Differential Transform Method Research Articles

Numerical treatment for time fractional order phytoplankton-toxic phytoplankton-zooplankton system

<abstract><p>The study of time-fractional problems with derivatives in terms of Caputo is a recent area of study in biological models. In this article, fractional differential equations with phytoplankton-toxic phytoplankton-zooplankton (PTPZ) system were solved using the Laplace transform method (LTM), the Adomain decomposition method (ADM), and the differential transform method (DTM). This study demonstrates the good agreement between the results produced by using the specified computational techniques. The numerical results displayed as graphs demonstrate the accuracy of the computational methods. The approaches that have been established are thus quite relevant and suitable for solving nonlinear fractional models. Meanwhile, the impact of changing the fractional order of a time derivative and time $ t $ on populations of phytoplankton, toxic-phytoplankton, and zooplankton has been examined using graphical representations. Furthermore, the stability analysis of the LTM approach has been discussed.</p></abstract>

Mathematical analysis of urea amperometric biosensor with Non-Competitive inhibition for Non-Linear Reaction-Diffusion equations with Michaelis-Menten kinetics

This article examines a mathematical model for stable urea amperometric biosensors with non-competitive inhibition under homogenous conditions. The system is based on reaction and diffusion equations with a non-linear component associated with the Michaelis-Menten kinetics of the enzyme reaction. Theoretical findings in this study can be used to examine the impact of various factors, including the Michaelis-Menten constant and the Thiele modulus. The steady-state non-linear reaction–diffusion equations were solved using two effective and widely used analytical approaches, the Akbari-Ganji method (AGM) and the differential transform method (DTM). The generalized approximation analytical solution for the concentrations of the substrate, inhibitor, and product, as well as the current for the experimental values of the parameter, is developed. The current can also be used to analyze the resistance and sensitivity of biosensors. The digital simulation was carried out utilizing MATLAB software. The numerical results help to validate the analytical results. The impact of membrane thickness, maximum enzymatic rate, and substrate concentration on biosensor response was evaluated.

LOCAL TIME FRACTIONAL REDUCED DIFFERENTIAL TRANSFORM METHOD FOR SOLVING LOCAL TIME FRACTIONAL TELEGRAPH EQUATIONS

In this paper, we seek to find solutions of the local time fractional Telegraph equation (LTFTE) by employing the local time fractional reduced differential transform method (LTFRDTM). This method produces a numerical approximate solution having the form of an infinite series that converges to a closed form solution in many cases. We apply LTFRDTM on four different LTFTEs to examine the efficiency of the proposed method. The yielded results established the effectiveness of LTFRDTM as a reliable and solid approach for obtaining solutions of LTFTEs. The solutions coincided with the exact solution in the ordinary case when [Formula: see text]. It also required minimal amount of computational work and saved a lot of time.

Modification of the Differential Transform Method With Search Direction Along the Spatial Axis

This paper presents a modification of the Differential Transform Method (ModDTM) formulated so that the search direction of the solution is along a spatial axis (x). It is shown that the method is applicable to a wide spectrum of (1 + 1) partial differential equations, integro partial differential equations, and integral equations. The solutions obtained are in the form of a Taylor series, the coefficients of which are determined by recursively operating a differential transform of the equation. These solutions are either exact and in closed form, or are good approximations. To illustrate the application of the method, as well as to show its versatility, examples are chosen from all the categories mentioned earlier. These include equations such as the nonlinear Fisher, combined KdV‐mKdV, Hunter–Saxton (H‐S), Fornberg–Whitham (FW), coupled systems such as the Whitham–Broer–Kaup (WBK) and the sine‐Gordon (s‐G) equations, integral equations, Volterra partial integro‐differential equations (PIDE), and linear and nonlinear versions of the (complex) Schröedinger equation. The Tables and 3D‐plots show the good rate of convergence of the obtained solutions with the exact ones. It is thus found that the new scheme is successful in producing satisfactory results as any other method in which the conventional approach of searching for solutions along the temporal axis is followed. Further, the procedures involved are simple, and hence may be operated with great ease.

