Search results for: first integrals

Authors: Baghdad Said

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Differential equations of fractional order have proved to be important tools to describe many physical phenomena and have been used in diverse fields such as engineering, mathematics as well as other applied sciences. On the other hand, the theory of differential equations involving the Stieltjes derivative (SD) with respect to a non-decreasing function is a new class of differential equations and has many applications as a unified framework for dynamic equations on time scales and differential equations with impulses at fixed times. The aim of this paper is to investigate the existence, uniqueness, and generalized Ulam-Hyers-Rassias stability (UHRS) of solutions for a boundary value problem of sequential fractional differential equations (SFDE) containing (SD). This study is based on the technique of noncompactness measures (MNCs) combined with Monch-Krasnoselski fixed point theorems (FPT), and the results are proven in an appropriate Banach space under sufficient hypotheses. We also give an illustrative example. In this work, we introduced a class of (SFDE) and the results are obtained under a few hypotheses. Future directions connected to this work could focus on another problem with different types of fractional integrals and derivatives, and the (SD) will be assumed under a more general hypothesis in more general functional spaces.

Keywords: SFDE, SD, UHRS, MNCs, FPT

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Authors: T. J. Hasanova, C. N. Imamalieva

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The problem about the movement of the rigid cylinder keeping the vertical position under the influence of running superficial waves in a liquid is considered. The indignation of a falling wave caused by the presence of the cylinder which moves is thus considered. Special decomposition on a falling harmonious wave is used. The problem dares an operational method. For a finding of the original decision, Considering that the image denominator represents a tabular function, Voltaire's integrated equation of the first sort which dares a numerical method is used. Cylinder movement in the continuous environment under the influence of waves is considered in work. Problems are solved by an operational method, thus originals of required functions are looked for by the numerical definition of poles of combinations of transcendental functions and calculation of not own integrals. Using specificity of a task below, Decisions are under construction the numerical solution of the integrated equation of Volter of the first sort that does not create computing problems of the complex roots of transcendental functions connected with search.

Keywords: rigid cylinder, linear interpolation, fluctuations, Voltaire's integrated equation, harmonious wave

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Authors: Yingjie Zhang, Ming Li, Ying Zhang, Jing Zhang, Zuolei Hu

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In the actual start-up process of DC motors, the DC drive system often faces a conflict between energy consumption and acceleration performance. To resolve the conflict, this paper proposes a comprehensive performance index that energy consumption index is added on the basis of classical control performance index in the DC motor starting process. Taking the comprehensive performance index as the cost function, particle swarm optimization algorithm is designed to optimize the comprehensive performance. Then it conducts simulations on the optimization of the comprehensive performance of the DC motor on condition that the weight coefficient of the energy consumption index should be properly designed. The simulation results show that as the weight of energy consumption increased, the energy efficiency was significantly improved at the expense of a slight sacrifice of fastness indicators with the comprehensive performance index method. The energy efficiency was increased from 63.18% to 68.48% and the response time reduced from 0.2875s to 0.1736s simultaneously compared with traditional proportion integrals differential controller in energy saving.

Keywords: comprehensive performance index, energy consumption, acceleration performance, particle swarm optimal control

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Authors: M. Asl. Dehghan, M. H. Daemi, S. Radmard, S. H. Nabavi

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Thin disk lasers are engineered for efficient thermal cooling and exhibit superior performance for this task. However the disk thickness and large pumped area make the use of this gain format in a resonator difficult when constructing a single-mode laser. Choosing an unstable resonator design is beneficial for this purpose. On the other hand, the low gain medium restricts the application of unstable resonators to low magnifications and therefore to a poor beam quality. A promising idea to enable the application of unstable resonators to wide aperture, low gain lasers is to couple a fraction of the out coupled radiation back into the resonator. The output coupling gets dependent on the ratio of the back reflection and can be adjusted independently from the magnification. The excitation of the converging wave can be done by the use of an external reflector. The resonator performance is numerically predicted. First of all the threshold condition of linear, V and 2V shape resonator is investigated. Results show that the maximum magnification is 1.066 that is very low for high quality purposes. Inserting an additional reflector covers the low gain. The reflectivity and the related magnification of a 350 micron Yb:YAG disk are calculated. The theoretical model was based on the coupled Kirchhoff integrals and solved numerically by the Fox and Li algorithm. Results show that with back reflection mechanism in combination with increasing the number of beam incidents on disk, high gain and high magnification can occur.

