Equations Of Order Research Articles

EXISTENCE AND UNIQUENESS OF SOLUTIONS OF NONLINEAR FUNCTIONAL INTEGRAL ITOˆ EQUATIONS

A new class of Ito^ integral equations is considered, which contains many classical problems, for example, the Cauchy problem for differential equations of integer and fractional order with and without stochastic perturbations, as well as some less known and little-studied types of equations that have been introduced recently. The purpose of the study is to find sufficiently general conditions that guarantee the existence and the uniqueness of solutions to such equations, taking into account their specific features. The article therefore proposes to use a special generalized Lipschitz condition, which, due to its flexibility, allows one to obtain effective solvability criteria in terms of the right-hand sides of equations. Numerous examples are considered, covering in particular Ito^ differential equations of fractional order with aftereffect and without aftereffect, equations with fractional Wiener processes, Ito^ equations with several time scales, as well as their generalizations.

Energy absorption capacity and vibration-control responses of smart perovskite solar cells under external excitation via intelligent velocity feedback control algorithm

Smart perovskite solar cells offer engineers a highly efficient, cost-effective, and flexible solution for renewable energy. Their tunable properties, ease of fabrication, and potential for integration into various surfaces make them ideal for sustainable power generation, driving innovation in clean energy technologies and energy-efficient designs. This comprehensive approach ensures that smart perovskite solar cells can withstand external mechanical excitations more effectively and maintain their integrity over time. Due to the mentioned problem, using a data-driven solution approach, this work presents an intelligent controller for reducing transitory vibrations in smart perovskite solar systems outfitted with functionally-graded graphene origami-enabled auxetic metamaterial (FG-GOEAM) layer in the metal position, sensor, and actuator face sheets. In order to effectively detect and react to vibrations, the system integrates sensor and actuator face sheets. The controller applies the converse and direct piezoelectric equations together with interpolation functions to coupled motion equations in order to efficiently decrease vibration amplitudes. By integrating intelligent control algorithms, system parameters may be monitored and adjusted in real-time to maximize stability and performance. The piezoelectric actuator applies a control force to promptly halt the vibration of the structure. This study determines the controlled dynamic response of the smart perovskite solar cell construction by the use of a velocity feedback control algorithm. Numerical simulations and validation show that the suggested method is effective in mitigating transient vibrations and improving structural integrity. Through the development of intelligent control systems for smart perovskite solar cells, this study offers interesting solutions for engineering applications by reducing transitory vibrations.

Kinetic model of thermal decomposition of nitric acid solutions of hydrazine nitrate

The kinetic parameters of thermal decomposition of an aqueous solution containing 0.029 mass fractions of hydrazine nitrate and 0.44 mass fractions of nitric acid have been determined: onset temperature and specific thermal effect of exothermic reactions, activation energy, pre-exponential factor, reaction rate and order. Based on the data obtained, a mathematical model of thermal decomposition of nitric acid solutions of hydrazine nitrate has been proposed. It has been found that thermolysis of the solution has a multi-stage nature and can be described by a system of 5 equations of 1st and 2nd order. A comparison of two approaches to creating a mathematical model has been carried out: based on adiabatic calorimetry and differential scanning calorimetry data.

Investigation of the adsorption process of methylene blue on diatomite from a kinetic and thermodynamic perspective

The adsorption of methylene blue on diatomite has been studied. Kinetics, thermodynamics and adsorption isotherms were used to identify retention mechanisms. The experiments, carried out in batch mode, for the determination of the kinetics, were carried out after adjustment of the parameters influencing the system, such as the pH, the temperature, the mass of adsorbent. The adsorption process of methylene blue onto diatomite is effectively represented by the Langmuir model, which indicates an adsorption capacity of roughly 166.66 mg/g. This is confirmed by the fact that the correlation coefficient exceeds that obtained with the Freundlich model; furthermore, the separation factor indicates that this adsorption is favorable. The results obtained were modeled according to the kinetic equations of the pseudo-first order, pseudo-second order, as well as that related to intraparticle diffusion. The experimental results of the global reaction are perfectly tunable to the pseudo-second order, with very large regression coefficients. Thermodynamics and adsorption isotherms predict the passage of a spontaneous endothermic surface reaction.

