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AbstractThis article aims to analyze the free vibration of closed and open horizontal circular cylindrical shells based on the concept of state-space transformation in conjunction with the differential transform method (DTM). The governing equations of the cylinders were developed based upon Sanders’ thin shell theory. Then, the set of coupled governing differential equations transformed into nine first-order differential equations. Afterwards, the well-established semi-analytical scheme so-called DTM was employed to solve the differential equations. The numerical investigation of the present method was also comprehensively carried out by analyzing some closed and open cylindrical shells with different boundary and compatibility conditions. To ensure the correctness of the results, the outcomes were compared with experimental and other numerical results, especially with the well-known finite element method. Furthermore, some plots of two-dimensional and three-dimensional mode shapes for the first and second axial modes are depicted. [ABSTRACT FROM AUTHOR]

The study focuses on the free vibration analysis of beams made of axially functionally graded materials (AFGM) with curvilinear variable cross-sections along their length. The beams encompass various shapes, including concave and convex conic sections, with axial material properties varying according to polynomial and exponential laws. The equations of motion are derived using Hamilton's principle within the framework of Timoshenko beam theory. These governing equations, subjected to various boundary conditions, are solved using the differential transform method (DTM). The proposed solution technique is validated by comparing computed natural frequencies with the existing literature and results obtained using three-dimensional finite element analysis in ABAQUS. The incorporation of material gradients into the beam finite element models was achieved using the user-defined material subroutine (UMAT). Additionally, a comprehensive study is conducted to examine the influence of various factors on the natural frequencies of functionally graded beams. These factors include parameters of material laws, types of variable beam shapes, slenderness ratio, and specific boundary conditions. This study provides a thorough understanding of the modal dynamics of the considered beams, offering valuable insights into the behavior of FGM structures. [ABSTRACT FROM AUTHOR]

The history of variational calculus dates back to the late 17th century when Johann Bernoulli presented his famous problem concerning the brachistochrone curve. Since then, variational calculus has developed intensively as many problems in physics and engineering are described by equations from this branch of mathematical analysis. This paper presents two non-classical, distinct methods for solving such problems. The first method is based on the differential transform method (DTM), which seeks an analytical solution in the form of a certain functional series. The second method, on the other hand, is based on the physics-informed neural network (PINN), where artificial intelligence in the form of a neural network is used to solve the differential equation. In addition to describing both methods, this paper also presents numerical examples along with a comparison of the obtained results.Comparingthe two methods, DTM produced marginally more accurate results than PINNs. While PINNs exhibited slightly higher errors, their performance remained commendable. The key strengths of neural networks are their adaptability and ease of implementation. Both approaches discussed in the article are effective for addressing the examined problems. [ABSTRACT FROM AUTHOR]

This article presents an innovative approach using the Differential Transform Method (DTM) to analyze the vibration characteristics of cylindrical shells, integrating Taylor's series with Sander's classical theory. It demonstrates DTM's efficiency, accuracy, and potential as an alternative method. The study introduces a novel application of the DTM in exploring the free vibration of cylindrical shells, detailing a technique to address challenges such as normalization, linear solution methodologies, and parameter derivative modifications. A dimensionless parameter analysis evaluates the impact of length, radius, thickness, and modulus of elasticity. Comparative analysis with Hybrid Finite Element Method (FEM) data and validation against existing literature highlights DTM's precision and reliability. In conclusion, DTM offers a robust solution for the eigenvalue problem in coupled differential equations, providing accurate vibration parameters. Additionally, an important relationship between the modulus of elasticity and frequency in the dimensionless state was obtained. [ABSTRACT FROM AUTHOR]

Abstract This article presents an innovative approach using the Differential Transform Method (DTM) to analyze the vibration characteristics of cylindrical shells, integrating Taylor's series with Sander's classical theory. It demonstrates DTM's efficiency, accuracy, and potential as an alternative method. The study introduces a novel application of the DTM in exploring the free vibration of cylindrical shells, detailing a technique to address challenges such as normalization, linear solution methodologies, and parameter derivative modifications. A dimensionless parameter analysis evaluates the impact of length, radius, thickness, and modulus of elasticity. Comparative analysis with Hybrid Finite Element Method (FEM) data and validation against existing literature highlights DTM's precision and reliability. In conclusion, DTM offers a robust solution for the eigenvalue problem in coupled differential equations, providing accurate vibration parameters. Additionally, an important relationship between the modulus of elasticity and frequency in the dimensionless state was obtained.