Keywords: unstable resonators, thin disk lasers, gain, external reflector

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Authors: Abdulrahman Abdulrahman

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A side weir is a hydraulic structure set into the side of a channel. This structure is used for water level control in channels, to divert flow from a main channel into a side channel when the water level in the main channel exceeds a specific limit and as storm overflows from urban sewerage system. Computation of water surface over the side weirs is essential to determine the flow rate of the side weir. Analytical solutions for water surface profile along rectangular side weir are available only for the special cases of rectangular and trapezoidal channels considering constant specific energy. In this paper, a rectangular side weir located in a combined (trapezoidal with exponential) channel was considered. Expanding binominal series of integer and fraction powers and the using of reduction formula of cosine function integrals, a general analytical formula was obtained for water surface profile along a side weir in a combined (trapezoidal with exponential) channel. Since triangular, rectangular, trapezoidal and parabolic cross-sections are special cases of the combined cross section, the derived formula, is applicable to triangular, rectangular, trapezoidal cross-sections as analytical solution and semi-analytical solution to parabolic cross-section with maximum relative error smaller than 0.76%. The proposed solution should be a useful engineering tool for the evaluation and design of side weirs in open channel.

Keywords: analytical solution, combined channel, exponential channel, side weirs, trapezoidal channel, water surface profile

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Authors: Farouk Metiri, Halim Zeghdoudi, Mohamed Riad Remita

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Credibility theory is an experience rating technique in actuarial science which can be seen as one of quantitative tools that allows the insurers to perform experience rating, that is, to adjust future premiums based on past experiences. It is used usually in automobile insurance, worker's compensation premium, and IBNR (incurred but not reported claims to the insurer) where credibility theory can be used to estimate the claim size amount. In this study, we focused on a popular tool in credibility theory which is the Bayesian premium estimator, considering Lindley distribution as a claim distribution. We derive this estimator under entropy loss which is asymmetric and squared error loss which is a symmetric loss function with informative and non-informative priors. In a purely Bayesian setting, the prior distribution represents the insurer’s prior belief about the insured’s risk level after collection of the insured’s data at the end of the period. However, the explicit form of the Bayesian premium in the case when the prior is not a member of the exponential family could be quite difficult to obtain as it involves a number of integrations which are not analytically solvable. The paper finds a solution to this problem by deriving this estimator using numerical approximation (Lindley approximation) which is one of the suitable approximation methods for solving such problems, it approaches the ratio of the integrals as a whole and produces a single numerical result. Simulation study using Monte Carlo method is then performed to evaluate this estimator and mean squared error technique is made to compare the Bayesian premium estimator under the above loss functions.

Keywords: bayesian estimator, credibility theory, entropy loss, monte carlo simulation

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Authors: Luiz G. Véras, Felipe L. Medeiros, Lamartine F. Guimarães

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This work approaches the automatic planning of paths for Unmanned Aerial Vehicles (UAVs) through the application of the Rapidly Exploring Random Tree Star-Smart (RRT*-Smart) algorithm. RRT*-Smart is a sampling process of positions of a navigation environment through a tree-type graph. The algorithm consists of randomly expanding a tree from an initial position (root node) until one of its branches reaches the final position of the path to be planned. The algorithm ensures the planning of the shortest path, considering the number of iterations tending to infinity. When a new node is inserted into the tree, each neighbor node of the new node is connected to it, if and only if the extension of the path between the root node and that neighbor node, with this new connection, is less than the current extension of the path between those two nodes. RRT*-smart uses an intelligent sampling strategy to plan less extensive routes by spending a smaller number of iterations. This strategy is based on the creation of samples/nodes near to the convex vertices of the navigation environment obstacles. The planned paths are smoothed through the application of the method called quintic pythagorean hodograph curves. The smoothing process converts a route into a dynamically-viable one based on the kinematic constraints of the vehicle. This smoothing method models the hodograph components of a curve with polynomials that obey the Pythagorean Theorem. Its advantage is that the obtained structure allows computation of the curve length in an exact way, without the need for quadratural techniques for the resolution of integrals.

