Solution Of Equation Research Articles

Soliton solutions and sensitive analysis to nonlinear wave model arising in optics

In this study, we use analytical algorithms, specifically the auxiliary equation (AE) approach, the improved F-expansion method, and the modified Sardar sub-equation (MSSE) method to investigate complex wave structures for plentiful solutions associated with the fractional perturbed Gerdjikov-Ivanov (PGI) model with the M-fractional operator. The investigated model is a well-established mathematical model used to represent a variety of physical events in nonlinear dynamics and mathematical physics. By using the aforementioned techniques, we scrutinize some new optical wave solutions in the framework of dark, bright, periodic, combo, W-shaped, M-shape, V-shape, kink type, singular rational, exponential, trigonometric, and hyperbolic solutions. The acquired solutions address a wide range of optical solutions in the form of 3D plots, contour plots, and 2D plots, declaring the free parameters of such optical soliton solutions and comprehending their dynamic behavior. Also, the sensitive analysis of the selected model is analyzed. The main contribution of this study is to extract diverse solitary wave solutions of the adopted model. Some of the solutions are similar and some diverge from the previous solutions which justifies the novelty of the study. Finally, we discovered that the current technique provides a reliable instrument for investigating the analytic solutions of fractional differential equations. The proposed PGI model can be used to transmit ultra-fast pulses across optical fibers. This research goes beyond to the advancement of mathematical techniques for solving fractional differential equations and broadens their application to a wide range of real-world scientific and engineering problems.

Approximate analytical solution of a soil water movement equation under different ponding radii on the basis of numerical simulation

AbstractThis study addresses the problem of 2D soil water movement under ponding radii of 1, 2, and 3 cm. The soil water movement characteristics (shape parameters of the water content profile, ratio of horizontal wetting front to vertical wetting front, relationship between infiltration time and horizontal wetting front, and relationship between infiltration time and cumulative infiltration) under the above three kinds of water ponding radius were analyzed. On the basis of the assumption that the soil wetting body is a semi‐ellipse and the analytical solution of the 1D soil water movement equation at any angle, the approximate analytical solution of the 2D soil water movement equation under ponding conditions is optimized. The function relationships between infiltration time, wetting front, and cumulative infiltration are established. We applied the numerical data simulated by HYDRUS‐3D to validate the parameters in proposed analytical solutions and evaluated the relationships between the wetting front and hydraulic parameters. The results indicate that as the water ponding radius increases, the wetting body and 2D water content distribution becomes larger. When the water ponding radius was 2 cm, the numerical and analytical solution of 1D soil water distribution showed the best comparison results, and the model error was the smallest. The ratio of wetting fronts was linearly increased with the increase of air‐entry suction with R2 = 0.9969.

Towards to Theory of the X-ray Diffraction Tomography of Crystals with Nano-Sized Defects

X-ray diffraction tomography is an innovative method that is widely used to obtain 2D-phase-contrast diffraction images and their subsequent 3D-reconstruction of structural defects in crystals. The most frequent objects of research are linear and helical dislocations in a crystal, for which plane wave diffraction images are the most informative, since they do not contain additional interference artifacts unrelated to the images of the defects themselves. In this work the results of modeling and analysis of 2D plane wave diffraction images of a nano-dimensional Coulomb-type defect in a Si(111) thin crystal are presented based on the construction of numerical solutions of the dynamic Takagi-Taupin equations. An adapted physical expression for the elastic displacement field of the point defect, which excludes singularity at the defect location in the crystal, is used. A criterion for evaluating the accuracy of numerical solutions of the Takagi-Taupin equations is proposed and used in calculations. It is shown that in the case of the Coulomb-type defect elastic displacement field, out of the two difference algorithms for solving the Takagi-Taupin equations used in their numerical solution, only the algorithm for solving the Takagi-Taupin equations where the displacement field function enters in exponential form is acceptable in terms of the required accuracy-duration of the calculations.

