Analytical Schemes Research Articles

Superabundant novel solutions of the long waves mathematical modeling in shallow water with power-law nonlinearity in ocean beaches via three recent analytical schemes

Superabundant novel solutions of the long waves mathematical modeling in shallow water with power-law nonlinearity in ocean beaches via three recent analytical schemes

Multi GPU parallelization of maximum likelihood expectation maximization method for digital rock tomography data

Digital rock is an emerging area of rock physics, which involves scanning reservoir rocks using X-ray micro computed tomography (XCT) scanners and using it for various petrophysical computations and evaluations. The acquired micro CT projections are used to reconstruct the X-ray attenuation maps of the rock. The image reconstruction problem can be solved by utilization of analytical (such as Feldkamp–Davis–Kress (FDK) algorithm) or iterative methods. Analytical schemes are typically computationally more efficient and hence preferred for large datasets such as digital rocks. Iterative schemes like maximum likelihood expectation maximization (MLEM) are known to generate accurate image representation over analytical scheme in limited data (and/or noisy) situations, however iterative schemes are computationally expensive. In this work, we have parallelized the forward and inverse operators used in the MLEM algorithm on multiple graphics processing units (multi-GPU) platforms. The multi-GPU implementation involves dividing the rock volumes and detector geometry into smaller modules (along with overlap regions). Each of the module was passed onto different GPU to enable computation of forward and inverse operations. We observed an acceleration of sim 30 times using our multi-GPU approach compared to the multi-core CPU implementation. Further multi-GPU based MLEM obtained superior reconstruction compared to traditional FDK algorithm.

Plenty accurate soliton wave solutions of the prototype of an excitable system

In this paper, the nonlinear fractional Lotka–Volterra model is analyzed and numerically studied. This research is based on applying the three latest analytical schemes and three other numerical schemes to construct rich wave solutions. In different forms, many novel solitary wave solutions are built and presented in two-dimensional, three-dimensional, and contour plots. The numerical method conditions are evaluated through the obtained analytical solutions, and the accuracy of the analytical solutions is studied. Many numerical solutions are constructed based on the employed schemes. Additionally, the analytical, semi-analytical, numerical, and absolute values of error between the values of obtained solutions are calculated with the different values of the given variable in the solutions. Furthermore, the match between the obtained analytical solution and the numerical solution has been explained through some two-dimensional distributed radar charts. The contribution of this article is demonstrated by comparing the obtained solution with the recently published results of the same model.

Semi–analytical and numerical simulations of the modified Benjamin–Bona–Mahony model

In this article, semi-analytical and numerical simulations of the well-known modified Benjamin–Bona–Mahony (mBBM) equation are processed. This study targets to check the accuracy of the obtained analytical solutions of the mBBM model that have been obtained in [1] through three recent analytical schemes (extended simplest equation (ESE) method, modified kudryashov (mKud) method, and Sech-Tanh (ST) expansion method). The considered model describes the propagation of long waves in the nonlinear dispersive media in a visual illusion. The Homotopy iteration (HI) method, exponential cubic - B-spline (ECBS) method, and trigonometric- quantic-B-spline (TQBS) method are employed to construct novel semi-analytical and accurate numerical solutions. The obtained solutions’ accuracy has been checked through some different types of two-dimensional graphs

Seismic behaviors of free-spanning submarine pipelines subjected to multi-support earthquake motions within offshore sites

Due to the shortage of offshore seismic records and inhere difficulty in simulating artificial offshore ground motions, numerous existing works have been performed to investigate the seismic responses of free-spanning submarine pipelines, not only using onshore seismic records as inputs but also adopting uniform seismic excitations. These analytical schemes may result in severe misestimates of the seismic responses. In this content, this paper numerically investigates the seismic behaviors of a free-spanning submarine pipeline by using multi-support earthquake motions within offshore sites (MEMOSs) as inputs. The soil-pipe and water-pipe interactions are modeled using the nonlinear soil spring and added mass methods, respectively. The effects of spatially varying and overlying seawater are explicitly taken into account in the simulation of MEMOSs. The seismic behavior law of the free-spanning submarine pipeline under the synthesized MEMOSs is analyzed and summarized. Moreover, the impacts of the seismic excitation type, offshore seismic motion, coherence loss, local site condition and free span length on the Mises stress, vertical displacement and failure PGA of the submarine pipeline are examined and discussed comprehensively. Numerical results show that the middle and ends of free span are the dangerous regions; and the MEMOSs, coherence loss, local site condition and free span length have significant influences on the seismic responses and operational capacity of the pipeline. Based on the obtained results, some meaningful advices are recommended to the pipeline designers that (1) more attentions should be paid to the middle and ends of free span in the pipeline design; (2) the MEMOSs should be adopted as seismic inputs to accurately predict the seismic behaviors of pipelines; (3) the construction of submarine pipelines in soft clay and formation of large-dimensional free span are adverse for the safe operation of pipelines under earthquake actions, which should be avoided in engineering practice.

