients. These findings suggest that bilingualism may contribute to cognitive reserve and potentially delay the onset of clinical symptoms associated with PD. This study adds to the existing literature supporting the relationship between bilingualism and cognitive reserve. Further research in this area could provide valuable insights into the potential protective effects of bilingualism in neurodegenerative disorders.

Keywords: bilingualis, cogntiive reserve, diagnosis, parkinson's disease

Procedia PDF Downloads 58

Authors: Hakiki Kheira, Belhamiti Omar

Abstract:

In this paper, we propose a new compartmental fractional order model for the simulation of epidemic obesity dynamics. Using the Legendre wavelet method combined with the decoupling and quasi-linearization technique, we demonstrate the validity and applicability of our model. We also present some fractional differential illustrative examples to demonstrate the applicability and efficiency of the method. The fractional derivative is described in the Caputo sense.

Keywords: Caputo derivative, epidemiology, Legendre wavelet method, obesity

Procedia PDF Downloads 378

Authors: Jorge Gonzalez Camus, Valentin Keyantuo, Mahamadi Warma

Abstract:

In this work, we obtain representation results for solutions of a time-fractional differential equation involving the discrete fractional Laplace operator in terms of generalized Wright functions. Such equations arise in the modeling of many physical systems, for example, chain processes in chemistry and radioactivity. The focus is on the linear problem of the simplified Moore - Gibson - Thompson equation, where the discrete fractional Laplacian and the Caputo fractional derivate of order on (0,2] are involved. As a particular case, we obtain the explicit solution for the discrete heat equation and discrete wave equation. Furthermore, we show the explicit solution for the equation involving the perturbed Laplacian by the identity operator. The main tool for obtaining the explicit solution are the Laplace and discrete Fourier transforms, and Stirling's formula. The methodology mainly is to apply both transforms in the equation, to find the inverse of each transform, and to prove that this solution is well defined, using Stirling´s formula.

Keywords: discrete fractional Laplacian, explicit representation of solutions, fractional heat and wave equations, fundamental

Procedia PDF Downloads 179

Authors: A. Zerarka, W. Djoudi

Abstract:

We use some fractional transformations to obtain many types of new exact solutions of nonlinear lattice equation. These solutions include rational solutions, periodic wave solutions, and doubly periodic wave solutions.

Keywords: fractional transformations, nonlinear equation, travelling wave solutions, lattice equation

Procedia PDF Downloads 627

Authors: Ayub Khan, Net Ram Garg, Geeta Jain

Abstract:

In this paper, backstepping method is proposed to synchronize two fractional-order systems. The simulation results show that this method can effectively synchronize two chaotic systems.

Keywords: backstepping method, fractional order, synchronization, chaotic system

Procedia PDF Downloads 433

Authors: Truong Nguyen Luan Vu, Le Hieu Giang, Le Linh

Abstract:

The fractional–order proportional integral (FOPI) controller tuning rules based on the fractional calculus for the cascade control system are systematically proposed in this paper. Accordingly, the ideal controller is obtained by using internal model control (IMC) approach for both the inner and outer loops, which gives the desired closed-loop responses. On the basis of the fractional calculus, the analytical tuning rules of FOPI controller for the inner loop can be established in the frequency domain. Besides, the outer loop is tuned by using any integer PI/PID controller tuning rules in the literature. The simulation study is considered for the stable process model and the results demonstrate the simplicity, flexibility, and effectiveness of the proposed method for the cascade control system in compared with the other methods.

Keywords: Bode’s ideal transfer function, fractional calculus, fractional–order proportional integral (FOPI) controller, cascade control system

Procedia PDF Downloads 350

Authors: Haniye Dehestani, Yadollah Ordokhani

Abstract:

In this work, we present an efficient approach for solving variable-order time-fractional partial differential equations, which are based on Legendre and Laguerre polynomials. First, we introduced the pseudo-operational matrices of integer and variable fractional order of integration by use of some properties of Riemann-Liouville fractional integral. Then, applied together with collocation method and Legendre-Laguerre functions for solving variable-order time-fractional partial differential equations. Also, an estimation of the error is presented. At last, we investigate numerical examples which arise in physics to demonstrate the accuracy of the present method. In comparison results obtained by the present method with the exact solution and the other methods reveals that the method is very effective.