Studies on Bianchi type-III cosmological model with parametrization of deceleration parameter in a SBST theory

In this paper, we delve into the cosmological assessments concerning the deceleration parameter [Formula: see text] in the context of the Bianchi type-III cosmological model within the framework of the Saez–Ballester theory of gravitation. By postulating a proportional relationship between the shear scalar ([Formula: see text]) and the expansion scalar ([Formula: see text]), we derive solutions for the corresponding field equations. Additionally, we conduct an examination and evaluation of the model’s physical and kinematic parameters, including the jerk parameter and snap parameter, while also discussing the energy condition of the model. Through the depiction of graphical representations, we illustrate the variations of the cosmological parameters for specific sets of constants.

Domain wall and holographic dark energy in f(Q) theory of gravity

In this paper, we have studied the flat Friedmann–Robertson–Walker (FRW) space time with domain wall and holographic dark energy in the context of [Formula: see text] theory of gravity. The exact solutions of the field equations are obtained using two different forms of varying deceleration parameters: (i) [Formula: see text] and (ii) [Formula: see text]. It is observed that the universe is accelerating and expanding. Furthermore, we have also discussed some physical parameters of the investigated model.

Analytical solution of fractional oscillation equation with two Caputo fractional derivatives

Analytical solution of initial value problem for the fractional oscillation equation with two Caputo fractional derivatives , where the coefficients and orders satisfy and , is investigated by using the Laplace transform and complex inverse integral method on the principal Riemann surface. It is proved by using the argument principle that the characteristic equation has a pair of conjugated simple complex roots with a negative real part on the principal Riemann surface under the assumption that and are not both integers. Then three fundamental solutions, the unit impulse response, the unit initial displacement response, and the unit initial rate response, are derived analytically. Each of these solutions is expressed into a superposition of a classical damped oscillation decaying exponentially and a real Laplace integration decaying in a negative power law. Finally, the asymptotic behaviors of these analytical solutions for sufficiently large are determined as monotonous decays in a power of negative exponent.

Oscillatory behavior of solutions of second-order non-linear differential equations with mixed non-linear neutral terms

This study primarily seeks to expand upon these developments by encompassing neutral differential equations of mixed type, incorporating both delay and advanced terms, particularly in the case of the canonical operator. The presented results are derived from the application of the comparison method, Riccati transformation, and integral averaging technique. These methodologies lead to substantial improvements and extensions of existing results found in the literature. Additionally, illustrative examples are provided to demonstrate the practical implications of the developed criteria.

A brief review on quantum computing based drug design

AbstractDesign and development of new drug molecules are essential for the survival of human society. New drugs are designed for therapeutic purposes to combat new diseases. Besides treating new diseases, new drug development is also needed to treat pre‐existing diseases more effectively and reduce the existing drugs' side effects. The design of drugs involves several steps, from the discovery of the drug molecule to its commercialization in the market. One of the most critical steps in drug design is to find the molecular interactions between the target (infected) molecule and the drug molecule. Several complex chemical equations need to be solved to determine the molecular interactions. In the late 20th Century, the advancement of computational technologies has made the solution of chemical equations relatively easier and faster. Moreover, the design of drug molecules involves multi‐criteria optimization. Classical computational methodologies have been used for drug design since the end of the 20th Century. However, nowadays, more advanced computational methodologies are inevitable in designing drugs for new diseases and drugs with fewer side effects. In this context, the quantum computing paradigm has proved beneficial in drug design due to its advanced computational capabilities. This paper presents a state‐of‐the‐art comprehensive review of the quantum computing‐based methodologies involved in drug design. A comparative study is made about the different quantum‐aided drug design methods, stating each methodology's merits and demerits. The review work presented in this manuscript will help new researchers assess the present state‐of‐the‐art concept of quantum‐based drug design.This article is categorized under: Technologies > Structure Discovery and Clustering Technologies > Computational Intelligence Application Areas > Health Care

Existence and uniqueness of neutral functional differential equations with sequential fractional operators.

In this research paper, we investigate the existence and uniqueness of solutions for neutral functional differential equations with sequential fractional orders, specifically involving the [Formula: see text]-Caputo operator. To obtain the desired results, we employ the Banach fixed point theorem (BFPT), a nonlinear variation of the Leray-Schauder fixed point theorem (SFPT), and the Krasnoselski fixed point theorem (KFPT). Additionally, we provide illustrative examples that demonstrate the key findings. Furthermore, we address a scenario where an initial value integral condition is considered.