Transient State Modelling and Experimental Investigation of the Thermal Behavior of a Vapor Compression System

The main objective of this work is to establish a detailed modelling technique to predict the refrigerant conditions such as pressure and enthalpy of a VC system. The transient state modelling techniques suggested in many research works are usually not easy to reproduce due to lack of detailed methodology and the multitude of analytical or computational schemes that could not be assessed objectively. This work has addressed this issue by introducing a modelling method developed from conservation equations of mass and energy represented with Navier–Stokes equations. A finite volume scheme has been used to discretize the governing equations along the heat exchanger models. Transient state modelling matrices have been established after dividing the condenser as well as the evaporator into 3 andncontrol volumes. The model validation with experiments was satisfactory. The model outputs such as the refrigerant pressure across the condenser and evaporator are in agreement with experiments. The proposed modelling technique could be adopted to predict optimal parameters during start-up. The modelling results could be used to design VC systems with optimal performance.

On the Cole–Hopf transformation and integration by parts formulae in computational methods within fractional differential equations and fractional optimal control theory

The treatment of fractional differential equations and fractional optimal control problems is more difficult to tackle than the standard integer-order counterpart and may pose problems to non-specialists. Due to this reason, the analytical and numerical methods proposed in the literature may be applied incorrectly. Often, such methods were established for the classical integer-order operators and are then applied directly without having in mind the restrictions posed by their fractional-order versions. It was recently reported that the Cole–Hopf transformation can be used to convert the time-fractional nonlinear Burgers’ equation into the time-fractional linear heat equation. In this article, we show that, unlike integer-order differential equations, employing the Cole–Hopf transformation for reducing the nonlinear time-fractional Burgers’ equation into the time-fractional heat equation is wrong from two different perspectives. Indeed, such a reduction is accomplished using incorrect transcripts of the Leibniz or chain rules. Hence, providing numerical or analytical schemes based on the Cole–Hopf transformation leads to erroneous results for the nonlinear time-fractional Burgers’ equation. Regarding constant-order, variable-order, and distributed-order Caputo fractional optimal problems, we note an inconsistency in the necessary optimality conditions derived in the literature. The transversality conditions were introduced as identical to those for the integer-order case, with a vanishing multiplier at the terminal of the interval. The correct condition should involve a constant-order, variable-order, or distributed-order fractional integral operator. We also deduce that if the control system is defined with a Caputo derivative, then the adjoint equations should be expressed in the Riemann–Liouville sense and vice versa. In fact, neglecting some terms in the integration by parts formulae, during the derivation of the optimality conditions, causes some confusion in the literature.

Numerical Investigation of Fractional-Order Swift–Hohenberg Equations via a Novel Transform

In this paper, the Elzaki transform decomposition method is implemented to solve the time-fractional Swift–Hohenberg equations. The presented model is related to the temperature and thermal convection of fluid dynamics, which can also be used to explain the formation process in liquid surfaces bounded along a horizontally well-conducting boundary. In the Caputo manner, the fractional derivative is described. The suggested method is easy to implement and needs a small number of calculations. The validity of the presented method is confirmed from the numerical examples. Illustrative figures are used to derive and verify the supporting analytical schemes for fractional-order of the proposed problems. It has been confirmed that the proposed method can be easily extended for the solution of other linear and non-linear fractional-order partial differential equations.

4D-Printed Temperature-Controlled Flow-Actuated Solid-Phase Extraction Devices.

