Differential Transform Method Research Articles

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.

Combined effects of rotating and thermal radiation on heat and mass transfer of MHD flow in converging and diverging channels

ABSTRACT Current study deals with influence of rotating and thermal radiative fluxing on fluid’s thermal, hydrodynamic, and concentration behavior flowing between two nonparallel walls in existence of uniform magnetized force. The modeling is based on nonlinear partial differential equations (PDEs) such as continuity, momentum, temperature, and concentricity equations. These equations are converted into a system of nonlinear ordinary differential equations (ODEs) utilizing similarity transformations. Analytical solution has been constructed using the Differential Transform Method (DTM). The present results are compared to results obtained by the HAM package and by the numerical procedure (RKF-45) for validation. The impacts of active variables like the rotational factor, Hartmann, Schmidt numbers, and thermal radiation parameter, are investigated on the rapidity, temperature, and concentricity profiles. Moreover, frictional force factor, Nusselt, and Sherwood quantities. It is found that speed and concentricity of the fluid flow increase with the growth of Hartmann number, whereas a reverse behaviour is observed for thermal distribution. Results obtained also reveal an improvement in frictional force factor, Nusselt and Sherwood numbers with the enhancement of Reα parameter. It is found that an increase in the rotational parameter Ro and the radiation factor Rd leads to a decrease in the Nusselt number.

Multistep Reduced Differential Transform Method in Solving Nonlinear Schrodinger Equations

This paper obtains semi-analytical solutions for the nonlinear Schrodinger equations (NLSEs) using the multistep reduced differential transform method (MsRDTM). The implemented method yields an analytical approximate solution over a longer time frame, in which the method applied is treated as an algorithm in a sequence of small sub-division of intervals of identical length compared to the traditional reduced differential transform method (RDTM). Excluding the need of perturbation, linearization, or discretization, this method offers the benefit and reliability of the multistep algorithm. The outcomes show that the MsRDTM generated highly accurate solutions of NLSEs than the RDTM. In addition, the results show that the suggested method is straightforward to use, saves a significant amount of computing work when solving NLSEs, and has potential for broad application in other complex partial differential equations (PDEs) in the fields of engineering and science. The accuracy of the method is shown through the tables and graphical illustrations provided.

The improved residual power series method for the solutions of higher-order linear and nonlinear boundary value problems

AbstractIRPSM is the extended form of RPSM that gives an approximate solution to boundary value problems without requiring an exact solution. This method creates a truncated series for the determination of the missing initial conditions. The present study is conducted to evaluate semi-numerical solutions to differential equations using the Improved Residual Power Series Method (IRPSM). Our main objective is to check whether the proposed scheme is efficient for solving such equations or not. The solution of differential equations has been approximated using truncated residual power series. The method is used to solve differential equations for higher-order linear and non-linear boundary value problems. Absolute errors for the solved problems are calculated to ensure accuracy. It is also compared with some known methods, like the Differential Transform Method (DTM) and the Homotopy Perturbation Method (HPM). For the calculations, Mathematica 12.0 software is used. The computed results are compared to the exact solutions as well as the available results in the literature. The results showed that the results obtained by IRPSM are closely related to the exact solution when compared to the DTM and HPM results.

Unsteady tetra hybrid nanofluid in MHD flow between two parallel rotating stretchable disks under chemical reaction and activation energy effects

This article aims to investigate the dynamics of unsteady tetra hybrid nanofluid magnetohydrodynamic (MHD) flow occurring between two parallel rotating stretchable disks with convective boundary conditions. The study employs blood as the base fluid, incorporating quadrant hybrid nanoparticles comprising ZrO2, MoS2, MWCNTs, and UO2. The influence of parameters such as shape factor, chemical reaction, and activation energy on the flow characteristics is examined. By employing similarity transformations, the governing partial differential equations are converted into a set of ordinary differential equations. Analytical solutions are obtained utilizing the differential transform method (DTM) and validated against results obtained through the HAM-package and numerical techniques (Explicit Runge-Kutta Method). Velocity profiles in the axial, radial, and tangential directions, as well as temperature and concentration profiles, are scrutinized across various fluid parameters. The tetra hybrid nanofluid demonstrates a higher Nusselt number compared to other fluid compositions. However, it is noted that skin friction is also elevated for the tetra hybrid nanofluid, underscoring the importance of carefully considering optimal nanoparticle utilization in engineering applications. This study offers valuable insights into the behavior of hybrid nanofluids within MHD flow systems and their potential industrial applications.