Keywords: collocation method, fractional partial differential equations, legendre-laguerre functions, pseudo-operational matrix of integration

Procedia PDF Downloads 142

Authors: Ahmed Soltan, Ahmed G. Radwan, Ahmed M. Soliman

Abstract:

This work aims to generalize the integer order Sallen-Key filters into the fractional-order domain. The analysis in the case of two different fractional-order elements introduced where the general transfer function becomes four terms which are unusual in the conventional case. In addition, the effect of the transfer function parameters on the filter poles and hence the stability is introduced and closed forms for the filter critical frequencies are driven. Finally, different examples of the fractional order Sallen-Key filter design are presented with circuit simulations using ADS where a great matching between the numerical and simulation results is obtained.

Keywords: Sallen-Key, fractance, stability, low-pass filter, analog filter

Procedia PDF Downloads 676

Authors: A. Cherifi, B. S. Bouazza, A. O. Dahman, B. Yagoubi

Abstract:

Transmission over Optical channels will introduce inter-symbol interference (ISI) as well as inter-channel (or inter-carrier) interference (ICI). To decrease the effects of ICI, this paper proposes equalizer for the Optical OFDM system based on the fractional Fourier transform (FrFFT). In this FrFT-OFDM system, traditional Fourier transform is replaced by fractional Fourier transform to modulate and demodulate the data symbols. The equalizer proposed consists of sampling the received signal in the different time per time symbol. Theoretical analysis and numerical simulation are discussed.

Keywords: OFDM, fractional fourier transform, internet and information technology

Procedia PDF Downloads 377

Authors: Maria Sciammarella, Uttam M. Shrestha, Youngho Seo, Grant T. Gullberg, Elias H. Botvinick

Abstract:

Background: Myocardial perfusion imaging (MPI) is typically performed with static imaging protocols and visually assessed for perfusion defects based on the relative intensity distribution. Dynamic cardiac SPECT, on the other hand, is a new imaging technique that is based on time varying information of radiotracer distribution, which permits quantification of myocardial blood flow (MBF). In this abstract, we report a progress and current status of dynamic cardiac SPECT using conventional gamma camera (Infinia Hawkeye 4, GE Healthcare) for estimation of myocardial blood flow and coronary flow reserve. Methods: A group of patients who had high risk of coronary artery disease was enrolled to evaluate our methodology. A low-dose/high-dose rest/pharmacologic-induced-stress protocol was implemented. A standard rest and a standard stress radionuclide dose of ⁹⁹ᵐTc-tetrofosmin (140 keV) was administered. The dynamic SPECT data for each patient were reconstructed using the standard 4-dimensional maximum likelihood expectation maximization (ML-EM) algorithm. Acquired data were used to estimate the myocardial blood flow (MBF). The correspondence between flow values in the main coronary vasculature with myocardial segments defined by the standardized myocardial segmentation and nomenclature were derived. The coronary flow reserve, CFR, was defined as the ratio of stress to rest MBF values. CFR values estimated with SPECT were also validated with dynamic PET. Results: The range of territorial MBF in LAD, RCA, and LCX was 0.44 ml/min/g to 3.81 ml/min/g. The MBF between estimated with PET and SPECT in the group of independent cohort of 7 patients showed statistically significant correlation, r = 0.71 (p < 0.001). But the corresponding CFR correlation was moderate r = 0.39 yet statistically significant (p = 0.037). The mean stress MBF value was significantly lower for angiographically abnormal than that for the normal (Normal Mean MBF = 2.49 ± 0.61, Abnormal Mean MBF = 1.43 ± 0. 0.62, P < .001). Conclusions: The visually assessed image findings in clinical SPECT are subjective, and may not reflect direct physiologic measures of coronary lesion. The MBF and CFR measured with dynamic SPECT are fully objective and available only with the data generated from the dynamic SPECT method. A quantitative approach such as measuring CFR using dynamic SPECT imaging is a better mode of diagnosing CAD than visual assessment of stress and rest images from static SPECT images Coronary Flow Reserve.