Analysis of the n-Term Klein-Gordon Equations in Cantor Sets

The effectiveness of the local fractional reduced differential transformation method (LFRDTM) for the approximation of the solution related to the extended n-term local fractional Klein-Gordon equation is the main aim of this paper in which fractional complex transform and local fractional derivative have been employed to analyze the n-term Klein-Gordon equations, and Cantor sets. The proposed method, along with the existence of the solutions demonstrated through some examples, provides a powerful mathematical means in solving fractional linear differential equations. Considering these points, the paper also provides an accurate and effective method to solve complex physical systems that display fractal or self-similar behavior across various scales. In conclusion, the fractional complex transform with the local fractional differential transform method has been proven to be a robust and flexible approach towards obtaining effective approximate solutions of local fractional partial differential equations.

Free vibration of cylindrical shells and panels with cutout via the differential transform method

Semi-numerical methods hold significant value due to their efficiency in time-consuming calculations. Accordingly, this article proposes an effective strategy to derive the vibration parameters of cylindrical shells (CSs) with cutout utilizing a semi-analytical approach entitled differential transform method (DTM). To achieve this, the governing equations of motion are first introduced based on Sander’s theory of shells. These equations give rise to a coupled system of linear ordinary differential equations, which pose a challenge in solving the eigenvalue problem. It should be emphasized that the classical version of the differential transform method could not be directly applied to solve the eigenvalue problem related to the above-mentioned coupled differential equation. An innovative technique is employed and discussed in the paper to remedy these difficulties, which could be widely used in similar problems. As a result, the natural frequencies and mode shapes of some CSs with cutouts, various internal angles, and diverse material characteristics are attained. The outcomes undergo validation through the hybrid finite element method, and the proposed method’s accuracy is rigorously proved. In conclusion, the proposed DTM provides an efficient solution to the eigenvalue problem of coupled differential equations and offers a valuable tool for obtaining accurate vibration parameters of CSs.

Differential Transform Method for Solving Volterra q-Integral Equations

In this paper, Volterra q-integral equations are solved by using the method of q-differential transformation. Exact solutions of linear and nonlinear q-integral equations have been investigated. To illustrate the method, several problems are discussed for the effectiveness and performance of the proposed method.

Computing neural network to analyze heat and mass transfer in the flow of nanofluid between two disks

The model of copper nanoparticles which are suspended in the engine oil (EO) and rotated between two stretchable disks is analyzed. The flow, heat, and mass transmission phenomena of nanofluid with magnetohydrodynamics (MHD) have a vital role in many industries. A magnetic field in the vertical direction is imposed in the flow of the nanofluid and Dufour and Soret (DS) effects are discussed in the equations of energy and concentration. The main equations of motion and energy are converted into a set of nonlinear ordinary differential equations (ODEs) after applying the similarity conversions. A popular semi-analytical approach, namely the differential transform method (DTM) is used to get the solution of velocity, temperature, and concentration profiles. The effect of the various parameters on all profiles is graphically presented and explained. The present data of shear stress, the Nusselt, and the Sherwood numbers calculated by DTM are matched and verified by numerical method data and literature for the novelty of the work. The strength of the work is to analyze the validation, training, and testing by using Levenberg-Marquardt artificial neural network (ANN). This ANN is verified by mean square error, error histogram, and regression analysis.

MHD flow of a rotating micropolar fluid between two plates by using DTM

The primary purpose of this work is to investigate the influence of a magnetic field on a micropolar fluid, which is rotating between two horizontally arranged parallel plates. A uniform magnetic field acts perpendicularly to the porous plate, which consumes fluid with a steady suction velocity. The obtained higher-order nonlinear partial differential equations (PDEs) are converted into nonlinear ordinary differential equations (ODEs) by using similarity transformation. A semi-analytical methodology has been applied to obtain the result to the problem. The influence of various flow parameters of the flow field like coupling parameter, viscosity parameter, Reynolds number, porosity parameter, rotation parameter and magnetic field parameter has been examined and explained by graphs. A resemblance between the results obtained by solving by the differential transform method (DTM) and the outcomes of the former research works has also been displayed which shows the correctness and efficiency of the used analytical method. The numerical values of the coefficient of skin friction at the lower plate are also displayed in tabular form and compared with the literature.