Two-dimensional Coupled Asymmetric van der Pol Oscillator

A reliable algorithm based on an adaptation of the standard differential transform method (DTM) is presented, which is the multi-step differential transform method (Ms-DTM) since it may be difficult to directly apply differential transform method (DTM) to obtain the series solutions for the present two- dimensional coupled asymmetric van der Pol oscillator. The solutions of a two dimensional coupled asymmetric van der Pol oscillator were obtained by Ms-DTM. Figurative comparisons between the Ms-DTM and the classical fourth order Runge-Kutta method (RK4) are given. The obtained results reveal that the proposed technique is a promising tool to solve the considered van der Pol oscillator and yield same information on the phase portrait confirming the stability of the system, effectively. It can be said that the considered approach can be easily extended to other nonlinear van der Pol oscillator systems and therefore is widely applicable in engineering and other sciences.

Free vibration analysis of soil-beam-fluid interaction with variable Winkler backfill using DTM

This study aims to analyze the free vibrations of the Euler-Bernoulli beam vibrating in the vicinity of an inviscid, irrotational, and incompressible fluid, as well as Winkler soil, using the differential transformation method. To show the robustness of the proposed formulation, it is used in different numerical tests in which the free vibration of different soil-beam-fluid systems is assessed. The differences between these systems are rooted in their supports and the soil characteristics. Furthermore, the results achieved from the proposed method are compared with those of the finite element strategy. It is emphasized that the differential transformation method employed in this study showcases sufficient precision and reliability throughout the analysis process.

Vibrational Analysis of Axially Loaded Rayleigh Beams on Variable Winkler Foundations with Implications for Structural Design

In this study, we explore the complex vibration problem of a Rayleigh beam under axial loads, positioned over a variable Winkler foundation with elastic properties changing along its length. The investigation delves into how factors like rotational inertia, axial forces, and foundation properties affect the beam’s dimensionless eigenvalues and eigenmodes. Employing the Differential Transform Method (DTM), we tackle the governing equations across various foundation conditions, including constant, linear, and parabolic variations. The results align well with the previous research, affirming their precision between scenarios considering or neglecting the elastic foundation’s effects. Notably, we emphasize the influence of linear spring stiffness on vibration characteristics, overshadowing rotary spring stiffness. Thus, the important novelty of the study lies in its comprehensive investigation into the vibration dynamics of a Rayleigh beam under axial loads on a variable Winkler foundation with elastic properties varying along its length. The findings illuminate a previously unexplored dynamic: the intricate interplay between linear and rotary spring stiffness, showcasing the dominant impact of linear stiffness on vibration characteristics. Of significant note is the profound effect of axial forces on fundamental eigenvalues, underscoring their critical role in structural behavior. These developments possess substantial practical implications, providing invaluable guidance for the accurate modeling of beams supported by elastic foundation, thereby enhancing structural design and performance.

Using a Mix of Finite Difference Methods and Fractional Differential Transformations to Solve Modified Black–Scholes Fractional Equations

This paper discusses finding solutions to the modified Fractional Black–Scholes equation. As is well known, the options theory is beneficial in the stock market. Using call-and-pull options, investors can theoretically decide when to sell, hold, or buy shares for maximum profits. However, the process of forming the Black–Scholes model uses a normal distribution, where, in reality, the call option formula obtained is less realistic in the stock market. Therefore, it is necessary to modify the model to make the option values obtained more realistic. In this paper, the method used to determine the solution to the modified Fractional Black–Scholes equation is a combination of the finite difference method and the fractional differential transformation method. The results show that the combined method of finite difference and fractional differential transformation is a very good approximation for the solution of the Fractional Black–Scholes equation.