Keywords: dynamic SPECT, clinical SPECT/CT, selective coronary angiograph, ⁹⁹ᵐTc-Tetrofosmin

Procedia PDF Downloads 133

Authors: Akmal E. Egamberdiev, Alim M. Nigmatullaev, Trobjon Kh. Makhkamov

Abstract:

The results of scientific research on the biomorphological features of Ajuga turkestanica (Regel) Brig., its role in plant communities, modern distribution areas, and raw material reserves are presented. Plant ontogeny is divided into 3 periods and 9 growth stages. Information on its seasonal and diurnal flowering and seed productivity is provided. As a result of the research, the participation of the studied species in plant communities, its place, the structure and floristic composition of communities were determined, and as a result, for the first time, the description of 11 new associations in 7 formations of Ajuga turkestanica, and a schematic map of the geolocation of formations and associations of plants in Uzbekistan is given. A. turkestanica (within the range) are divided into 3 categories and 21 massifs. Its current biological reserve is 93.5±35.3 tons, its usable reserve is 46.2±13.8 tons, and the reserve that can be prepared in 1 year is 28.4±5.42 tons.

Keywords: ontogeny, seed productivity, seasonal flowering, formation, association, dominant, subdominant, areal, biological reserve, operational reserve, annual reserve, GIS map

Procedia PDF Downloads 66

Authors: Guy Joseph Eyebe, Gambo Betchewe, Alidou Mohamadou, Timoleon Crepin Kofane

Abstract:

In the present study, the dynamics of nanobeam resting on fractional order softening nonlinear viscoelastic pasternack foundations is studied. The Hamilton principle is used to derive the nonlinear equation of the motion. Approximate analytical solution is obtained by applying the standard averaging method. The Melnikov method is used to investigate the chaotic behaviors of device, the critical curve separating the chaotic and non-chaotic regions are found. It is shown that appearance of chaos in the system depends strongly on the fractional order parameter.

Keywords: chaos, fractional-order, Melnikov method, nanobeam

Procedia PDF Downloads 132

Authors: Artion Kashuri, Rozana Liko

Abstract:

In this paper, some generalization integral inequalities of Hermite–Hadamard type for functions whose derivatives are s–convex in modulus are given by using k–fractional integrals. Some applications to special means are obtained as well. Some known versions are recovered as special cases from our results. We note that our inequalities can be viewed as new refinements of the previous results. Finally, our results have a deep connection with various fractional integral operators and interested readers can find new interesting results using our idea and technique as well.

Keywords: Hermite-Hadamard's inequalities, Hölder's inequality, k-Riemann-Liouville fractional integral, special means

Procedia PDF Downloads 104

Authors: Shubham Jaiswal

Abstract:

In the present article, a drive is taken to compute the solution of spatial fractional order advection-dispersion equation having source/sink term with given initial and boundary conditions. The equation is converted to a system of ordinary differential equations using second-kind shifted Chebyshev polynomials, which have finally been solved using finite difference method. The striking feature of the article is the fast transportation of solute concentration as and when the system approaches fractional order from standard order for specified values of the parameters of the system.

Keywords: spatial fractional order advection-dispersion equation, second-kind shifted Chebyshev polynomial, collocation method, conservative system, non-conservative system

Procedia PDF Downloads 232

Authors: Samuel Hyellamada Jerry

Abstract:

This study focused on assessing conservation priorities within the Karlahi Forest Reserve of Fufore Local Government in Adamawa State. The main objective was to identify specific areas within the forest reserve that require immediate conservation attention. The research employed remote sensing and GIS techniques to achieve this goal. By overlaying the IDRIS Silva module results, a spatial distribution map was generated, highlighting the cumulative priority areas within and outside the forest. Among the total vegetated area of 26.38 km² in the Karlahi Forest Reserve, the analysis revealed that 16.16 km² were classified as high-priority conservation zones. Additionally, 4.59 km² and 5.63 km² were identified as medium and low-priority areas, respectively. In light of these findings, it is recommended that conservation efforts incorporate detailed land cover information and regular assessments of species diversity. Furthermore, strict adherence to national and state policies regarding forest reserves and parks is crucial for effective conservation management.

