Search results for: Convergence

Authors: Sachin Aggarwal, Sarah Kassinger, Nicholas Hoffman

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Geometry simplification is a key step in performing conjugate heat transfer analysis using CFD. This paper proposes a standard methodology for the geometry simplification of rotating machines, such as electrical generators and electrical motors (both air and liquid-cooled). These machines are extensively deployed throughout the aerospace and automotive industries, where optimization of weight, volume, and performance is paramount -especially given the current global transition to renewable energy sources and vehicle hybridization and electrification. Conjugate heat transfer analysis is an essential step in optimizing their complex design. This methodology will help in reducing convergence issues due to poor mesh quality, thus decreasing computational cost and overall analysis time.

Keywords: CFD, electrical machines, Geometry simplification, heat transfer

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Authors: Slimane Ouhmad, Abdellah Halimi

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In this work, we have applied neural networks method MLP type to a database from an array of six sensors for the detection of three toxic gases. As the choice of the number of hidden layers and the weight values has a great influence on the convergence of the learning algorithm, we proposed, in this article, a mathematical formulation to determine the optimal number of hidden layers and good weight values based on the method of back propagation of errors. The results of this modeling have improved discrimination of these gases on the one hand, and optimize the computation time on the other hand, the comparison to other results achieved in this case.

Keywords: MLP Neural Network, back-propagation, number of neurons in the hidden layer, identification, computing time

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Authors: Abraham Motlhabane

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In most practical-related science lessons, the focus is on completing the experimental procedure as directed by the teacher. However, the scientific discourse among learners themselves and teacher–learner discourse about scientific processes, scientific inquiry and the nature of science should play an important role in the teaching and learning of science. This means the incorporation of inquiry-based activities aimed at sparking debates about scientific concepts. This article analyses a science lesson presented by a teacher to his colleagues acting as learners. Six lessons were presented and transcribed. One of the lessons has been used for this study as the basis for the events as they unfolded during the lesson. Data was obtained through direct observations and the use of a predetermined observation schedule. Field notes were compiled during teacher preparations and the presentation of the lessons.

Keywords: discourse, inquiry, practical work, science, scientific

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Authors: Aydin Teymourifar, Gurkan Ozturk

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In this paper, a new hybrid Hopfield neural network is proposed for the dynamic, flexible job shop scheduling problem. A new heuristic based and easy to implement energy function is designed for the Hopfield neural network, which penalizes the constraints violation and decreases makespan. Moreover, for enhancing the performance, several heuristics are integrated to it that achieve active, and non-delay schedules also, prevent early convergence of the neural network. The suggested algorithm that is designed as a generalization of the previous studies for the flexible and dynamic scheduling problems can be used for solving real scheduling problems. Comparison of the presented hybrid method results with the previous studies results proves its efficiency.

Keywords: dynamic flexible job shop scheduling, neural network, heuristics, constrained optimization

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Authors: Seyedmahdi Mousavihashemi

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One of the most important subjects of interest in researches is 'improving' which result in various algorithms. In so many geometrical problems we are faced with target functions which should be optimized. In group practices, all the functions’ cooperation lead to convergence. In the study, the optimization algorithm of dense particles is used. Usage of the algorithm improves the given performance norms. The results reveal that usage of swarm algorithm for reinforced particles in designing state feedback improves the given performance norm and in optimized designing of multi-target state feedback controlling, the network will maintain its bearing structure. The results also show that PSO is usable for optimization of state feedback controllers.

Keywords: multi-objective, enhanced, feedback, optimization, algorithm, particle, design

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Authors: Anteneh Getachew Gebrie, Rabian Wangkeeree

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The purpose of this paper is to introduce iterative algorithm solving split system of minimization problem given as a task of finding a common minimizer point of finite family of proper, lower semicontinuous convex functions and whose image under a bounded linear operator is also common minimizer point of another finite family of proper, lower semicontinuous convex functions. We obtain strong convergence of the sequence generated by our algorithm under some suitable conditions on the parameters. The iterative schemes are developed with a way of selecting the step sizes such that the information of operator norm is not necessary. Some applications and numerical experiment is given to analyse the efficiency of our algorithm.