Solution of Laplace Equation by Modified Differential Transform Method

In this paper, we applied the modified two-dimensional differential transform method to solve Laplace equation. Laplace equation is one of Elliptic partial differential equations. These kinds of differential equations have specific applications models of physics and engineering. We consider four models with two Dirichlet and two Neumann boundary conditions. The simplicity of this method compared to other iteration methods is shown here. It is worth mentioning that here only a few number of iterations are required to reach the closed form solutions as series expansions of some known functions.

External and internal resonances of thin-walled curved beams under three-directional moving harmonic loads

This paper deals with the internal and external resonances of simply supported curved beams subjected to three-directional harmonic moving loads. Firstly, based on the differential transformation methods and the Glerkin methods, the analytical solutions are newly proposed for vertical, torsional, radial and axial response of thin-walled curved beams under three-directional harmonic moving loads. Then, in the proposed analytical solutions, this study attains the conditions for the external resonance for in-plane and out-of-plane vibrations by letting the denominator of the proposed analytical solutions equal to zero. Besides, the semi-analytical solution for the internal resonance of curved beams is first proposed in this paper by the coincidence of both two mentioned-above resonant frequencies of beams. The internal resonance will occur when the beam is designed with the critical length for the given cross-section. The derived condition for the internal resonance can offer a basis for structural design to avoid the resonant appearance. All the analytical solutions are in good agreement with the results addressed in existing literature and numerical ones obtained from the finite element method (FEM). The results reveal that the corresponding motion of curved beams diverges, reaching a maximum value during the duration of the harmonic moving loads, when in the external resonance. Moreover, under the internal resonance condition, the curved beams are in repetitive transition between the out-of-plane mode and the in-plane mode without any external energy input. Additionally, in this scenario, the excitation frequency can coincide with the two resonance frequencies of beams resulting in simultaneous occurrences of out-of-plane and in-plane resonances on curved beams. The aim of this study highlights the potential resonant appearances on beams and provide reference in the design phase to avoid the internal resonance appearance and ensure the safety of structures.

Irreversibility of Al2O3-Ag hybrid nanoparticles in mixture base fluid on microchannel with variable viscosity, buoyancy forces, and suction/injection effects: An analytical study

The development of inherent irreversibility in the system is caused by single phase Poiseuille flow considering hybrid nanoparticles (Al2O3-Ag) and mixture fluid (water and ethylene glycol H2O50%-C2H6O250%) in the upright microchannel with unequal viscosity. Taking into account the buoyancy force, suction/injection at the walls, and the form factor and geometry of the nanoparticles. The modeling is based on nonlinear PDEs such as continuity, momentum, and heat equations, which are then transformed to a system of nonlinear ODEs using similarity transformations and solved numerically and analytically. The analytical solution was built using the Differential Transform Method (DTM), and the current results in specific cases are compared to results obtained by the HAM-based Mathematica package, the Runge-Kutta Fehlberg 4th-5th order (RKF45), and those available in literature. The effects of active parameters are investigated on the velocity and temperature, entropy generation and Bejan number. The outcomes of the present analysis reveal that the nanoparticles volume fraction, thermal radiation and Biot number acts to enhance the cooling of the system through the release of thermal energy. in addition, the enhancement in variable viscosity parameter, causes a rise in the irreversibility rate, which in turn boosts the rates of entropy generation and Bejan number. On the other hand, Irreversibility due to heat transfer is dominant in the centerline of the microchannel, while fluid friction irreversibility dominates its walls.

Improve Solutions of Differential Transform Method for Jeffrey Hamel Magneto-hydrodynamic Non-Newtonian Casson Fluid Flow Problem

In this article, Magneto-hydrodynamic flow of a non-Newtonian Casson fluid in converging and diverging channels subject to stretching -shrinking cases is an important problem in the field of fluid flow that has been highlighted by the combination of the differential transformation method and the optimal method so-called optimal differential transformation method (ODTM). In this synthesis that is used achieves improved analytical results for this problem. This study examines the impact of several developing characteristics, including channel angle, stretching-shrinking parameter, Casson fluid parameter, Hartmann number, and Reynolds number with the velocity profiles. Results obtained from the DTM and the numerical method BVP4c are compared to those resulting from ODTM which consist of the collocation differential transform method (CDTM), the sub-domain differential transform method (SDTM), the least squares differential transform method (LSDTM), and the Galerkin differential transform method (GDTM). The comparison shows that the percentage errors of the optimal methods (ODTM) are more accurate than those of the DTM for a similar number of parts of the repeated series over a wide range of physical parameters. Finally, it can be said that the results of the proposed method for third-order nonlinear Jeffrey Hamel flow in the non-Newtonian Casson fluid problem are high accuracy.