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Purpose: The study aims to analyze the free vibration behavior of functionally graded porous beams with non-uniform rectangular cross-sections, investigating four distinct porosity distribution across the beam's thickness. Methods: Utilizing the Euler–Bernoulli beam model and Hamilton's principle, the equation of motion is derived. Four different boundary conditions (clamped–clamped, clamped-free, clamped–pinned, and pinned–pinned) are considered, and the resulting equation is solved using the differential transform method. Verification of accuracy is ensured through comparison with solutions for natural frequencies found in existing literature. Results and Conclusion: The study provides validated natural frequency solutions for functionally graded porous beams with non-uniform rectangular cross-sections, confirming the accuracy of the proposed method through literature comparison. A comprehensive parametric study reveals significant insights into the influence of various factors on natural frequencies, including porosity volume fractions, types of porosity distribution, taper ratios, and boundary conditions. These findings deepen our understanding of free vibration analysis for functionally graded porous beams, offering valuable guidance for engineering design and structural optimization in relevant applications. [ABSTRACT FROM AUTHOR]

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.

Recently, an approximation technique was presented for solving strong nonlinear oscillators modeled by second-order differential equations. Due to the arising of an algebraic complicity, the method fails to determine suitable solution of some important nonlinear problems such as quadratic oscillator, cubical Duffing oscillator of softening springs, and pendulum equation. However, suitable solutions of these oscillators are found by rearranging only an algebraic equation related to amplitude and frequency. The determination of solutions is simpler than the original version. [ABSTRACT FROM AUTHOR]

In multi-storey buildings, unforeseen foundation rotations can change the building's behavior. Therefore, these effects should be taken into account in the analysis. In the studies conducted in the literature, the impact of the rotation of the foundation on the building behavior under static loads has been investigated. In this study, an approach is proposed to determine the dynamic behavior of buildings under the effect of foundation rotation, regardless of the type of bearing system. The multi-storey building was modeled as an equivalent flexural-shear beam in the study. In the study, the axial displacements of the columns, which are neglected in the flexural-shear beam model, are also considered, which is different from the literature. The equation of motion representing the dynamic analysis of the equivalent flexural-shear beam was solved with the help of the Differential Transform Method. The period coefficients, effective mass ratio and peak displacement coefficient for five cases were determined and plotted. A code was prepared using Matlab for the analysis with the Differential Transform Method. As a result of the study, the dynamic characteristics obtained depending on the dimensionless building behavior coefficient have been graphed. Using the given graphs, the dynamic characteristics of the buildings under the effect of foundation rotation can be determined quickly and practically. The method presented in this study can be used for Response spectrum analysis of all systems with pure shear beam, pure bending beam and bending-shear beam behavior. In addition, an approach has been proposed to consider P − Δ effects within the scope of the study. The results obtained in the study were interpreted, and the impact of rotation on dynamic characteristics was discussed. At the end of the study, for the convenience of the presented method, two examples, one for wall-frame and the other for frame systems, were solved with the proposed method, and the results were compared with the SAP2000 program. [ABSTRACT FROM AUTHOR]

AbstractIn 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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