Four-dimensional printing (4DP) technologies can extend the functionality and applicability of manufactured analytical devices through employing stimuli-responsive materials. In this study, we used a photocurable resin of stimuli-responsive shape-memory polymers and digital light processing three-dimensional printing (3DP) to fabricate a smart sample pretreatment device featuring a solid-phase extraction (SPE) column and a temperature-controlled flow-actuated valve. Through manipulation of the temperatures and flow rates of the sample, eluent, and rinsing streams, we used this 4D-printed SPE device to extract Mn, Co, Ni, Cu, Zn, Cd, and Pb ions from high-salt content samples and remove the sample matrix prior to their determination by inductively coupled plasma mass spectrometry. After optimizing the valve design and operation and the analytical scheme, this device displayed competitive analytical performance-the method detection limits (MDLs) ranged from 0.7 to 22.1 ng L-1 for these metal ions (the MDLs ranged from 0.5 to 18.8 ng L-1 when validating the same printed SPE column using an online automatic system equipped with electric switching valves). Furthermore, we performed analyses of these metal ions in three reference materials (CASS-4, 1643f, and 2670a) and spike analyses of collected samples (seawater, ground water, river water, and human urine) to confirm the reliability and applicability of this analytical method. For the first time, 4DP has been used to fabricate a multi-functional, stimuli-responsive sample pretreatment device displaying analytical performance equal to that of a commercial apparatus. This novel approach builds upon the functionality and diversity of 3DP-enabling devices with the goal of developing more efficient analytical schemes.

Two improved techniques for the perturbed nonlinear Biswas–Milovic equation and its optical solitons

In this article, abundant optical soliton solutions of the Biswas–Milovic equation with Kudryashov’s law and nonlinear perturbation terms in polarization preserving fibers is constructed by employing two improved analytical schemes, namely, the improved Sardar sub-equation method (IMSSEM) and the improved generalized Riccati equation mapping method (IGREMM). As a result of these improved approaches, many constraint conditions emerge that are required for the existence of soliton solutions. These solutions are the rational, exponential, trigonometric hyperbolic, and trigonometric which can be classified into the bright soliton, W-shaped soliton, dark soliton, singular soliton, periodic, and the mixed complex soliton. All solutions obtained have been verified using symbolic computations. Moreover, the dynamics of some of the achieved solutions are presented by plotting two and three-dimensional graphs.

Abundant Traveling Wave and Numerical Solutions of Weakly Dispersive Long Waves Model

In this article, plenty of wave solutions of the (2 + 1)-dimensional Kadomtsev–Petviashvili–Benjamin–Bona–Mahony ((2 + 1)-D KP-BBM) model are constructed by employing two recent analytical schemes (a modified direct algebraic (MDA) method and modified Kudryashov (MK) method). From the point of view of group theory, the proposed analytical methods in our article are based on symmetry, and effectively solve those problems which actually possess explicit or implicit symmetry. This model is a vital model in shallow water phenomena where it demonstrates the wave surface propagating in both directions. The obtained analytical solutions are explained by plotting them through 3D, 2D, and contour sketches. These solutions’ accuracy is also tested by calculating the absolute error between them and evaluated numerical results by the Adomian decomposition (AD) method and variational iteration (VI) method. The considered numerical schemes were applied based on constructed initial and boundary conditions through the obtained analytical solutions via the MDA, and MK methods which show the synchronization between computational and numerical obtained solutions. This coincidence between the obtained solutions is explained through two-dimensional and distribution plots. The applied methods’ symmetry is shown through comparing their obtained results and showing the matching between both obtained solutions (analytical and numerical).

Abundant breather and semi-analytical investigation: On high-frequency waves’ dynamics in the relaxation medium

This paper investigates the high-frequency waves’ dynamical behavior in the relaxation medium through two recent analytical schemes. This study depends on the Vakhnenko–Parkes (VP) equation that has been reduced from the well-known Ostrovsky equation. The modified Khater (MKhat) and the extended simplest equation (ESE) methods are used to handle the considered model. As a result, many novel solitary wave solutions have been obtained to construct the initial and boundary conditions. These conditions allow employing the variational iteration (VI) method to study the semi-analytical solutions of the considered model. The accuracy of solutions is explained along with showing the matching between analytical and semi-analytical solutions and comparing our obtained solutions with the previous results that have been obtained in published research papers. Moreover, the high-frequency waves’ behavior relaxation medium is illustrated through some distinct sketches. The methods’ performance shows their effectiveness, direct, easy, and consequential for studying many nonlinear evolution equations.