Keywords: Hilbert Space, minimization problems, Moreau-Yosida approximate, split feasibility problem

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Authors: Yan Wang, Yuan George Shan

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This study examines whether earnings management reduces firm performance and how other comprehensive income (OCI) disclosure and strong corporate governance restrain earnings management. Using a data set comprising 6,260 firm-year observations from listed companies on the Shanghai and Shenzhen Stock Exchanges during 2009–2015, the results indicate that OCI disclosure generally improves firm performance, but earnings management lowers firm performance. The study also finds that OCI disclosure and corporate governance are complementary in restraining earnings manipulation and promote firm performance. The implications of the findings are relevant policy-makers and regulators in assisting them evaluate the consequences of convergence of Chinese Accounting Standards with the International Financial Reporting Standards.

Keywords: other comprehensive income, corporate governance, earnings management, firm performance, China

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Authors: Semeh Ben Salem, Sami Naouali, Moetez Sallami

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The increasing amount of collected data has limited the performance of the current analyzing algorithms. Thus, developing new cost-effective algorithms in terms of complexity, scalability, and accuracy raised significant interests. In this paper, a modified effective k-means based algorithm is developed and experimented. The new algorithm aims to reduce the computational load without significantly affecting the quality of the clusterings. The algorithm uses the City Block distance and a new stop criterion to guarantee the convergence. Conducted experiments on a real data set show its high performance when compared with the original k-means version.

Keywords: pattern recognition, global terrorism database, Manhattan distance, k-means clustering, terrorism data analysis

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Authors: Safa Ougoujil, Sidi Mohamed Rigar

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Economics is not an exact science. It cannot be from the moment it is a social science that concerns society organization, a human science that depends on the behavior of the men and women who make a part of this society. Therefore, it cannot ignore morality, the instinctive sense of good and evil, the natural order which place us between certain values, and which religion often sheds light on. In terms of finance, the reference to ethics is becoming more popular than ever. This is naturally due to the growing financial crises. Finance is less and less ethical, but some financial practices have continued to do so. This is the case of ethical finance and Islamic finance. After attempting to define the concepts of ethical finance and Islamic finance, in a period when financial innovation seeks to encourage differentiation in order to create more profit margins, this article attempts to expose the particularities, the convergences and the potentialities of development of these two sensibilities.

Keywords: convergences, ethical finance, Islamic finance, potential development

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Authors: Aman Sharma, Sonali Sengupta

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The study explores the utility of Bayesian optimization in tuning the physical and chemical parameters of reactions in an offline experimental setup. A comparative analysis of the influence of the acquisition function on the optimization performance is also studied. For proxy first and second-order reactions, the results are indifferent to the acquisition function used, whereas, while studying the parameters for oxidative desulphurization of thiophene in an offline setup, upper confidence bound (UCB) provides faster convergence along with a marginal trade-off in the maximum conversion achieved. The work also demarcates the critical number of independent parameters and input observations required for both sequential and offline reaction setups to yield tangible results.

Keywords: acquisition function, Bayesian optimization, desulfurization, kinetics, thiophene

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Authors: Anila Syed, Manzoor Ahmad

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The relationship between United States and India has transformed from estrangement to wider engagement since 2004. With the convergence of interests and shared values both the US and India came close towards each other and evolved strategic partnership through civil nuclear cooperation. This paper analyze the cost and benefit of strategic partnership with India for US, the impact of India’s emergence as regional power on South Asian balance of power and its impact on Pak-US relationship. It also focuses on security structure of the region and challenges for the US to maintain strategic partnership with two rival states (India and Pakistan). The work also gives some recommendations for balancing power in the region in order to ensure durable peace not only between India and Pakistan but also in south Asia.

Keywords: US-India strategic partnership, civil-nuclear cooperation, balance of power, impacts on Pak-US relationship

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Authors: Chivon Choeung, Heng Tang, Panha Soth, Vichet Huy

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This paper presents a robust control strategy for a single-phase DC-AC inverter with an output LC-filter. An all-pass filter is utilized to create an artificial β-signal so that the proposed controller can be simply used in dq-synchronous frame. The proposed robust controller utilizes a state feedback control with integral action in the dq-synchronous frame. A linear matrix inequality-based optimization scheme is used to determine stabilizing gains of the controllers to maximize the convergence rate to steady state in the presence of uncertainties. The uncertainties of the system are described as the potential variation range of the inductance and resistance in the LC-filter.