This work proposes a semi-analytical new hybrid approach, so-called differential -transform method (D TM), to evaluate the behavior of n-space dimensional fractional-nonlinear hyperbolic-like wave equations, where time-fractional derivative is considered in Caputo format. The D TM is the hybrid method in which projected differential transform is implemented after imposing the recently introduced integral transform, i.e., so-called transform [W. Zhao and S. Maitama, J. Appl. Anal. Comput. 10, 1223 (2020)]. The efficiency and applicability of the proposed D TM had been tested by considering three different test examples of the Caputo time-fractional nonlinear hyperbolic-like wave equations in terms of absolute error norms, and the different order D TM solutions are compared with exact solution behaviors and the existing results, for the large time level τ ∈ [ 0 , 1 0 ]. In addition, the convergence analysis of D TM is studied theoretically and verified it numerically as well as graphically, which confirms that the numerical experiments via D TM for distinct fractional orders support the theoretical findings excellently, and the presented D TM results converge to their exact solution behavior, very fast. The evaluated series approximations are expressed in the compact form of Mittag-Leffler functions. [ABSTRACT FROM AUTHOR]

In this paper, we have first discussed that groundwater is very important, especially as it is getting scarce. We see rain waters on the puddles but it is not going underground. That has to do with the increased saturated soil, which gives less space for water to move deeper. We studied the nonlinear problem that arises in infiltration phenomenon with External effect (Input and output). The infiltration phenomenon with External effect (Input and output) is defined by a non-linear partial differential equation, which gives the soil moisture content under suitable conditions. The moisture content was determined and was analyzed using the Differential transform method. We investigated groundwater infiltration with an external impact (Input and output) using the Differential Transform method. The analysis is also explained, as well as its numerical and graphical depiction. For this, we utilized MATLAB for that. [ABSTRACT FROM AUTHOR]

AbstractThe heat transfer aspects are considered in the Jeffery–Hamel hybrid nanofluid flow between non-parallel plates. In a flow regime, molybdenum disulfide nanoparticles are suspended with a viscous dissipation, magnetic field, and thermal radiation effects. The flow is theoretically modeled, and the differential transformed approach is used to solve the problem. The flow field of the hybrid nanofluid is investigated for different parameters. For the Nusselt number, a sample of 75 values is gathered and compared to five inputs. Fifty-three (70%) values are taken into account for training the neural network. For testing and validating the artificial neural network (ANN) model, 11 (15%) each is used. Using Levenberg–Marquardt backpropagation, the training is carried out. It is noticed that the temperature of MoS2–MWCNTs/H2O is higher in comparison with MoS2/H2O. Further, according to ANN prediction, Pr, Ec, and M have a growing relationship with the Nusselt number; however, Re and Nr have an opposite relationship. [ABSTRACT FROM AUTHOR]

In this paper, two numerical methods has been applied to numerically solve the generalized Zakharov system (GZS). The spectral collocation method, which based on two dimensional Legendre polynomials (LCM) and the well-known differential trans-form method (DTM). Both of the proposed methods have high accuracy and have been successfully compared with Adomian decomposition method. [ABSTRACT FROM AUTHOR]

This paper processes numerically and analytically the magnetohydrodynamic flow among two porous solid plate intersections at an angle or through non-parallel porous walls, which can be interpreted as a mix of Jeffery Hamel fluid flow additives from suction-injection parameters. In fluid mechanics, the equations of governing, which are transferred to the non-linear ordinary differential equation of third order, are modeled instead of the conventional Navier-Stokes equations by Maxwell's electromagnetism. Results obtained from the DTM and the numerical method BVP4c are compared to those resulting from the optimal methods (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). A comparison is made between the approximate analytical and numerical solutions for different parameters like open angles (ε), Reynolds numbers (Re), injection parameters (S), and Hartmann numbers (Ha). The results demonstrate that the two approaches are very similar.

The study focuses on the free vibration analysis of beams made of axially functionally graded materials (AFGM) with curvilinear variable cross-sections along their length. The beams encompass various shapes, including concave and convex conic sections, with axial material properties varying according to polynomial and exponential laws. The equations of motion are derived using Hamilton’s principle within the framework of Timoshenko beam theory. These governing equations, subjected to various boundary conditions, are solved using the differential transform method (DTM). The proposed solution technique is validated by comparing computed natural frequencies with the existing literature and results obtained using three-dimensional finite element analysis in ABAQUS. The incorporation of material gradients into the beam finite element models was achieved using the user-defined material subroutine (UMAT). Additionally, a comprehensive study is conducted to examine the influence of various factors on the natural frequencies of functionally graded beams. These factors include parameters of material laws, types of variable beam shapes, slenderness ratio, and specific boundary conditions. This study provides a thorough understanding of the modal dynamics of the considered beams, offering valuable insights into the behavior of FGM structures.