Monodromy of Picard–Fuchs system for a family of K3 toric hypersurfaces and fixed points of the Hilbert modular group for Q(2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {Q}(\\sqrt{2})$$\\end{document}

We solve the monodromy problem about the Picard–Fuchs system for the two-parameter family of the anticanonical hypersurfaces in the toric three-fold arising from a specific reflexive polytope. The members of the family are resolved to be K3 surfaces polarized by a lattice of signature (1, 17) with discriminant -8\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$-8$$\\end{document}. The Picard–Fuchs system consists of two hypergeometric differential equations of order 2 with the four-dimensional solution space. The independent variables are affine coordinates of the punctured weighted projective space P(1,2,3)\\{(1,0,0)}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {P}(1,2,3)\\hspace{1.111pt}{\\setminus }\\hspace{1.111pt}\\{(1,0,0)\\}$$\\end{document} which forms a moduli space of the family. We construct the isomorphism from P(1,2,3)\\{(1,0,0)}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {P}(1,2,3)\\hspace{1.111pt}{\\setminus }\\hspace{1.111pt}\\{(1,0,0)\\}$$\\end{document} to the symmetric Hilbert modular orbifold H2/⟨PSL2(OK),τ1⟩\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {H}^2/\\langle \ ext {PSL}_2(\\mathscr {O}_K),\ au _1\\rangle $$\\end{document} for K=Q(2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$K=\\mathbb {Q}(\\sqrt{2})$$\\end{document} and its inverse as a triplet of symmetric Hilbert modular forms. Here (1, 0, 0) corresponds to the cusp. As the fundamental set of solutions, we take a set of series convergent near the cusp. We obtain the generators of the monodromy group in GL4(Z)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ext {GL}_4(\\mathbb {Z})$$\\end{document} by deforming the domains for the period integrals expressing the solutions. We construct the isomorphism from the monodromy group to ⟨PSL2(OK),τ1⟩\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\langle \ ext {PSL}_2(\\mathscr {O}_K),\ au _1\\rangle $$\\end{document}. Using the monodromy, we obtain fixed points of the Hilbert modular group PSL2(OK)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ext {PSL}_2(\\mathscr {O}_K)$$\\end{document} and their respective isotropy groups, which are consistent with the results of the prior research taking an arithmetic approach.

Black hole to grey hole metamorphosis in the deep quantum regime

In order to search for new solutions for collapsed objects in quantum gravity, we consider in this paper a Kantowski–Sachs metric labelled by parameters that have no classical significance. In addition, we include a Klein–Gordon field to represent in a simple manner the inevitable zero-point vacuum fluctuations that permeate the spacetime. With this framework, we quantize the system and obtain the Wheeler–DeWitt equation in order to focus upon the deep quantum regime of the interior and to analyze any kind of transition that the black hole may undergo. The Wheeler–DeWitt equation reveals the existence of new solutions of different nature, designated herein as “quantum grey holes,” in addition to the existence of quantum black holes, with all solutions satisfying the DeWitt boundary condition. The existence of new solutions gives rise to the novel possibility of a quantum black hole making a transition to a quantum grey hole. We find that there exists non-zero probability of quantum black-to-grey hole transition. These transition probabilities exhibit resonances for a continuous range of eigenvalues of the system.

Advancing non-linear PDE's solutions: modified Yang transform method with Caputo-Fabrizio fractional operator

Abstract Nonlinear partial differential equations have a crucial rule in many physical processes. In this paper, a novel approach is used to study nonlinear partial differential equations of fractional order, which is named as Modified Yang Transform (MYT) method. This approach combines Yang transform with the Adomian decomposition method. The fractional order is considered in the Caputo-Fabrizio sense. Convergence analysis of the modified Yang transform to nonlinear fractional order partial differential equations is presented. Additionally, a solution framework for the solution of nonlinear partial differential equation is carried out and some examples are provided to highlight the application of the current method. To illustrate that how the solution behaves for various fractional orders, 2D and 3D graphs are plotted. Various tables are also provided to show the difference between exact and approximate solutions and the values are compared with other methods in the literature. Results and discussion sections are included for each example to explain the graphs, tables and their results.