Diverse Novel Stable Traveling Wave Solutions of the Advanced or Voltage Spectrum of Electrified Transmission Through Fractional Non-linear Model

This study analyzes the exact solutions of the compliance fractional non-linear time–space telegraph (FNLTST) equation by Oliver Heaviside in 1880 via three non-applied analytical schemes. The solutions obtained to define the advanced or voltage spectrum of electrified transmission with day-to-day distance from electrical communication or the application of electromagnetic waves. Many new solutions are obtained, and three distinct styles of drawings are introduced (two-dimensional, three-dimensional, and density plots). Furthermore, stability characterization of the solutions is addressed using the properties of the Hamiltonian system. The originality of this study is shown by matching the solutions built with solutions produced previously using various analytical methods. Overall, the success of the three systems demonstrates their quality, intensity, and capacity to cope with several different types of non-linear evolutionary equations.

Periodic stick–slip mode of ice surface premelting during friction

The rheological nonlinear model of ice surface softening during friction is developed. Analytical and numerical schemes describing stick–slip frictional oscillations for three boundary relations between the shear strain, stress, and temperature relaxation times are built. The phase portraits and time dependencies of friction force are calculated. The random force (additive uncorrelated noise) effect on found damped oscillation mode is revealed. It is shown that white noise influence leads to an undamped oscillation mode corresponding to a periodic intermittent (stick–slip) regime of friction that is basically responsible for destruction of rubbing surfaces. The most pronounced such oscillatory behavior occurs if relaxation time of the temperature is much longer that its value for shear strain and stress.

Sub-10-fs-pulse propagation between analytical and numerical investigation

This paper investigates the analytical solutions of the well-known nonlinear Schrödinger (NLS) equation with the higher-order through three members of Kudryashov methods (the original Kudryashov method, modified Kudryashov method, and generalized Kudryashov method). The considered model is also known as the sub-10-fs-pulse propagation model used to describe these measurements’ implications for creating even shorter pulses. We also discuss the problem of validating these measurements. Previous measurements of such short pulses using techniques. This paper’s aim exceeds the idea of just finding the traveling wave solution of the considered model. Still, it researches to compare the used schemes’ accuracy by applying the quintic-B-Spline scheme and the convergence between three methods. Many distinct and novel solutions have been obtained and sketched, along with different techniques to show more details of the model’s dynamical behavior. Finally, the matching between analytical and numerical schemes has been shown through some tables and figures.

Energy-Based Criterion for Testing the Nonlinear Response Strength of Strong Nonlinear Oscillators

This article proposes a simple energy-based criterion developed to characterize four commonly identified responses, namely: linear, weakly nonlinear, moderately nonlinear and strongly nonlinear regimes. The response of the nonlinear simple pendulum was used for benchmarking the boundary conditions for each of the four response regimes and the test criterion was demonstrated using relevant examples. The test presented in this article is important for clarifying the obscurity surrounding the accuracy and range of validity of recent approximate analytical schemes used to investigate strong nonlinear oscillators. Furthermore, it is meant to create awareness of the need to develop more robust testing criteria.
 Keywords: strong nonlinear oscillation; periodic oscillation; approximate analytical solution

Verification of ansys and matlab Heat Conduction Results Using an “Intrinsically” Verified Exact Analytical Solution