The history of variational calculus dates back to the late 17th century when Johann Bernoulli presented his famous problem concerning the brachistochrone curve. Since then, variational calculus has developed intensively as many problems in physics and engineering are described by equations from this branch of mathematical analysis. This paper presents two non-classical, distinct methods for solving such problems. The first method is based on the differential transform method (DTM), which seeks an analytical solution in the form of a certain functional series. The second method, on the other hand, is based on the physics-informed neural network (PINN), where artificial intelligence in the form of a neural network is used to solve the differential equation. In addition to describing both methods, this paper also presents numerical examples along with a comparison of the obtained results.Comparingthe two methods, DTM produced marginally more accurate results than PINNs. While PINNs exhibited slightly higher errors, their performance remained commendable. The key strengths of neural networks are their adaptability and ease of implementation. Both approaches discussed in the article are effective for addressing the examined problems.

In this study, we investigate a system of coupled nonlinear Lane-Emden equations that arise from catalytic diffusion reactions. A highly effective algorithm, which heavily relies on the differential transform method, is proposed to solve this system. The algorithm produces a convergent series solution with components that can be easily computed. The Adomian polynomials corresponding to the given system are utilized for calculating the differential transforms of its nonlinearities with multiple variables. A practical numerical example is provided to validate the effectiveness and accuracy of the present scheme. The numerical results obtained by our developed approach show a significantly lower error rate compared to other existing approaches.—In this study, we investigate a system of coupled nonlinear Lane-Emden equations that arise from catalytic diffusion reactions. A highly effective algorithm, which heavily relies on the differential transform method, is proposed to solve this system. The algorithm produces a convergent series solution with components that can be easily computed. The Adomian polynomials corresponding to the given system are utilized for calculating the differential transforms of its nonlinearities with multiple variables. A practical numerical example is provided to validate the effectiveness and accuracy of the present scheme. The numerical results obtained by our developed approach show a significantly lower error rate compared to other existing approaches. [ABSTRACT FROM AUTHOR]

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.

In this study, free vibration of stepped beam which is parallel to a uniform beam with same length and elastically connected to it, is considered. Euler-Bernoulli beam theory has been applied to drive equations of motion, abrupt change in height of beam considered as step and Winkler-type elastic layer model serve as connection between beams. The differential transform method (DTM) is applied to determine dimensionless frequencies and mode shapes. In the case of two uniform parallel beams accuracy of solution is verified by comparing with results reported by other methods. It is assumed all supports have one type and fully clamped and fully hinged supports considered for boundary conditions. The effects of different parameters such as: step location and ratio, connecting layer coefficient and boundary conditions on dimensionless frequencies and mode shapes investigated and discussed. This problem handled for first time in present study and results are completely new. [ABSTRACT FROM AUTHOR]

In this study, the differential transform method (DTM) associated with the Hahn difference operator ( D q , ω) has been examined. The definitions and theorems of the one and two-dimensional q , ω -DTM are introduced. Then, the presented q , ω -DTMs are applied to obtain the analytical or approximation solution of linear or non-linear q , ω -initial value problems consisting of the q , ω -ordinary or q , ω -partial differential equation. [ABSTRACT FROM AUTHOR]

In this work, We are looking at the characteristics of micropolar flow in a porous channel that's being driven by suction or injection. The working of the fluid is described in the flow model. We can reduce the governing nonlinear partial differential equations (PDEs) to a model of coupled systems of nonlinear ordinary differential equations using similarity variables (ODEs). In order to obtain the results of a coupled system of nonlinear ODEs, we discuss a method which is known as the differential transform method (DTM). The concern transform is an excellent mathematical tool to obtain the analytical series solution to the nonlinear ODEs. To observe beast agreement between analytical method and numerical method, we compare our result with the Rung-Kutta method of order four (RK4).We also provide simulation plots to the obtained result by using Mathematica. On these plots, we discuss the effect of different parameters which arise during the calculation of the flow model equations. [ABSTRACT FROM AUTHOR]