Comparative study on curing kinetics of MDI-based polyurethanes with different chain length diol curing agents

The difference in the reaction between thermosetting polyurethane (CPU) and binary alcohol system curing agents is crucial for obtaining accurate curing kinetics models and selecting curing agents. Based on a systematic comparison of the reaction activation energies and kinetic models of three different chain length fatty alcohol curing agents, ethylene glycol, 1,4-butanediol, and 1,6-hexanediol, the effect of curing agent chain length on the curing reaction was elucidated. Using differential and integral transformation methods to process non-isothermal DSC tests results, the effect of different chain lengths curing agents on the activation energy of MDI based polyurethane were analyzed. Based on the self-catalytic model, the segmented curing mechanism of the MDI based polyurethane systems were studied, and applicable high-precision segmented kinetic models’ parameters were calibrated. The optimal curing conditions for these systems were determined using extrapolation. The research results indicate that when the curing degree increases to 95 %, the overall activation energy of the MDI/1,4-butanediol system decreases by 47.1 %, accompanied by a faster self-catalytic. However, the activation energies of the other two systems, MDI/propylene glycol system and MDI/1,6-hexanediol system increased by 62. % and 295.0 %, respectively. This slows down the reaction rate of the two systems, which was 7.8 % and 10.9 % slower than the MDI/1,4 butanediol. heir molecular cross-linking network limits the diffusion ability of their chains and reduces the self-catalysis, resulting in slow reactions. These results will provide theoretical guidance and reference for the material formulation, process optimization of MDI based polyurethane actual curing production.

Double diffusive MHD squeezing copper water nanofluid flow between parallel plates filled with porous medium and chemical reaction

Purpose This paper aims to explore the double diffusive magneto-hydrodynamic (MHD) squeezed flow of (Cu–water) nanofluid between two analogous plates filled with Darcy porous material in existence of chemical reaction and external magnetic field. Design/methodology/approach The governing nonlinear equations are transformed into ordinary differential equations by means of similarity transforms, and the coupled mass and heat transference equations are resolved analytically with the application of differential transform method (DTM). The effects of different relevant parameters on velocity, temperature and concentration, including the squeeze number, magnetic parameter, Biot number, Darcy number and chemical reaction parameter, are illustrated with figures. In addition, for various parameters, the local skin friction coefficient, local Nusselt number and local Sherwood number are computed and are graphically displayed. Findings It is observed that the squeeze number has a direct relationship with Sherwood number and an inverse relationship with skin friction as Biot number increases. With enhanced Biot numbers, the temperature value increases during both squeeze and non-squeeze moments, but the temperature values are higher for squeeze moments compared to the other case. Practical implications This research has potential applications in various large-scale enterprises that might benefit from increased productivity. Social implications The results are useful to thermal science community. Originality/value Unique and valuable insights are provided by studying the impact of chemical reaction on double diffusive MHD squeezing copper–water nanofluid flow between parallel plates filled with porous medium. In addition, this research has potential applications in various large-scale enterprises that might benefit from increased productivity.

Solution of Two-Dimensional Solute Transport Model for Heterogeneous Porous Medium Using Fractional Reduced Differential Transform Method

This study contains a two-dimensional mathematical model of solute transport in a river with temporally and spatially dependent flow, explicitly focusing on pulse-type input point sources with a fractional approach. This model is analyzed by assuming an initial concentration function as a declining exponential function in both the longitudinal and transverse directions. The governing equation is a time-fractional two-dimensional advection–dispersion equation with a variable form of dispersion coefficients, velocities, decay constant of the first order, production rate coefficient for the solute at the zero-order level, and retardation factor. The solution of the present problem is obtained by the fractional reduced differential transform method (FRDTM). The analysis of the initial retardation factor has been carried out via plots. Also, the influence of initial longitudinal and transverse dispersion coefficients and velocities has been examined by graphical analysis. The impact of fractional parameters on pollution levels is also analyzed numerically and graphically. The study of convergence for the FRDTM technique has been conducted to assess its efficacy and accuracy.