Keywords: path planning, path smoothing, Pythagorean hodograph curve, RRT*-Smart

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Authors: Kizito Ugochukwu Nwajeri

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This paper presents a distinct approach to solving fractional dynamical systems using hybrid block methods (HBMs). Fractional calculus extends the concept of derivatives and integrals to non-integer orders and finds increasing application in fields such as physics, engineering, and finance. However, traditional numerical techniques often struggle to accurately capture the complex behaviors exhibited by these systems. To address this challenge, we develop HBMs that integrate single-step and multi-step methods, enabling the simultaneous computation of multiple solution points while maintaining high accuracy. Our approach employs polynomial interpolation and collocation techniques to derive a system of equations that effectively models the dynamics of fractional systems. We also directly incorporate boundary and initial conditions into the formulation, enhancing the stability and convergence properties of the numerical solution. An adaptive step-size mechanism is introduced to optimize performance based on the local behavior of the solution. Extensive numerical simulations are conducted to evaluate the proposed methods, demonstrating significant improvements in accuracy and efficiency compared to traditional numerical approaches. The results indicate that our hybrid block methods are robust and versatile, making them suitable for a wide range of applications involving fractional dynamical systems. This work contributes to the existing literature by providing an effective numerical framework for analyzing complex behaviors in fractional systems, thereby opening new avenues for research and practical implementation across various disciplines.

Keywords: fractional calculus, numerical simulation, stability and convergence, Adaptive step-size mechanism, collocation methods

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Authors: Steffen Illig, Alexander S. Eggeman, Alessandro Troisi, Stephen G. Yeates, John E. Anthony, Henning Sirringhaus

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Large-amplitude intermolecular vibrations in combination with complex shaped transfer integrals generate a thermally fluctuating energetic landscape. The resulting dynamic disorder and its intrinsic presence in organic semiconductors is one of the most fundamental differences to their inorganic counterparts. Dynamic disorder is believed to govern many of the unique electrical and optical properties of organic systems. However, the low energy nature of these vibrations makes it difficult to access them experimentally and because of this we still lack clear molecular design rules to control and reduce dynamic disorder. Applying a novel technique based on electron diffraction we encountered strong intermolecular, thermal vibrations in every single organic material we studied (14 up to date), indicating that a large degree of dynamic disorder is a universal phenomenon in organic crystals. In this paper a new molecular design strategy will be presented to avoid dynamic disorder. We found that small molecules that have their side chains attached to the long axis of their conjugated core have been found to be less likely to suffer from dynamic disorder effects. In particular, we demonstrate that 2,7-dioctyl[1]benzothieno[3,2-b][1]benzothio-phene (C8-BTBT) and 2,9-di-decyl-dinaphtho-[2,3-b:20,30-f]-thieno-[3,2-b]-thiophene (C10DNTT) exhibit strongly reduced thermal vibrations in comparison to other molecules and relate their outstanding performance to their lower dynamic disorder. We rationalize the low degree of dynamic disorder in C8-BTBT and C10-DNTT with a better encapsulation of the conjugated cores in the crystal structure which helps reduce large amplitude thermal motions. The work presented in this paper provides a general strategy for the design of new classes of very high mobility organic semiconductors with low dynamic disorder.