Rational Solutions to the Fourth Equation of the Nonlinear Schrödinger Hierarchy

This study concerns the research of rational solutions to the hierarchy of the nonlinear Schrödinger equation. In particular, we are interested in the equation of order 4. Rational solutions to the fourth equation of the NLS hierarchy are constructed and explicit expressions of these solutions are given for the first order. These solutions depend on multiple real parameters. We study the associated patterns of these solutions in the (x,t) plane according to the different values of their parameters. This work allows us to highlight the phenomenon of rogue waves, such as those seen in the case of lower-order equations such as the nonlinear Schrödinger equation, the mKdV equation, or the Hirota equation.

Theory of thin elastic plate: history and current state of the problem

The article is an analytical review and is devoted to the theory of thin, isotropic elastic plates. There are basic relations of the theory based on the kinematic hypothesis confirming that the tangential displacements are distributed linearly along the thickness of the plate and its deflection does not depend on the normal coordinate. As a result, a system of equations of the sixth order with respect to two potential functions – the penetrating potential, which determines the plate deflection, and the boundary potential which makes it possible to set three boundary conditions on the plate edge and eliminate the known contradiction of Kirchhoff's plate theory was obtained. Problems that have no correct solution in the framework of Kirchhoff's theory – cylindrical bending of a plate with a free edge, bending of a rectangular plate with non-classical hinge fixing, torsion of a square plate by moments distributed along the contour, and bending of a plate by a rigid die – are considered. In conclusion, a brief historical review of the papers devoted to the theory of plate bending was presented.

Rapid Estimation of Soil Moisture Using Spectroscopic Sensor

ABSTRACT Soil moisture plays a significant role in the global hydrological cycle in the field of agriculture, environment science, urban planning and construction, forestry and many more. Soil moisture can be determined using various approaches, however some of the limitations in these approaches are longer time duration, labor intensive, require installation and maintenance, expensive, lower resolution, etc. This research work proposes a soil moisture detection method which is portable, economic and time efficient. Spectroscopic sensor enables rapid soil moisture estimation using photodiodes of varying sensitivities and wavelengths across a broad spectrum of light, allowing precise identification and analysis of various samples based on their unique spectral fingerprints. In this study, a relation between sensor output data and soil moisture percentage value obtained from gravimetric method has been established. Using various regression methods, the highest value of correlation coefficient was found in case of polynomial function of order 3. Residual values of all wavelengths were then calculated by comparing the calculated with the expected moisture values obtained using the polynomial function. Result showed that 705 nanometers (nm) wavelength has the least residual value, thus signifying that the polynomial equation of order 3 at 705 nm wavelength perfectly demonstrates the relation between soil moisture and sensor reading. Furthermore, the results were verified using optimization method which showed that 705 nm wavelength has the highest closeness coefficient, thus verifying this model and observations. This model can be helpful in determining the soil moisture instantaneously instead of hours as seen in traditional gravimetric methods.

Structural Decomposition of the Passivity-Based Control System of Wind–Solar Power Generating and Hybrid Battery-Supercapacitor Energy Storage Complex

Wind–solar power generating and hybrid battery-supercapacitor energy storage complex is used for autonomous power supply of consumers in remote areas. This work uses passivity-based control (PBC) for this complex in accordance with the accepted energy management strategy (EMS). Structural and parametric synthesis of the overall PBC system was carried out, which was accompanied by a significant amount of research. In order to simplify this synthesis, a structural decomposition of the overall dynamic system of the object presented in the form of a port-Hamiltonian system, which was described by a system of differential equations of the seventh order, into three subsystems was applied. These subsystems are a wind turbine, a PV plant, and a hybrid battery-supercapacitor system. For each of the subsystems, it is quite simple to synthesize the control influence formers according to the interconnections and damping assignment (IDA) method of PBC, which locally performs the tasks set by the EMS. The results obtained by computer simulation of the overall and decomposed systems demonstrate the effectiveness of this approach in simplifying synthesis and debugging procedures of complex multi-physical systems.