Keywords: single-phase inverter, linear matrix inequality, robust control, all-pass filter

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Authors: Labar Hocine, Kelaiaia Mounia Samira, Mesbah Tarek, Kelaiaia Samia

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Maximum power point tracking (MPPT) techniques are used in photovoltaic (PV) systems to maximize the PV array output power by tracking continuously the maximum power point(MPP) which depends on panels temperature and on irradiance conditions. The main drawback of P&O is that, the operating point oscillates around the MPP giving rise to the waste of some amount of available energy; moreover, it is well known that the P&O algorithm can be confused during those time intervals characterized by rapidly changing atmospheric conditions. In this paper, it is shown that in order to limit the negative effects associated to the above drawbacks, the P&O MPPT parameters must be customized to the dynamic behavior of the specific converter adopted. A theoretical analysis allowing the optimal choice of such initial set parameters is also carried out. The fast convergence of the proposal is proven.

Keywords: P&O, Taylor’s series, MPPT, photovoltaic panel

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Authors: K. Raghuwaiya, S. Singh, B. Sharma, J. Vanualailai

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This paper presents a set of artificial potential field functions that improves upon; in general, the motion planning and posture control, with theoretically guaranteed point and posture stabilities, convergence and collision avoidance properties of 3-trailer systems in a priori known environment. We basically design and inject two new concepts; ghost walls and the Distance Optimization Technique (DOT) to strengthen point and posture stabilities, in the sense of Lyapunov, of our dynamical model. This new combination of techniques emerges as a convenient mechanism for obtaining feasible orientations at the target positions with an overall reduction in the complexity of the navigation laws. The effectiveness of the proposed control laws were demonstrated via simulations of two traffic scenarios.

Keywords: artificial potential fields, 3-trailer systems, motion planning, posture, parking and collision, free trajectories

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Authors: Talha A. Taj, Talha A. Khan, M. Imran Khalid

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Optimization techniques attract researchers to formulate a problem and determine its optimum solution. This paper presents an Enhanced Harmony Search (ENHS) algorithm for solving optimization problems. The proposed algorithm increases the convergence and is more efficient than the standard Harmony Search (HS) algorithm. The paper discusses the novel techniques in detail and also provides the strategy for tuning the decisive parameters that affects the efficiency of the ENHS algorithm. The algorithm is tested on various benchmark functions, a real world optimization problem and a constrained objective function. Also, the results of ENHS are compared to standard HS, and various other optimization algorithms. The ENHS algorithms prove to be significantly better and more efficient than other algorithms. The simulation and testing of the algorithms is performed in MATLAB.

Keywords: optimization, harmony search algorithm, MATLAB, electronic

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Authors: Aymen Laadhari

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We present in this paper a fully implicit finite element method tailored for the numerical modeling of inextensible fluidic membranes in a surrounding Newtonian fluid. We consider a highly simplified version of the Canham-Helfrich model for phospholipid membranes, in which the bending force and spontaneous curvature are disregarded. The coupled problem is formulated in a fully Eulerian framework and the membrane motion is tracked using the level set method. The resulting nonlinear problem is solved by a Newton-Raphson strategy, featuring a quadratic convergence behavior. A monolithic solver is implemented, and we report several numerical experiments aimed at model validation and illustrating the accuracy of the proposed method. We show that stability is maintained for significantly larger time steps with respect to an explicit decoupling method.

Keywords: finite element method, level set, Newton, membrane

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Authors: Hesham A. Elkaranshawy, Amr M. Abdelrazek, Hosam M. Ezzat

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The system of ordinary nonlinear differential equations describing sliding velocity during impact with friction for a three-dimensional rigid-multibody system is developed. No analytical solutions have been obtained before for this highly nonlinear system. Hence, a power series solution is proposed. Since the validity of this solution is limited to its convergence zone, a suitable time step is chosen and at the end of it a new series solution is constructed. For a case study, the trajectory of the sliding velocity using the proposed method is built using 6 time steps, which coincides with a Runge-Kutta solution using 38 time steps.