Keywords: charge transport, C8-BTBT, C10-DNTT, dynamic disorder, organic semiconductors, thermal vibrations

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Authors: Wedad Albalawi

Abstract:

The mathematical inequalities have been the core of mathematical study and used in almost all branches of mathematics as well in various areas of science and engineering. The inequalities by Hardy, Littlewood and Polya were the first significant composition of several science. This work presents fundamental ideas, results and techniques and it has had much influence on research in various branches of analysis. Since 1934, various inequalities have been produced and studied in the literature. Furthermore, some inequalities have been formulated by some operators; in 1989, weighted Hardy inequalities have been obtained for integration operators. Then, they obtained weighted estimates for Steklov operators that were used in the solution of the Cauchy problem for the wave equation. They were improved upon in 2011 to include the boundedness of integral operators from the weighted Sobolev space to the weighted Lebesgue space. Some inequalities have been demonstrated and improved using the Hardy–Steklov operator. Recently, a lot of integral inequalities have been improved by differential operators. Hardy inequality has been one of the tools that is used to consider integrity solutions of differential equations. Then dynamic inequalities of Hardy and Coposon have been extended and improved by various integral operators. These inequalities would be interesting to apply in different fields of mathematics (functional spaces, partial differential equations, mathematical modeling). Some inequalities have been appeared involving Copson and Hardy inequalities on time scales to obtain new special version of them. A time scale is defined as a closed subset contains real numbers. Then the inequalities of time scales version have received a lot of attention and has had a major field in both pure and applied mathematics. There are many applications of dynamic equations on time scales to quantum mechanics, electrical engineering, neural networks, heat transfer, combinatorics, and population dynamics. This study focuses on double integrals to obtain new time-scale inequalities of Copson driven by Steklov operator. They will be applied in the solution of the Cauchy problem for the wave equation. The proof can be done by introducing restriction on the operator in several cases. In addition, the obtained inequalities done by using some concepts in time scale version such as time scales calculus, theorem of Fubini and the inequality of H¨older.

Keywords: time scales, inequality of Hardy, inequality of Coposon, Steklov operator

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Authors: Lukas Vierus, Thomas Schuster

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A general framework is examined for dynamic tensor field tomography within an inhomogeneous medium characterized by refraction and absorption, treated as an inverse source problem concerning the associated transport equation. Guided by Fermat’s principle, the Riemannian metric within the specified domain is determined by the medium's refractive index. While considerable literature exists on the inverse problem of reconstructing a tensor field from its longitudinal ray transform within a static Euclidean environment, limited inversion formulas and algorithms are available for general Riemannian metrics and time-varying tensor fields. It is established that tensor field tomography, akin to an inverse source problem for a transport equation, persists in dynamic scenarios. Framing dynamic tensor tomography as an inverse source problem embodies a comprehensive perspective within this domain. Ensuring well-defined forward mappings necessitates establishing existence and uniqueness for the underlying transport equations. However, the bilinear forms of the associated weak formulations fail to meet the coercivity condition. Consequently, recourse to viscosity solutions is taken, demonstrating their unique existence within suitable Sobolev spaces (in the static case) and Sobolev-Bochner spaces (in the dynamic case), under a specific assumption restricting variations in the refractive index. Notably, the adjoint problem can also be reformulated as a transport equation, with analogous results regarding uniqueness. Analytical solutions are expressed as integrals over geodesics, facilitating more efficient evaluation of forward and adjoint operators compared to solving partial differential equations. Certainly, here's the revised sentence in English: Numerical experiments are conducted using a Nesterov-accelerated Landweber method, encompassing various fields, absorption coefficients, and refractive indices, thereby illustrating the enhanced reconstruction achieved through this holistic modeling approach.