Analytical Study of Interstitial Fluid Extractive Microneedle Arrays Using Differential Transform Method

The computational studies of a predictive mathematical model for the extraction of interstitial (ISF) for transdermal and non-invasive diagnosis using biodegradable and hollow microneedle patch is presented in this paper. Rapid Diagnostic Tests diagnosis, which is non-invasive, affordable, straightforward, and provides results promptly and reliably, has increased access to parasite-based analysis on a global scale. Microneedle arrays are a rapidly evolving and promising technology for transdermal interstitial fluid extraction, which is used for many clinical diagnostic procedures. Hence, a developed mathematical predictive model used to optimize the design of microneedle patch for transdermal ISF extraction and subsequent diagnosis using dissolvable microneedle arrays was applied in this study. The model's solutions were obtained using the Differential Transform Method. The numerical Runge-Kutta method of fourth order was used to validate it. An experimental test result was also used to further validate the analytical results in the absence of the extracted velocity parameter. And there was a good agreement among them. Influence of dissolution rate constant, microneedle height, diffusion coefficient, velocity of ISF, microneedle ISF drug load, and density of the microneedle; were investigated. Increase in diffusion coefficient and density led to an increase in concentration of ISF extracted over time, an increase in dissolution rate led to a decrease in concentration extracted, while decrease in drug load and height, led to increase in ISF concentration extracted. A negligible effect was observed by varying the velocity of ISF extracted. The approximate analytical approach utilized in the current work has given us a more precise strategy for creating a mathematical model that predicts how ISF will be extracted from skin for use in transdermal and non-invasive rapid diagnostic tests.

Transverse Vibration Analysis of a Self-Excited Beam Subjected to Delayed Distributed and a Singular Load Using Differential Transformation Method

Transverse Vibration Analysis of a Self-Excited Beam Subjected to Delayed Distributed and a Singular Load Using Differential Transformation Method

Fractional order ATR-FTIR differential spectroscopy for detection of weak bands and assessing the radiation modifications in gamma sterilized UHMWPE.

This study presents a new method for identifying radiation modifications in UHMWPE polymer samples. The method involves using a mathematical technique called fractional order differential transformation on IR spectra obtained through ATR-FTIR spectroscopy. This new method was compared to existing techniques such as FTIR, XRD, and DSC, and it was found to be more sensitive and accurate in detecting radiation-induced changes in the polymer. The study focused on identifying changes in weak IR bands in the UHMWPE samples caused by gamma sterilization while simulating IR spectra using different orders of fractional derivatives and compared them to experimental spectra. It was found that applying a lower order of differentiation was more suitable for identifying radiation-induced changes in the UHMWPE samples. Using this method, they were able to identify specific changes in the gamma irradiated structure, such as the splitting of a single absorption peak into a doublet, which was only present in the 50 kGy irradiated sample. The study also used correlation index analysis, principal component analysis, and hierarchy cluster analysis to analyze the simulated and experimental spectra. These techniques allowed to confirm the effectiveness of the fractional order differential transformation method and to identify the specific regions of the IR spectra that were affected by radiation-induced changes in the UHMWPE samples. Overall, this study presents a new method for identifying radiation-induced changes in UHMWPE polymer samples that is more sensitive and accurate than existing techniques. By identifying these changes, researchers can better understand the effects of gamma sterilization on medical equipment and potentially develop new methods for sterilization that do not damage the equipment.

Modified α -Parameterized Differential Transform Method for Solving Nonlinear Generalized Gardner Equation

In this article, we present a novel enhancement to the α -parameterized differential transform method (PDTM) for solving nonlinear boundary value problems. The proposed method is applied to solve the generalized Gardner equation by utilizing genetic algorithms to obtain optimal parameter values. Our proposed approach extends the general differential transformation method, allowing for the use of various values for the coefficient α . Our solution procedure offers a distinct advantage by allowing the original differential transformation method to be divided into multiple steps, thereby illustrating specific solution properties for nonlinear boundary value problems. Additionally, possible alternative solutions based on varying parameter values are also explored and discussed. The results with those obtained through the DTM method and exact solutions are compared to confirm the accuracy of our method and its efficiency in reaching the exact solution quickly.