Behaviour of 1-butyl-3-methylimidazolium chloride, 1-butyl-3-methylimdazolium thiocyanate and 1-butyl-1-methypyrrolidium chloride in dimethyl sulfoxide: Experimental, NCI and QTAIM analysis

The aim of this study is to examine the cation and anion influence on intermolecular interactions of mixtures comprising ionic liquids (ILs): 1-butyl-3-methylimidazolium chloride [Bmim][Cl], 1-butyl-3-methylimdazolium thiocyanate [Bmim][SCN] and 1-butyl-1-methylpyrrolidium chloride [Bmpym][Cl] with dimethyl sulfoxide (DMSO) at various temperatures. To understand the influence of cation and anion as well as temperature on interactions between the mixtures, thermodynamics parameters: apparent molar properties were calculated from the measured density (ρ) and sound velocity (u) data and fitted to the Redlich-Mayer type equation in order to obtained limiting apparent molar properties of the studied mixtures. A detailed non-covalent interaction (NCI) and quantum theory of atoms in molecules (QTAIM) analysis was done using density functional theory calculations. These calculations provided information on the electrostatic interactions between the anion, cation, and solvent. The experimental and computational results show that there is hydrogen bonding and strong ionic interactions in the studied systems.

Improved Carboxylate Density in Functionalized Biochar for Methylene Blue Adsorption

AbstractCarboxylate appended biochar BCKA was synthesized by a two‐step functionalization of leached rice straw biochar using ethyl 2‐bromoacetate/K2CO3/DMF and hydrolysis to furnish the modfied biochar. The modified biochar was characterized using proximate analysis, CHNS, SS CPMAS 13C NMR, FESEM, FTIR, XPS, BET, and zeta potential. Biochar BCKA was optimized for pH and dose, respectively, as 7 and 0.2 g/L for methylene blue (MB) adsorption with a contact time of 6 h at 298 K. MB removal up to 98% from 20 mg/L dye solution, using BCKA as adsorbent pH 7, 298 K. The nonlinear fit to Langmuir, Freundlich, Temkin, SIPS, and D‐R adsorption isotherm showed a good fit with correlation coefficients (R2) 0.970, 0.969, 0.988, 0.984, and 0.983, respectively, and maximum MB adsorption (qm) of 261 mg/g. Thermodynamic parameters for the adsorption of MB using BCKA were 38.72 kJ/mol (∆H°), 225.42 J/mole K (∆S°), and negative ∆G° confirming the spontaneity driven by entropy. Kinetic studies showed good nonlinear fit to pseudo‐second order (R2 0.967), intraparticle diffusion (R2 0.977), and Elovich equation (R2 0.958) with chemisorption‐based interaction between BCKA and MB. Chemisorption, particularly ion exchange on the surface of BCKA with MB, is inferred from the D‐R isotherm and kinetic analysis. Column breakthrough studies showed Thomas model fit with qt 267.09 mg/g, and efficiency >80% was retained for two cycles of MB adsorption.

Devising an analytical method for solving the eighth-order Kolmogorov equations for an asymmetric Markov chain

The object of research is a complex system of three subsystems, which function independently of each other and are in a working or failed state. There is a need to analytically model and manage the Markov random process in the system, varying the intensity of their development-restoration and degradation-destruction flows. In the study, an analytical method for solving Kolmogorov equations of the eighth order for an asymmetric Markov chain was devised. The corresponding Kolmogorov equations of the eighth order have an ordered transition probability matrix. The distribution of the eight roots of this equation in the complex plane has central symmetry. The results are analytical solutions for the probabilities of the eight states of the Markov chain in time in the form of ordered determinants with respect to the indices of the eight roots and the indices of the eight states, including the column vector of the initial conditions. Symmetry has been established in the distribution on the complex plane of eight real, negative roots of the characteristic Kolmogorov equation centered at the point defined as Re ϑ = –a7/8, where a7 is the coefficient of the characteristic equation of the eighth degree at the seventh power. Formulas expressing eight roots of the characteristic Kolmogorov equation have been heuristically derived, one of which is zero, due to the intensities of failures and recovery of three subsystems, the eight states of which in general make up an asymmetric Markov chain. For structures consisting of three independently functioning processes, the random process of the transition of the structure through eight possible states with a known initial state is determined in time. An analytical solution to Kolmogorov differential equations of the eighth order for an asymmetric state graph is proposed in harmonic form for the purpose of analysis and synthesis of a random Markov process in a triple system.