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This study aims at analyzing the effect of variable foundation parameters on the natural frequencies of a prestressed tapered Rayleigh beam having general elastically restrained ends. In this work, the elastic coefficients of the foundations are assumed varying along the beam major axis. In particular, the constant, linear and parabolic variations of the Pasternak foundation are considered. A semi-analytical approach known as differential transform method (DTM) is applied to the non-dimensional form of the governing equations of motion of the prestressed tapered Rayleigh beam and a set of recurrence algebraic equations are determined. Performing some direct algebraic operations on these derived equations and using some computer codes developed and implemented in MAPLE 18, the dimensionless natural frequencies and the associated mode shapes of the beam are obtained, the effects of these Pasternak foundation variations for various values of the slenderness ratio on the natural frequencies are investigated. It is found among others that : (i) an increase in foundation stiffness led generally to an increase in the natural frequencies; (ii) the constant elastic variations of Pasternak foundation produced highest values of natural frequencies; and (iii) the natural frequencies of tapered Rayleigh beam resting on Pasternak foundation are higher than those from the same beam on Winkler foundation. Finally, the efficiency and accuracy of differential transform method are illustrated by solving two numerical examples of vibration problems and validating the results obtained with those in the open literature, and are found to compare favorably well.

This study proposes a novel computational method for solving the fractional SIS epidemic model involving the Caputo and Caputo-Fabrizio fractional derivatives, that called Elzaki differential transform method (EDTM) which is a coupling of two powerful methods: the Elzaki transform method and the differential transform method. To demonstrate the effectiveness and advantage of the proposed method, a numerical example is presented. The results obtained by the EDTM are compared with well-known exact solutions. This results show that this method is very effective and more accurate for solving this type of problem. Therefore, our proposed method can be employed to study the solutions of a wide range of real problems arising in engineering and natural sciences, which can be modeled by a fractional differential equations. [ABSTRACT FROM AUTHOR]

In this paper, we have used gener-alized differential transform method to solve different types of integral equations of fractional order. All the fractional integrals are written in the Riemann-Liouville sense and fractional derivatives are written in the Caputo sense. [ABSTRACT FROM AUTHOR]

The paper presents an application of the semi‐analytical differential transform method (DTM) to thermofluidics of generalized Couette flow of couple stress fluid inside a composite channel. The channel comprises a fluid‐saturated porous layer and a clear fluid region. The flow is caused by a uniformly favorable pressure gradient applied at the mouth of the channel and the shear of the moving upper wall of the channel. The matching conditions are taken such that the velocity, temperature, and respective gradients are continuous at a clear fluid–porous medium. The problem is novel because this is the first mathematical endeavor to treat fluidic systems involving composite channels. Besides the mathematical rigor, it presents an analysis of thermodynamic irreversibility and other quantities of interest, viz, skin friction and Nusselt number. The results derived from DTM are compared with those obtained by exact solution showing excellent agreement. The plots and tables are drawn and discussed. [ABSTRACT FROM AUTHOR]

The complexity of solving differential equations in real-world applications motivates researchers to extend numerical methods. Among different numerical and semi-analytical methods for solving initial and boundary value problems, the differential transform method (DTM) has received no-table attention. It has developed and experienced generalizations for implementing other types of mathematical problems such as optimal control, calculus of variations, and integral equations. This review aims to provide insight into DTM. History, theoretical base, applications, computational aspects, and its revisions are reviewed. The present study helps to understand the theory, capabilities, and features of the DTM, as well as its drawbacks and limitations.