Fast Power System Transient Stability Simulation

Power system transient stability simulation is of critical importance for utilities to assess dynamic security. Most of the commercially available tools use the traditional numerical integration method to simulate power system transient stability, which is computationally intensive and has low simulation speed. This makes it difficult to identify any insecure contingency before it happens. It is already proven that power system transient stability simulation achieved using the differential transformation method (DTM) requires less computational effort and has improved simulation speed, but it still requires further improvement regarding its accuracy and performance efficiency. This paper introduces a novel power system transient stability simulation method based on the adaptive step-size differential transformation method. Using the proposed method, the step size is varied based on the estimated local solution error at each time step. The accuracy and speed of the proposed simulation approach are investigated in comparison with the classical differential transformation method and the traditional numerical integration method using the IEEE 9 bus and 39 bus test systems. The simulation results reveal that the proposed method increases the simulation speed by 20–44.57% and 83–92% when compared with the classical DTM and traditional numerical-integration-based simulation methods, respectively. It is also proved that compared with the DTM-based simulation, the proposed method provides 45.27% to 58.85% and more than 90% accurate simulation results for IEEE 9 and IEEE 39 test systems, respectively. Therefore the proposed power system transient stability simulation method is faster and relatively more accurate and can be applied for online transient stability monitoring of power system networks.

Numerical solution of rotavirus model using Runge-Kutta-Fehlberg method, differential transform method and Laplace Adomian decomposition method

This paper explores the numerical analysis of a model for the challenging childhood disease named Rotavirus, by examining the impact of reducing risk. Specifically, the numerical approximation solution of the rotavirus model is investigated using three different numerical methods; the Runge-Kutta-Fehlberg technique, differential transformation method, and the Laplace Adomian decomposition method. The effectiveness and accuracy of these methods are compared, and conclusions are drawn regarding their suitability for obtaining approximate solutions to modeling problems. Additionally, the effect of reducing the risk of infection on the susceptible and vaccinated populations has been studied, leading to interesting findings. This research provides valuable insights into the application of numerical methods to model infectious diseases and may be useful for researchers.

Free vibration of porous FG magneto-electro-elastic microbeams in the hygrothermal environment based on differential transformation method

Free vibration of the porous functionally graded magneto-electro-elastic (FGMEE) microbeams in hygrothermal environment is investigated. Four kinds of distributions of porosities including uniform porosity, “O,”“X,” and “V” type porosities are assumed in this paper. For the functionally graded microbeam, the physical neutral surface is considered. The modified couple stress theory is adopted to capture the size effect. The equations of motion for the porous FGMEE microbeam are derived and solved by differential transformation method. In the numerical examples, the effect of electric and magnetic loadings, material length scale parameter, the temperature rise, the moisture concentration, material gradient index, and porosity volume fraction on the natural frequency are analyzed and discussed in detail. The results indicate that different porous distribution modes have different effects on natural frequency. Moisture concentration/temperature rise can reduce the natural frequency regardless of what the temperature and moisture modes is. These results will be useful for the design of the porous FGMEE structures.

Thermoelastic creep evolution in a variable thickness functionally graded piezoelectric rotating annular plate considering convection and radiation heat transfer

The thermoelastic time-dependent creep progress of a rotating annular plate with nonuniform thickness made of functionally graded piezoelectric material considering convection heat transfer and solar radiation was investigated. The nonlinear differential equation of heat transfer considering conduction, convection, and solar radiation was solved using the differential transformation method. Then, the differential equation of equilibrium for the annular plate containing creep strains was derived. Firstly, the differential equation was solved analytically for the zero time . Then, the stress and strain rates were achieved using Norton’s law with Prandtl–Reuss equations. Finally, the time-dependent creep stresses and deformation redistributions were obtained employing an iterative method.