Keywords: impact with friction, nonlinear ordinary differential equations, power series solutions, rough collision

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Authors: Noga Bregman

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Accurate and efficient earthquake detection and phase picking are crucial for seismic hazard assessment and emergency response. This study introduces EQMamba, a deep-learning method that combines the strengths of the Earthquake Transformer and the Mamba model for simultaneous earthquake detection and phase picking. EQMamba leverages the computational efficiency of Mamba layers to process longer seismic sequences while maintaining a manageable model size. The proposed architecture integrates convolutional neural networks (CNNs), bidirectional long short-term memory (BiLSTM) networks, and Mamba blocks. The model employs an encoder composed of convolutional layers and max pooling operations, followed by residual CNN blocks for feature extraction. Mamba blocks are applied to the outputs of BiLSTM blocks, efficiently capturing long-range dependencies in seismic data. Separate decoders are used for earthquake detection, P-wave picking, and S-wave picking. We trained and evaluated EQMamba using a subset of the STEAD dataset, a comprehensive collection of labeled seismic waveforms. The model was trained using a weighted combination of binary cross-entropy loss functions for each task, with the Adam optimizer and a scheduled learning rate. Data augmentation techniques were employed to enhance the model's robustness. Performance comparisons were conducted between EQMamba and the EQTransformer over 20 epochs on this modest-sized STEAD subset. Results demonstrate that EQMamba achieves superior performance, with higher F1 scores and faster convergence compared to EQTransformer. EQMamba reached F1 scores of 0.8 by epoch 5 and maintained higher scores throughout training. The model also exhibited more stable validation performance, indicating good generalization capabilities. While both models showed lower accuracy in phase-picking tasks compared to detection, EQMamba's overall performance suggests significant potential for improving seismic data analysis. The rapid convergence and superior F1 scores of EQMamba, even on a modest-sized dataset, indicate promising scalability for larger datasets. This study contributes to the field of earthquake engineering by presenting a computationally efficient and accurate method for simultaneous earthquake detection and phase picking. Future work will focus on incorporating Mamba layers into the P and S pickers and further optimizing the architecture for seismic data specifics. The EQMamba method holds the potential for enhancing real-time earthquake monitoring systems and improving our understanding of seismic events.

Keywords: earthquake, detection, phase picking, s waves, p waves, transformer, deep learning, seismic waves

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Authors: Houda Abadlia, Nadia Smairi, Khaled Ghedira

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Multiobjective Particle Swarm Optimization (MOPSO) has shown an effective performance for solving test functions and real-world optimization problems. However, this method has a premature convergence problem, which may lead to lack of diversity. In order to improve its performance, this paper presents a hybrid approach which embedded the MOPSO into the island model and integrated a local search technique, Variable Neighborhood Search, to enhance the diversity into the swarm. Experiments on two series of test functions have shown the effectiveness of the proposed approach. A comparison with other evolutionary algorithms shows that the proposed approach presented a good performance in solving multiobjective optimization problems.

Keywords: particle swarm optimization, migration, variable neighborhood search, multiobjective optimization

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Authors: Leonid Borisov, Yuri Orlov

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We obtain explicit formulas of finitely multiple approximations of the equilibrium density matrix for the case of the harmonic oscillator using Chernoff's theorem and the notion of semigroup which is Chernoff equivalent to average semigroup. Also we found explicit formulas for the corresponding approximate Wigner functions and average values of the observable. We consider a superposition of τ -quantizations representing a wide class of linear quantizations. We show that the convergence of the approximations of the average values of the observable is not uniform with respect to the Gibbs parameter. This does not allow to represent approximate expression as the sum of the exact limits and small deviations evenly throughout the temperature range with a given order of approximation.

Keywords: Chernoff theorem, Feynman formulas, finitely multiple approximation, harmonic oscillator, Wigner function

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Authors: Shruti Hajirnis

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In May 2014, FASB issued Accounting Standards Update (ASU) 2014-09, Revenue from Contracts with Customers (Topic 606), and the International Accounting Standards Board (IASB) issued International Financial Reporting Standards (IFRS) 15, Revenue from Contracts with Customers that will supersede virtually all revenue recognition requirements in IFRS and US GAAP. FASB and the IASB have basically achieved convergence with these standards, with only some minor differences such as collectability threshold, interim disclosure requirements, early application and effective date, impairment loss reversal and nonpublic entity requirements. This paper discusses the impact of five-step model prescribed in new revenue standard on the entities operating in Pharma industry. It also outlines the considerations for these entities while implementing the new standard.