The mathematical model proposed by Chapman and Antano (Electrochimica Acta, 56 (2010), 128–132) for the catalytic electrochemical–chemical (EC’) processes in an irreversible second-order homogeneous reaction in a microelectrode is discussed. The mass-transfer boundary layer neighbouring an electrode can contribute to the electrode’s measured AC impedance. This model can be used to analyse membrane-transport studies and other instances of ionic transport in semiconductors and other materials. Two efficient and easily accessible analytical techniques, AGM and DTM, were used to solve the steady-state non-linear diffusion equation’s infinite layers. Herein, we present the generalized approximate analytical solution for the solute, product, and reactant concentrations and current for the small experimental values of kinetic and diffusion parameters. Using the Matlab/Scilab program, we also derive the numerical solution to this problem. The comparison of the analytical and numerical/computational results reveals a satisfactory level of agreement.

The present article deals with the MHD flow of a Casson nanofluid between two disks. The lower disk was fixed as well as permeable. The upper disk was not permeable, but it could move perpendicularly up and down toward the lower disk. Titanium dioxide was selected as nanoparticles and water as a base fluid. The governing higher-order nonlinear partial differential equations were transformed into a set of nonlinear ordinary differential equations by using similarity transformation. The differential transform method (DTM) was applied to solve the nonlinear ODEs. The nature of the velocity profiles for the different values of the suction injection parameter, the squeeze number, the Casson fluid parameter, and the volume fraction parameter of the nanofluid are pictorially discussed in this paper. The coefficient of skin friction was tabulated for the novelty of the research. The comparison of the results was determined by the DTM and the numerical methods. The profile values were also compared with the literature work and found to agree. This comparative study proves the accuracy and efficiency of the method. It is concluded from this research that the flow properties behave oppositely for all parameters during suction and injection. [ABSTRACT FROM AUTHOR]

In this paper, an iterative method for obtaining the numerical solutions of fractional partial differential equations is proposed. This method is based on the combination of the differential transform method with fractional linear multistep methods. The proposed method has a very low computational cost, with the help of which fractional partial differential equations are converted into a system of ordinary differential equations. Then the resulting equations are solved with high accuracy by applying fractional linear multi-step methods such as fractional Euler method. The obtained series of solutions in the differential transform method has a slow convergence in large regions. In this study, by combining the mentioned method with linear multi-step methods, this shortcoming is solved. Numerical results show that the obtained solutions are in good agreement with the exact solution. The results confirm the proven stability and accuracy of the method. [ABSTRACT FROM AUTHOR]

This work is an experiment to solve the fractional diffusion equation in two dimensions with a novel hybrid method. The method involves an amalgamation of the well-known differential transform method and the differential quadrature method. This work is not about the superiority of one method over the other, instead this is an idea that can be worked upon for possible greatness. Numerical examples are discussed with tables and figures.

Williamson fluid with cilia motion helps to understand the non-Newtonian fluids' rheological characteristics and get better heat transmission rates. Thus, the current problem is intended to semi-numerically discuss the heat transmission in Williamson fluid flow through a ciliated channel under a Magnetic field and Porous Medium. Mathematical modeling of the intended problem complicates the PDE system in viscous dissipation. The complex system of PDEs is transformed from wave to fixed frame under the low Reynolds number and long wavelength approximation. Differential Transform Method (DTM) has been incorporated as an efficient semi-numerical technique to solve these dimensionless coupled nonlinear partial differential equations. The existence of a solution has been established with the DTM, and the outcomes of the boundary layer distribution are found in mathematical and graphical forms. The pertinent parameters are analyzed through graphs which are plotted by the software “Mathematica.” The current investigation suggests that the conduction process escalates heat transfer through the molecules of the liquid. It is further noticed that enhancing the values of Darcy's parameter magnitude of the velocity profile rises due to an increase in the number of pores. Finally, the present study will provide significant applications in bioengineering, medical sciences, and medical equipment for the clearance of viscoelastic fluid from dust and viruses.