Keywords: attenuated refractive dynamic ray transform of tensor fields, geodesics, transport equation, viscosity solutions

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Authors: S. Trafela, X. Xua, K. Zuzek Rozmana

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In this study, we used an accurate and precise method to measure the electrochemically active surface areas (Aecsa) of nickel electrodes. Calculated Aecsa is really important for the evaluation of an electro-catalyst’s activity in electrochemical reaction of different organic compounds. The method involves the electrochemical formation of Ni(OH)₂ and NiOOH in the presence of adsorbed oxalate in alkaline media. The studies were carried out using cyclic voltammetry with polycrystalline nickel as a reference material and electrodeposited nickel nanowires, homogeneous and heterogeneous nickel films. From cyclic voltammograms, the charge (Q) values for the formation of Ni(OH)₂ and NiOOH surface oxides were calculated under various conditions. At sufficiently fast potential scan rates (200 mV s⁻¹), the adsorbed oxalate limits the growth of the surface hydroxides to a monolayer. Although the Ni(OH)₂/NiOOH oxidation peak overlaps with the oxygen evolution reaction, in the reverse scan, the NiOOH/ Ni(OH)₂ reduction peak is well-separated from other electrochemical processes and can be easily integrated. The values of these integrals were used to correlate experimentally measured charge density with an electrochemically active surface layer. The Aecsa of the nickel nanowires, homogeneous and heterogeneous nickel films were calculated to be Aecsa-NiNWs = 4.2066 ± 0.0472 cm², Aecsa-homNi = 1.7175 ± 0.0503 cm² and Aecsa-hetNi = 2.1862 ± 0.0154 cm². These valuable results were expanded and used in electrochemical studies of formaldehyde oxidation. As mentioned nickel nanowires, heterogeneous and homogeneous nickel films were used as simple and efficient sensor for formaldehyde detection. For this purpose, electrodeposited nickel electrodes were modified in 0.1 mol L⁻¹ solution of KOH in order to expect electrochemical activity towards formaldehyde. The investigation of the electrochemical behavior of formaldehyde oxidation in 0.1 mol L⁻¹ NaOH solution at the surface of modified nickel nanowires, homogeneous and heterogeneous nickel films were carried out by means of electrochemical techniques such as cyclic voltammetric and chronoamperometric methods. From investigations of effect of different formaldehyde concentrations (from 0.001 to 0.1 mol L⁻¹) on electrochemical signal - current we provided catalysis mechanism of formaldehyde oxidation, detection limit and sensitivity of nickel electrodes. The results indicated that nickel electrodes participate directly in the electrocatalytic oxidation of formaldehyde. In the overall reaction, formaldehyde in alkaline aqueous solution exists predominantly in form of CH₂(OH)O⁻, which is oxidized to CH₂(O)O⁻. Taking into account the determined (Aecsa) values we have been able to calculate the sensitivities: 7 mA mol L⁻¹ cm⁻² for nickel nanowires, 3.5 mA mol L⁻¹ cm⁻² for heterogeneous nickel film and 2 mA mol L⁻¹ cm⁻² for heterogeneous nickel film. The detection limit was 0.2 mM for nickel nanowires, 0.5 mM for porous Ni film and 0.8 mM for homogeneous Ni film. All of these results make nickel electrodes capable for further applications.

Keywords: electrochemically active surface areas, nickel electrodes, formaldehyde, electrocatalytic oxidation