Keywords: revenue recognition, pharma industry, standard, requirements

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Authors: Raphael Zanella, Todd A. Oliver, Karl W. Schulz

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This work deals with the finite element approximation of axisymmetric compressible flows with swirl velocity. We are interested in problems where the flow, while weakly dependent on the azimuthal coordinate, may have a strong azimuthal velocity component. We describe the approximation of the compressible Navier-Stokes equations with H1-conformal spaces of axisymmetric functions. The weak formulation is implemented in a C++ solver with explicit time marching. The code is first verified with a convergence test on a manufactured solution. The verification is completed by comparing the numerical and analytical solutions in a Poiseuille flow case and a Taylor-Couette flow case. The code is finally applied to the problem of a swirling subsonic air flow in a plasma torch geometry.

Keywords: axisymmetric problem, compressible Navier-Stokes equations, continuous finite elements, swirling flow

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Authors: Belloufi Mohammed, Sellami Badreddine

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This paper deals with a new nonlinear modified three-term conjugate gradient algorithm for solving large-scale unstrained optimization problems. The search direction of the algorithms from this class has three terms and is computed as modifications of the classical conjugate gradient algorithms to satisfy both the descent and the conjugacy conditions. An example of three-term conjugate gradient algorithm from this class, as modifications of the classical and well known Hestenes and Stiefel or of the CG_DESCENT by Hager and Zhang conjugate gradient algorithms, satisfying both the descent and the conjugacy conditions is presented. Under mild conditions, we prove that the modified three-term conjugate gradient algorithm with Wolfe type line search is globally convergent. Preliminary numerical results show the proposed method is very promising.

Keywords: unconstrained optimization, three-term conjugate gradient, sufficient descent property, line search

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Authors: Pius W. Molo Chin

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Real life problems such as the Huxley equation are always modeled as nonlinear differential equations. These problems need accurate and reliable methods for their solutions. In this paper, we propose a nonstandard finite difference method in time and the Galerkin combined with the compactness method in the space variables. This coupled method, is used to analyze a two dimensional Huxley equation for the existence and uniqueness of the continuous solution of the problem in appropriate spaces to be defined. We proceed to design a numerical scheme consisting of the aforementioned method and show that the scheme is stable. We further show that the stable scheme converges with the rate which is optimal in both the L2 as well as the H1-norms. Furthermore, we show that the scheme replicates the decaying qualities of the exact solution. Numerical experiments are presented with the help of an example to justify the validity of the designed scheme.

Keywords: Huxley equations, non-standard finite difference method, Galerkin method, optimal rate of convergence

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Authors: Emmanuel Omokhuale

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A mathematical model is presented to study free convective boundary layer flow in a semi-infinite vertical cylinder with heat sink effect in a porous medium. The governing dimensional governing partial differential equations (PDEs) with corresponding initial and boundary conditions are approximated and solved numerically employing finite difference method (FDM) the implicit type. Stability and convergence of the scheme are also established. Furthermore, the influence of significant physical parameters on the flow characteristics was analysed and shown graphically. The obtained results are benchmarked with previously published works in order to access the accuracy of the numerical method and found to be in good agreement.

Keywords: free convection flow, vertical cylinder, implicit finite difference method, heat sink and porous medium

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Authors: Yunos Zahri, Ariffin Aswami

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Cyber terrorism represents the convergence of two worlds: virtual and physical. The virtual world is a place in which computer programs function and data move, whereas the physical world is where people live and function. The merging of these two domains is the interface being targeted in the incidence of cyber terrorism. To better understand why cyber terrorism acts are committed, this study presents the context of cyber terrorism from motivational perspectives. Motivational forces behind cyber terrorism can be social, political, ideological and economic. In this research, data are analyzed using a qualitative method. A semi-structured interview with purposive sampling was used for data collection. With the growing interconnectedness between critical infrastructures and Information & Communication Technology (ICT), selecting targets that facilitate maximum disruption can significantly influence terrorists. This work provides a baseline for defining the concept of cyber terrorism from motivational perspectives.