Keywords: priority, Karlahi, forest, reserve, IDRISI Silva, species diversity

Procedia PDF Downloads 122

Authors: Mrinal Jana, Geetanjali Panda

Abstract:

In this paper a methodology is developed to solve a nonlinear fractional programming problem in which the coefficients of the objective function and constraints are interval parameters. This model is transformed into a general optimization problem and relation between the original problem and the transformed problem is established. Finally the proposed methodology is illustrated through a numerical example.

Keywords: fractional programming, interval valued function, interval inequalities, partial order relation

Procedia PDF Downloads 490

Authors: Pablo Martin, Jorge Olivares, Fernando Maass

Abstract:

The use of fractional derivatives to different problems in Engineering and Physics has been increasing in the last decade. For this reason, we have here considered partial derivatives when the integral is a spherical Bessel function of the first kind in both regular and modified ones simple initial conditions have been also considered. In this way, the solution has been found as a combination of hypergeometric functions. The case of a general rational value for α of the fractional derivative α has been solved in a general way for alpha between zero and two. The modified spherical Bessel functions of the first kind have been also considered and how to go from the regular case to the modified one will be also shown.

Keywords: caputo fractional derivatives, hypergeometric functions, linear differential equations, spherical Bessel functions

Procedia PDF Downloads 295

Authors: Ayan Chakraborty, BV. Rathish Kumar

Abstract:

Off-late many models in viscoelasticity, signal processing or anomalous diffusion equations are formulated in fractional calculus. Tempered fractional calculus is the generalization of fractional calculus and in the last few years several important partial differential equations occurring in the different field of science have been reconsidered in this term like diffusion wave equations, Schr$\ddot{o}$dinger equation and so on. In the present paper, a time-dependent tempered fractional diffusion equation of order $\gamma \in (0,1)$ with forcing function is considered. Existence, uniqueness, stability, and regularity of the solution has been proved. Crank-Nicolson discretization is used in the time direction. B spline finite element approximation is implemented. Generally, B-splines basis are useful for representing the geometry of a finite element model, interfacing a finite element analysis program. By utilizing this technique a priori space-time estimate in finite element analysis has been derived and we proved that the convergent order is $\mathcal{O}(h²+T²)$ where $h$ is the space step size and $T$ is the time. A couple of numerical examples have been presented to confirm the accuracy of theoretical results. Finally, we conclude that the studied method is useful for solving tempered fractional diffusion equations.

Keywords: B-spline finite element, error estimates, Gronwall's lemma, stability, tempered fractional

Procedia PDF Downloads 156

Authors: A. Cherifi, B. Bouazza, A. O. Dahmane, B. Yagoubi

Abstract:

Transmission over Optical channels will introduce inter-symbol interference (ISI) as well as inter-channel (or inter-carrier) interference (ICI). To decrease the effects of ICI, this paper proposes equalizer for the Optical OFDM system based on the fractional Fourier transform (FrFFT). In this FrFT-OFDM system, traditional Fourier transform is replaced by fractional Fourier transform to modulate and demodulate the data symbols. The equalizer proposed consists of sampling the received signal in the different time per time symbol. Theoretical analysis and numerical simulation are discussed.

Keywords: OFDM, (FrFT) fractional fourier transform, optical OFDM, equalization algorithm

Procedia PDF Downloads 402

Authors: Neelam Singha

Abstract:

The present work intends to analyze the system dynamics of Hashimoto’s thyroiditis with the assistance of fractional calculus. Hashimoto’s thyroiditis or chronic lymphocytic thyroiditis is an autoimmune disorder in which the immune system attacks the thyroid gland, which gradually results in interrupting the normal thyroid operation. Consequently, the feedback control of the system gets disrupted due to thyroid follicle cell lysis. And, the patient perceives life-threatening clinical conditions like goiter, hyperactivity, euthyroidism, hyperthyroidism, etc. In this work, we aim to obtain the approximate solution to the posed fractional-order problem describing Hashimoto’s thyroiditis. We employ the Adomian decomposition method to solve the system of fractional-order differential equations, and the solutions obtained shall be useful to provide information about the effect of medical care. The numerical technique is executed in an organized manner to furnish the associated details of the progression of the disease and to visualize it graphically with suitable plots.

Keywords: adomian decomposition method, fractional derivatives, Hashimoto's thyroiditis, mathematical modeling

Procedia PDF Downloads 184