Abstract A two-dimensional (2D) transient thermal conduction problem is examined, and numerical solutions to the problem generated by ansys and matlab, employing the finite element (FE) method, are compared against an “intrinsically” verified analytical solution. Various grid densities and time-step combinations are used in the numerical solutions, including some as recommended by default in the ansys software, including coarse, medium, and fine spatial grids. The transient temperature solutions from the analytical and numerical schemes are compared at four specific locations on the body, and time-dependent error curves are generated for each point. Additionally, tabular values of each solution are presented for a more detailed comparison. Two different test cases are examined for the various numerical solutions using selected grid densities. The first case involves uniform constant heating on a portion of one surface for a long duration, up to a dimensionless time of 30. The second test case still involves uniform constant heating but for a dimensionless time of one, immediately followed by an insulated condition on that same surface for another duration of one dimensionless time unit. Although the errors at large times for both ansys and matlab are extremely small, the errors found within the short-duration test are more significant, in particular when the heating is suddenly set “on.” Surprisingly, very small errors occur when the heating is suddenly set “off.” The solution developed using the matlab differential equation solver is found to have errors an order of magnitude larger than those generated using ansys with a similar mesh size and same FE type (quadratic triangular). This occurs not only during the transient but also for steady-state problems. However, the matlab computational efficiency is superior. Also, when using ansys, at early times the numerical errors are very sensitive to the time-step chosen rather than the spatial discretization. Therefore, the time-steps have to be very small near a singularity.

Establishment of Deep-Eutectic-Solvent-Assisted Matrix Solid-Phase Dispersion Extraction for the Determination of Four Flavonoids in Scutellariae Radix Based on the Concept of Quality by Design.

Sample preparation is the most crucial step in analytical schemes. Micro-matrix solid-phase dispersion, as a method for microextraction of analytes, has prevailed recently for its low sample and extraction solvent consumption. However, small amounts of adsorbent or sample, or a short extraction time, always bring uncertainty to the result when using this method. The aim was to develop a simple and reliable method of deep-eutectic-solvent-assisted ultrasonic-synchronized matrix solid-phase dispersion microextraction for the analysis of four flavonoids in Scutellariae Radix based on the concept of quality by design. The ZSM-5 molecular sieve was used as a new adsorbent in the micro-matrix solid-phase dispersion process. Single-factor and Box-Behnken designs were used to construct the design space. Verification of the experiment demonstrated that the design space is robust. Under optimal conditions, all analytes showed good linearity (R2 > 0.999), high reproducibility (RSD < 2.24%) and stability (RSD < 2.87%), and satisfactory recoveries (95.90-102.31%), which indicated that the established method is reliable and reproducible. Moreover, it has been successfully applied to determine the flavonoids in nine batches of Scutellariae Radix. The results indicate a great potential for analyzing complicated samples especially with small amount and helping to promote the quality control of the sample preparation process for traditional Chinese medicines. A systematic approach using a facile deep-eutectic-solvent-assisted ultrasonic-synchronized matrix solid-phase dispersion extraction coupled with HPLC for the analysis of flavonoids in Scutellariae Radix has been developed based on the concept of quality by design.

Diverse accurate computational solutions of the nonlinear Klein–Fock–Gordon equation

This manuscript handles the nonlinear Klein–Fock–Gordon (KFG) equation by applying two recent computational schemes (generalized exponential function (GEF) and generalized Riccati expansion (GRE) methods) to construct abundant novel wave solutions The considered model is the generalized form of the well-known nonlinear Schrödinger equation which is considered a quantized version of the relativistic energy-momentum relation. The accuracy of the employed analytical schemes by showing the matching between computational and approximate solutions and calculating the absolute value of error between these solutions. This matching is investigated by employing the variational iteration (VI) method to show the precision of the used schemes with the previously published solutions. The physical characterization of the evaluated solutions has explained through some distinct sketches in 2D, 3D, contour, polar, and spherical plots. The originality and novelty of our investigation have been checked by comparing our solution’s accuracy with previous other solutions’ accuracy.

Computational technology for improving the quality of difference schemes based on moving nodes

In our papers by us [11],[12] a new method of moving nodes (MMN) was introduced to obtain an approximate analytical solution and construct compact schemes for a one-dimensional convective-diffusion problem. It is proposed to improve the scheme in a three-point pattern. As an initial scheme, a counter flow with one-sided differences is taken. In the QUICK scheme [5], quadratic up flow interpolation is used to determine convective flow. Here we use the solution obtained by the upwind scheme based on MMN. In our above works, the improvement in the accuracy of difference schemes was obtained on the basis of using the method of multi point MMN. In this work, it is proposed to improve the accuracy of the constructed difference schemes using the three point MMN. This is achieved using a more accurate solution to the unknown function on the edge of the considered control volume, the algorithm of which is described in detail in this work.