The present investigation is intended to follow the characteristics of thermal as well as solutal buoyancy on the flow of non‐Newtonian–Casson nanofluid flow over the exponential stretching sheet embedding with the permeable medium. The interaction of thermal radiation along with the chemical reaction enriches the flow phenomena significantly. These properties are useful due to their recent applications in various industries as well as the biomedical area. The blood flow phenomenon described in the human body is characterized by the use of Casson nanofluid which is helpful for the drug delivery system. The suitable transformation rules are employed for the transformation of the proposed governing equation into ordinary and nonlinear. Furthermore, an approximate analytical approach such as Differential Transform Method is used for the solution of transformed governing equations. The illustration of the physical behavior of the contributing parameters involved in the flow phenomena is presented via graphs and tabular form. [ABSTRACT FROM AUTHOR]

• High efficiency intermittent bistable states emerge from fractionally. • Combination of different fractional terms leads to new dynamics characteristics. • Fractional Duffing system investigations using differential transform method. • The energy dissipation depends on different combinations of fractional terms. • Fractionality of the system modifies energy dissipation efficiency. Fractional derivatives appear to be a convenient and effective tool for describing complex processes, systems, and material characteristics. In this study these mathematical operations are used to modify an exemplary non-linear system (the Duffing system) by adding additional fractional components. Numerical analyses were performed to check how the orders of derivatives in fractional terms affect the energy efficiency of the modified system. A numerical differential transform method (D T M) was applied to solve fractional-order differential equations quickly and effectively. This analysis focuses on fractional terms with small intensity coefficients and low values of the fractional orders of derivatives (i.e. close to integer values). The results show that in some cases the fractional elements clearly modify the system dynamics and significantly increase the system energy efficiency. Different fractional derivatives can have markedly different qualitative effects and we show that introducing multiple fractional terms can stabilise changes to the energy efficiency with, for example, high efficiency intermittent bistable solutions. [ABSTRACT FROM AUTHOR]

In this paper, the method of differential transform with two variables is combined the newly defined conformable derivative for fractional calculus. Some useful fundamental properties of presented method are given and used to get solutions of linear and non-linear differential problems with fractional order. To control the reliability, applicability and correctness of the technique we applied the method to distinct fractional partial differential problems which have exact solutions in the literature. The obtained results show that this suggested method is useful, appropriate and practical to linear and non-linear fractional differential problems. [ABSTRACT FROM AUTHOR]

In this paper, a numerical scheme so-called modified differential transfor-mation method (MDTM) based on differential transformation method (DTM), Laplace transform and Pade approximation will be used to obtain accurate approximate solution for a class of boundary value problems (BVP's). The MDTM is employed as an alter-native technique to overcome some difficulties in the behavior of the solution and to be valid for a large region. The numerical results obtained demonstrate the applicability and validity of this technique. Numerical comparison is made with existing exact solution. [ABSTRACT FROM AUTHOR]

The study focuses on the free vibration analysis of beams composed of functionally graded porous materials and characterized by a variable cross-section along their length. A broad range of beams is examined encompassing various tapered configurations, porosity profiles, and porosity content. The equations of motion are derived using Hamilton's principle within the framework of Timoshenko beam theory. These equations are solved semi-analytically using the differential transform method, which has been adapted to incorporate various boundary conditions such as clamped–clamped, clamped–free, clamped–pinned, and pinned–pinned constraints within a general formulation of the beam eigenvalue problem. To validate the proposed solution technique, computed natural frequencies are compared with existing literature results for tapered inhomogeneous beams and uniform porous beams. Notably, new results are obtained for tapered porous beams. In this regard, a comprehensive parametric study explores the influence of various factors on the natural frequencies and mode shapes of functionally graded porous beams with variable cross-sections. These factors include the type of porosity profiles, a range of porosity parameters, cross-section taper ratios, and specific boundary conditions. The findings deepen our understanding of the modal characteristics of functionally graded porous beams, providing valuable guidance for engineering design and structural optimization in relevant applications. Additionally, they may serve as benchmarks for other researchers. [ABSTRACT FROM AUTHOR]

In this study, the multi-step fractional differential transform method (MSFDTM) is employed to obtain approximate analytical solutions of stiff systems of fractional order. The fractional derivative is described in the Caputo sense. Several numerical experiments are carried out to illustrate the feasibility of MSFDTM as an approximate analytical solution technique for stiff systems of fractional order.