Keywords: cyber terrorism, terrorism, motivation, qualitative analysis

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Authors: Daphne Ashley Sze, Kenneth Santos, Ariane Gabrielle Lim

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As women are now given more freedom and choice to pursue employment, the world’s over-all fertility has been decreasing mainly due to the shift in time allocation between working and child rearing. As such, we study the case of the Philippines, where there exists a decreasing fertility rate and increasing openness for women labor participation. We focused on the distinction between fertility and fecundity, the former being the manifestation of the latter and aim to trace and compare the effects of both fecundity and fertility to women’s employment status through the estimation of the reproduction function and multinomial logistic function. Findings suggest that the perception of women regarding employment opportunities in the Philippines links the negative relationship observed between fertility, fecundity and women’s employment status. Today, there has been a convergence in the traditional family roles of men and women, as both genders now have identical employment opportunities that continue to shape their preferences.

Keywords: multinomial logistic function, tobit, fertility, women employment status, fecundity

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Authors: Fariborz Ahmadi, Reza Tati

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Genetic algorithm is a soft computing method that works on set of solutions. These solutions are called chromosome and the best one is the absolute solution of the problem. The main problem of this algorithm is that after passing through some generations, it may be produced some chromosomes that had been produced in some generations ago that causes reducing the convergence speed. From another respective, most of the genetic algorithms are implemented in software and less works have been done on hardware implementation. Our work implements genetic algorithm in hardware that doesn’t produce chromosome that have been produced in previous generations. In this work, most of genetic operators are implemented without producing iterative chromosomes and genetic diversity is preserved. Genetic diversity causes that not only do not this algorithm converge to local optimum but also reaching to global optimum. Without any doubts, proposed approach is so faster than software implementations. Evaluation results also show the proposed approach is faster than hardware ones.

Keywords: hardware, genetic algorithm, computer science, engineering

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Authors: Behzad Babaei, B. Gangadhara Prusty

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The objective of this study is to assess the hypotheses that a restored tooth with a class II occlusal-distal (OD) cavity can be strengthened by designing an optimized cavity geometry, as well as selecting the composite restoration with optimized elastic moduli when there is a sharp de-bonded edge at the interface of the tooth and restoration. Methods: A scanned human maxillary molar tooth was segmented into dentine and enamel parts. The dentine and enamel profiles were extracted and imported into a finite element (FE) software. The enamel rod orientations were estimated virtually. Fifteen models for the restored tooth with different cavity occlusal depths (1.5, 2, and 2.5 mm) and internal cavity angles were generated. By using a semi-circular stone part, a 400 N load was applied to two contact points of the restored tooth model. The junctions between the enamel, dentine, and restoration were considered perfectly bonded. All parts in the model were considered homogeneous, isotropic, and elastic. The quadrilateral and triangular elements were employed in the models. A mesh convergence analysis was conducted to verify that the element numbers did not influence the simulation results. According to the criteria of a 5% error in the stress, we found that a total element number of over 14,000 elements resulted in the convergence of the stress. A Python script was employed to automatically assign 2-22 GPa moduli (with increments of 4 GPa) for the composite restorations, 18.6 GPa to the dentine, and two different elastic moduli to the enamel (72 GPa in the enamel rods’ direction and 63 GPa in perpendicular one). The linear, homogeneous, and elastic material models were considered for the dentine, enamel, and composite restorations. 108 FEA simulations were successively conducted. Results: The internal cavity angles (α) significantly altered the peak maximum principal stress at the interface of the enamel and restoration. The strongest structures against the contact loads were observed in the models with α = 100° and 105. Even when the enamel rods’ directional mechanical properties were disregarded, interestingly, the models with α = 100° and 105° exhibited the highest resistance against the mechanical loads. Regarding the effect of occlusal cavity depth, the models with 1.5 mm depth showed higher resistance to contact loads than the model with thicker cavities (2.0 and 2.5 mm). Moreover, the composite moduli in the range of 10-18 GPa alleviated the stress levels in the enamel. Significance: For the class II OD cavity models in this study, the optimal geometries, composite properties, and occlusal cavity depths were determined. Designing the cavities with α ≥100 ̊ was significantly effective in minimizing peak stress levels. The composite restoration with optimized properties reduced the stress concentrations on critical points of the models. Additionally, when more enamel was preserved, the sturdier enamel-restoration interface against the mechanical loads was observed.

Keywords: dental composite restoration, cavity geometry, finite element approach, maximum principal stress

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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

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