Analytic solutions of fractional kinetic equations involving incomplete ℵ-function

This paper expressed the fractional kinetic equation (KE) in terms of the incomplete [Formula: see text]-function, I-function, Fox H-function, and Meijer’s G-function. It assesses the importance of fractional KE in diverse scientific and engineering contexts, highlighting its relevance across various scenarios. In this paper, we investigate the results of Katugampola’s kinetic fractional equations with the product of the generalized M-series and incomplete [Formula: see text]-function. The [Formula: see text]-Laplace transform is applied for obtaining the desired results. Utilizing an M-series, the universality of this series is harnessed to deduce solutions for a fractional KE. Furthermore, a graphical representation of the obtained solutions’ behaviors is presented, enhancing the impact of the findings.

Analysis and solution of unstable heat equation based on output feedback

In this paper, the output feedback-based bipartite consensus problem for MASs of unstable heat equations is investigated. The interactions of participating individuals are cooperative and competitive on an undirected graph. In order to achieve accurate estimation of system state, a distributed state observer is proposed. By introducing a reversible transformation, the observer system is transformed into a new system, and the controller based on the new system states is designed, which guarantees the well-posedness and stability of the considered control system and the observer system due to the invertibility of the presented transformation. Similarly, the bipartite consensus is analysed and proved. Finally, the validity of the proposed method is verified by simulation examples.

Expanding the Areas of Attraction of Solutions to Singularly Perturbed Equations

In this paper, a singularly perturbed equation of the first order is considered, the concept of a region of attraction (DO) of the solution of a singularly perturbed equation to the solution of an unperturbed equation is introduced, and the existence of an OA is proved. The problem has been set about the possibility of expanding the areas of attraction of VCA solutions. It has been proven that if there is a region of attraction, then it can be expanded to the boundary of the region under consideration. In the proof, geometric constructions were used, using the level line of conjugate-harmonic functions, the method of successive approximations and methods of asymptotic estimates.

The Effect of Nonlinear Sloshing Response of Water on Seismic Behavior of Reinforced Concrete Elevated Water Tanks

In this study, the fluid-structure interactive behavior by using the Westergaard approach and the Smoothed Particle Hydrodynamics (SPH) method of a 1000 m3 reinforced concrete (RC) elevated water tank under different seismic activities i.e., Kocaeli, Van, Kahramanmaraş and Kobe earthquakes was investigated. In the SPH method, the resulting hydrodynamic pressures were applied to the interior of the wall of the structure as an external force which is made as mass additions to the tank walls in Westergaard approach. With the help of software developed, the load carrying system of the elevated water tank was modeled using the MATLAB Partial Differential Equation Toolbox (PDE) software, based on the finite element method, and its verification was made by comparing with ANSYS software model. Dynamic condensation method was used to perform time history analyses for the cited earthquakes. The time-dependent solutions of partial differential equations are solved with the help of ODE45 functions based on Runge-Kutta method, by arranging the equation of motion in state-space format. The cited tank was analyzed linearly under different seismic activities for empty, half-filled (50%) and full-filled (100%) cases and the obtained results were compared with each other. It has been concluded that considering nonlinear behavior of the fluid during the earthquake with the SPH method provided consistency with the real behavior of RC elevated water tanks and gave more realistic results than the traditional Westergaard approach.

Transient nonlocal heat flux in simple metals at low excitations

In this paper we discuss the ultrafast transfer of charge and energy in a metal in the kinetic approximation. An exact solution of the Boltzmann transport equation for the electron distribution function in the relaxation time approximation is obtained, which is used to calculate transport coefficients for charge and heat fluxes. Different regimes of energy transfer under conditions of significant spatial dispersion and non-equilibrity of the electron distribution function are discussed.