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Authors: Temur Chilachava

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In work the new nonlinear mathematical model describing assimilation of the people (population) with some less widespread language by two states with two various widespread languages, taking into account demographic factor is offered. In model three subjects are considered: the population and government institutions with the widespread first language, influencing by means of state and administrative resources on the third population with some less widespread language for the purpose of their assimilation; the population and government institutions with the widespread second language, influencing by means of state and administrative resources on the third population with some less widespread language for the purpose of their assimilation; the third population (probably small state formation, an autonomy), exposed to bilateral assimilation from two rather powerful states. Earlier by us it was shown that in case of zero demographic factor of all three subjects, the population with less widespread language completely assimilates the states with two various widespread languages, and the result of assimilation (redistribution of the assimilated population) is connected with initial quantities, technological and economic capabilities of the assimilating states. In considered model taking into account demographic factor natural decrease in the population of the assimilating states and a natural increase of the population which has undergone bilateral assimilation is supposed. At some ratios between coefficients of natural change of the population of the assimilating states, and also assimilation coefficients, for nonlinear system of three differential equations are received the two first integral. Cases of two powerful states assimilating the population of small state formation (autonomy), with different number of the population, both with identical and with various economic and technological capabilities are considered. It is shown that in the first case the problem is actually reduced to nonlinear system of two differential equations describing the classical model "predator - the victim", thus, naturally a role of the victim plays the population which has undergone assimilation, and a predator role the population of one of the assimilating states. The population of the second assimilating state in the first case changes in proportion (the coefficient of proportionality is equal to the relation of the population of assimilators in an initial time point) to the population of the first assimilator. In the second case the problem is actually reduced to nonlinear system of two differential equations describing type model "a predator – the victim", with the closed integrated curves on the phase plane. In both cases there is no full assimilation of the population to less widespread language. Intervals of change of number of the population of all three objects of model are found. The considered mathematical models which in some approach can model real situations, with the real assimilating countries and the state formations (an autonomy or formation with the unrecognized status), undergone to bilateral assimilation, show that for them the only possibility to avoid from assimilation is the natural demographic increase in population and hope for natural decrease in the population of the assimilating states.

Keywords: nonlinear mathematical model, bilateral assimilation, demographic factor, first integrals, result of assimilation, intervals of change of number of the population

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Authors: I. M. L. Nadeesha Jayaweera, Adao Alex Trindade

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In choosing a candidate model in likelihood-based modeling via an information criterion, the practitioner is often faced with the difficult task of deciding just how far up the ranked list to look. Motivated by this pragmatic necessity, we construct an uncertainty band for a generalized (model selection) information criterion (GIC), defined as a criterion for which the limit in probability is identical to that of the normalized log-likelihood. This includes common special cases such as AIC & BIC. The method starts from the asymptotic normality of the GIC for the joint distribution of the candidate models in an independent and identically distributed (IID) data framework and proceeds by deriving the (asymptotically) exact distribution of the minimum. The calculation of an upper quantile for its distribution then involves the computation of multivariate Gaussian integrals, which is amenable to efficient implementation via the R package "mvtnorm". The performance of the methodology is tested on simulated data by checking the coverage probability of nominal upper quantiles and compared to the bootstrap. Both methods give coverages close to nominal for large samples, but the bootstrap is two orders of magnitude slower. The methodology is subsequently extended to two other commonly used model structures: regression and time series. In the regression case, we derive the corresponding asymptotically exact distribution of the minimum GIC invoking Lindeberg-Feller type conditions for triangular arrays and are thus able to similarly calculate upper quantiles for its distribution via multivariate Gaussian integration. The bootstrap once again provides a default competing procedure, and we find that similar comparison performance metrics hold as for the IID case. The time series case is complicated by far more intricate asymptotic regime for the joint distribution of the model GIC statistics. Under a Gaussian likelihood, the default in most packages, one needs to derive the limiting distribution of a normalized quadratic form for a realization from a stationary series. Under conditions on the process satisfied by ARMA models, a multivariate normal limit is once again achieved. The bootstrap can, however, be employed for its computation, whence we are once again in the multivariate Gaussian integration paradigm for upper quantile evaluation. Comparisons of this bootstrap-aided semi-exact method with the full-blown bootstrap once again reveal a similar performance but faster computation speeds. One of the most difficult problems in contemporary statistical methodological research is to be able to account for the extra variability introduced by model selection uncertainty, the so-called post-model selection inference (PMSI). We explore ways in which the GIC uncertainty band can be inverted to make inferences on the parameters. This is being attempted in the IID case by pivoting the CDF of the asymptotically exact distribution of the minimum GIC. For inference one parameter at a time and a small number of candidate models, this works well, whence the attained PMSI confidence intervals are wider than the MLE-based Wald, as expected.

Keywords: model selection inference, generalized information criteria, post model selection, Asymptotic Theory

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