On the Convergence of Multi-Stage Differential Transform Method

In many paper, Multi-stage Differential Transform Method (MsDTM) has been presented as a method to solve linear and nonlinear equations of various kinds. Multi-stage Differential Transform method has been used to find the exact or analytical approximate solution of the different problems. But only a few works have been considered of convergence of this method. In this paper, Convergence analysis of multi-stage differential transform method is presented. Some numerical examples are proposed to verify the convergence of the proposed method

Simulation of Fractional Order 2D-Mathematical Model Using α-Fractional Differential Transform Method

In this paper, we will introduce a well-known transformation technique, the modified α-fractional differential transform, to the differential equation of fractional order. We derive some new results with proof using new techniques that never existed before. By using this new technique, we are attempting to solve the nonlinear fractional-order mathematical epidemic model. Furthermore, the fractional epidemic model’s solution obtained by using this new technique is correlated with the solution of the same model calculated for a different fractional order by the modified α-fractional differential transform method. Moreover, using the Python software, we can numerically and graphically represent the solution of fractional differential equations.

Thermal and solutal stratified Heimanz flow of AA7072-deionized water over a wedge in the presence of bioconvection

The bioconvective Heimanz flow of nanofluid across a wedge with thermal stratification is analyzed in this article. The wedges are often seen in glider aircraft, rocket climbing frames, etc. The nanofluid considered in this study is composed of aluminum alloys of AA7072 and deionized water. The AA7072 alloys are specially manufactured materials composed of Aluminum and Zinc in the ratio of 98 : 1 along with metals like silicon, ferrous, and copper so that they possess enhanced heat transfer features. The mathematical model is formed using the modified Buongiorno’s model that includes the discussions related to slip mechanisms and volumetric analysis in terms of the weight of the nanoparticle. The model is in the form of partial differential equations and is later converted to ordinary differential equations with the assistance of similarity transformation. This set of equations is solved by the Differential Transformation Method (DTM) and the outcomes are discussed through graphs. ,

Darcy-Forchheimer nanofluid flow and heat transfer with convective boundary conditions: Insights from differential transform method

In this work, we use the sophisticated differential transform method (DTM) to derive solutions to the complicated momentum, heat, and mass transfer equations that characterize Darcy-Forchheimer nanofluid flow over a stretching surface with convective boundary conditions. The investigation includes the nuanced effects of heat sources and chemical reactions and also addresses the synergistic effects of Brownian motion and thermophoresis. By applying an appropriate similarity transformation, the governing partial differential equations (PDEs) are gracefully transformed into an elegant system of nonlinear coupled ordinary differential equations (ODEs), a problem that is ingeniously solved using DTM. The veracity of our results is underpinned by a careful validation against previous scientific work that illuminates the path to scientific clarity. As the mathematical tapestry unfolds, the enchanting interplay of physical parameters on fluid dynamics is revealed, manifesting in visually captivating graphs and illuminating tables. Of particular note is the transformative influence of the magnetic parameter, which stages a ballet in which an increase in skin friction and the Nusselt number causes a diminuendo. Meanwhile, the solutal concentration decreases as the Brownian motion, Lewis number, and chemical reaction parameters increase, while it waltzes in reverse harmony with the thermophoresis parameter. Beyond equations and diagrams, this investigation promises to cast a luminous beacon on practical realms, imparting invaluable insights for industries ensconced in the realm of heat exchangers, the alchemy of lubricant refining processes, and the grand tapestry of scientific and engineering domains.

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.

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.