Exact moments for trapped active particles: inertial impact on steady-state properties and re-entrance

Abstract In this study, we investigate the behavior of inertial active Brownian particles in a d-dimensional harmonic trap in the presence of translational diffusion. While the solution of the Fokker-Planck equation is generally challenging, it can be utilized to compute the exact time evolution of all time- dependent dynamical moments using a Laplace transform approach. We present the explicit form for several moments of position and velocity in d-dimensions. An interplay of time scales assures that the effective diffusivity and steady-state kinetic temperature depend on both inertia and trap strength, unlike passive systems. The distance from equilibrium, measured by the violation of equilibrium fluctuation-dissipation and the amount of entropy production, decreases with increasing inertia and trap strength. We present detailed ‘phase diagrams’ using kurtosis of velocity and position, showing possibilities of re-entrance to equilibrium.

On the solution of evolution p-Laplace equation with memory term and unknown boundary Dirichlet condition

On the solution of evolution p-Laplace equation with memory term and unknown boundary Dirichlet condition

Numerical and Experimental Determination of Selected Performance Indicators of the Liquid Flat-Plate Solar Collector under Outdoor Conditions

The paper proposes applying an in-house mathematical model of a liquid flat-plate solar collector to calculate the collector time constant. The described model, proposed for the first time in an earlier study, is a one-dimensional distributed parameter model enabling simulations of the collector operation under arbitrarily variable boundary conditions. The model is based on the solution of energy balance equations for all collector components. The formulated differential equations are solved iteratively using an implicit difference scheme. To obtain a stable numerical solution, it is necessary to use appropriate steps of time and spatial division. These were found by comparing the results obtained from the model with the results of the analytical solution available in the literature for the transient state, which constitutes the novelty of the present study. The accuracy of the results obtained from the model was verified experimentally by comparing the measured and calculated history of the fluid temperature at the outlet of the collector. The calculation of the collector time constant is proposed in the paper as an example of the model’s practical application. The results of the time constant calculation were compared with the values obtained experimentally on the test stand. This is another novelty of the presented research. The analysed collector instantaneous efficiency was then calculated for selected outdoor conditions. The presented mathematical model can also be used to verify the correctness of the collector operation. By comparing, on an ongoing basis, the measured and calculated values of the fluid temperature at the collector outlet, conclusions can be drawn about the process of solar glass fouling or glycol gelling. The simplicity of the model and the low computational demands enable such comparisons in an online mode.

One step inversion free iterative algorithm for computing generalised bisymmetric solution for nonlinear matrix equations

In this paper, we investigate one step inversion free iterative algorithm for computing generalised bisymmetric solution to the nonlinear matrix equations (NLME) X + A T X − q A + B T X − q B = H ( q ≥ 2 ) , where A , B , H are given matrices, while X is a generalised bisymmetric matrix to be computed. The necessary conditions for the existence of a generalised bisymmetric solution and the convergence of the suggested iterative method are derived. Some lemmas and theorems are provided and proved where the iterative solutions are obtained. Four numerical examples are presented to exhibit the effectiveness of the suggested method and to verify the theoretical findings of this paper.

Solution of Linear Ordinary Differential Equations with Polynomial Coefficients

Solution of Linear Ordinary Differential Equations with Polynomial Coefficients

℘ -adic quantum mechanics, the Dirac equation, and the violation of Einstein causality

This article studies the breaking of the Lorentz symmetry at the Planck length in quantum mechanics. We use three-dimensional ℘ -adic vectors as position variables, while the time remains a real number. In this setting, the Planck length is 1/℘ , where ℘ is a prime number, and the Lorentz symmetry is naturally broken. In the framework of the Dirac-von Neumann formalism for quantum mechanics, we introduce a new ℘ -adic Dirac equation that predicts the existence of particles and antiparticles and charge conjugation like the standard one. The discreteness of the ℘ -adic space imposes substantial restrictions on the solutions of the new equation. This equation admits localized solutions, which is impossible in the standard case. We show that an isolated quantum system whose evolution is controlled by the ℘ -adic Dirac equation does not satisfy the Einstein causality, which means that the speed of light is not the upper limit for the speed at which conventional matter or energy can travel through space. The new ℘ -adic Dirac equation is not intended to replace the standard one; it should be understood as a new version (or a limit) of the classical equation at the Planck length scale.