Generalized Linear Differential Equation using Hyers - Ulam Stability Approach

In this paper, We demonstrate the Hyers - Ulam stability of linear differential equation of fourth order. We interact with the differential equation\begin{align*}\gamma^{iv} (\omega) + \rho_1 \gamma{'''} (\omega)+ \rho_2 \gamma{''} (\omega) + \rho_3 \gamma' (\omega) + \rho_4 \gamma(\omega) = \chi(\omega),\end{align*}where $\gamma \in c^4 [\alpha,\beta], \chi \in [\alpha,\beta]$. Hyers-Ulam stability concerns the robustness of solutions of functional equations under small perturbations, ensuring that a solution approximately satisfying the equation is close to an exact solution. We extend this concept to fourth-order linear differential equations and continuous functions. Using fixed-point methods and various norms, we establish conditions under which such equations exhibit Hyers-Ulam stability. Several illustrative examples are provided to demonstrate the application of these results in specific cases, contributing to the growing understanding of stability in higher-order differential equations. Our findings have implications in both theoretical research and practical applications in physics and engineering.

On the Limited Role of Conservation Laws in the Double Reduction Routine for Partial Differential Equations

The double reduction method for finding invariant solutions of a given partial differential equation (PDE) provides for the reduction of a $ q $-th order PDE that admits a nontrivial Lie symmetry and an associated nontrivial conservation law to an ordinary differential equation (ODE) of order $ q-1. $ In all the articles we have seen where the method has been used, the algorithm has involved writing the conservation law in canonical variables determined by the associated symmetry. In this paper, we illustrate that it is not necessary to use or even have the associated conservation law. It is enough to know that there exists a conservation law associated with a given Lie symmetry. Canonical variables derived from the symmetry are sufficient to achieve double reduction. In the canonical variables, the PDE is transformed after routine calculations into an ODE of order one less than that of the PDE. We have outlined steps involved in this variation of the double reduction method and illustrated the routine using five PDEs.

On the Periodicity Solutions of Five Systems of Rational Systems of Difference Equations of Order Five

The primary aim of this paper is to explore the behavior of several nonlinear systems of difference equations following T_{n+1}=(E_{n}E_{n-4})/(T_{n-3}(\pm 1 \pm E_{n}E_{n-4})), E_{n+1}=(T_{n}T_{n-4})/(E_{n-3}(pm 1 \pm T_{n}T_{n-4})), and obtain solution expressions for them. Moreover, we utilize MATLAB programming to simulare the dynamics and validate our results.

Application of Sumudu Transform and New Iterative Method to Solve Certain Nonlinear Ordinary Caputo Fractional Differential Equations

By combining the new iteration method (NIM) and Sumudu transform (ST) methods, together known as the new iteration ST method (NISTM), a semi-analytical or series solution for various fractional differential equations, in which new iteration method (NIM) is used to decompose the nonlinear operator prior to performing the ST, is produced in this script. Using the Caputo sense to account for the fractional derivative, this method computes series solutions for a variety of nonlinear fractional differential equation instances. While in many convergence issues a solution is achieved for just a small number of series terms, our series solution merges with the exact solution differential equation of fractional order in many example problems.

Integrated Jacobi elliptic function solutions to the [formula omitted]-dimensional generalized Kadomtsev–Petviashvili equation by utilizing new solutions of the elliptic equation of order six

In this research, the generalized (3+1)-dimensional Kadomtsev–Petviashvili equation ((3+1)-GKPE) that expresses various nonlinear phenomena was studied. An extended Jacobi elliptic function expansion method (JEFEM) was developed by considering new solutions for the Jacobi elliptic equation of order six. Then the extended method was applied to the (3+1)-GKPE, where new exact Jacobi elliptic function solutions were obtained. This equation is of particular interest as it required a special transformation in order to apply the JEFEM. Moreover, some of the solutions are shown graphically.

Existence and regularity of strict solutions for a class of fractional evolution equations

AbstractWe study the existence and Hölder regularity of solutions for fractional evolution equations of order . By means of an analytic resolvent, we construct an interpolation space, which can effectively lower the regularity of initial data. By virtue of the interpolation space and some properties of the analytic resolvent, we derive the existence and Hölder regularity of strict solutions for an inhomogeneous problem, as well as the existence and Hölder regularity of a nonlinear problem.