Differential Transform Method Research Articles

A Study of Applied Reduced Differential Transform Method Using Volterra Integral Equations in Solving Partial Differential Equations

Nowadays, integration is one of the trending fields applied in calculus, especially in partial differential equations. Researchers are contributing to support useful utilities to solve partial differential equations in many kinds of methods. In this paper, we perform an application of Volterra Integral Equations in a reduced differential transform method (we call VIE-RDTM) to find the approximate solutions of partial differential equations. The aim is to find the approximate solutions approach to the exact solutions with more general forms. We also extend some new results for basic functions and compare the solutions using the reduced differential transform method and VIE-RDTM by depicting the approximate solutions in some partial differential equations. The results showed that the VIE-RDTM method gets the state-of-the-art general form of the solutions when the errors approach zero.

Estimating Organic Matter Content in Hyperspectral Wetland Soil Using Marine-Predators-Algorithm-Based Random Forest and Multiple Differential Transformations

To achieve a rapid and accurate estimation of the soil organic matter (SOM) content in wetland soil, we focused on surface soil samples from the Xianshan Lake wetland area in Zhejiang Province and proposed a novel method called Marine-Predators-Algorithm-Based Random Forest (MPARF) to establish a fast detection model for the SOM content. This study analyzed 85 soil samples from the study area with the aim of assessing the performance of various combinations of ten differential transformation methods and five regression algorithms in predicting the SOM content. Our research findings demonstrate that the combination of second-order differentiation (SD) and MPARF yields the best results, with the highest R2 value (0.92) and the lowest RMSE (1.32 g/kg). Furthermore, we determined that the average SOM content in the study area’s soil is 9.77 g/kg. Additionally, we confirmed that different differential transformation methods contribute to improving the correlation between spectral data and the SOM content, thereby enhancing the development of predictive models. This study provides a robust methodology and foundation for future soil organic matter monitoring in the region.

A Semi-Discretization Method Based on Finite Difference and Differential Transform Methods to Solve the Time-Fractional Telegraph Equation

The telegraph equation is a hyperbolic partial differential equation that has many applications in symmetric and asymmetric problems. In this paper, the solution of the time-fractional telegraph equation is obtained using a hybrid method. The numerical simulation is performed based on a combination of the finite difference and differential transform methods, such that at first, the equation is semi-discretized along the spatial ordinate, and then the resulting system of ordinary differential equations is solved using the fractional differential transform method. This hybrid technique is tested for some prominent linear and nonlinear examples. It is very simple and has a very small computation time; also, the obtained results demonstrate that the exact solutions are exactly symmetric with approximate solutions. The results of our scheme are compared with the two-dimensional differential transform method. The numerical results show that the proposed method is more accurate and effective than the two-dimensional fractional differential transform technique. Also, the implementation process of this method is very simple, so its computer programming is very fast.

The Fuzzy Differential Transform Method for the Solution of the System of Fuzzy Integro-Differential Equations Arising in Biological Model

This article deals with the implementation of fuzzy differential transform method for solving a system of nonlinear fuzzy integro-differential equations. This system appears in a model of biological species living together. Though the differential transform method is an iterative method, the current approach reduces this model to a set of nonlinear algebraic equations due to its delay terms. The basic definitions and theorems are first presented. The applicability and accuracy of the current methodologies have been demonstrated through the discussion of a few exemplary situations.

New optimum configuration for the FGM rotating disc in gas turbines using combined thermal-mechanical analysis: An analytical approach

The gas turbine is one of the most suitable equipment for various applications. Usually, a gas turbine comprises a set of discs to maintain turbine blades. The rotating disc in gas turbines is one of the critical components affected by various thermal and mechanical loads. The current article considers a new rotating disc with variable thickness (power-law profile). The functionally graded material has been used as the disc material, and the disc’s properties vary with temperature and location. Therefore, the mechanical and thermal behaviours of the disc are a function of temperature and location. The differential energy balance equation has been derived for analysing the new disc. In the performed analysis, the conduction and convection heat transfer coefficients vary with temperature and location; thus, the derived final heat transfer equation has high nonlinearity. One of the efficient expansion methods, the so-called Differential Transformation Method (DTM), is used for the governed equation’s analytical solution, and the numerical solution is used for validation. Then, the stress analysis is performed using analytical expressions according to the temperature profile. The impacts of geometrical and thermophysical characteristics on the temperature and stress distribution are evaluated using the analytical solution in the new disc. The comparison of displacement and stresses in the new FGM disc and old discs shows that the thermo-mechanical performance of the new proposed disc is improved significantly.

Effective Modified Fractional Reduced Differential Transform Method for Solving Multi-Term Time-Fractional Wave-Diffusion Equations

In this work, we suggest a new method for solving linear multi-term time-fractional wave-diffusion equations, which is named the modified fractional reduced differential transform method (m-FRDTM). The importance of this technique is that it suggests a solution for a multi-term time-fractional equation. Very few techniques have been proposed to solve this type of equation, as will be shown in this paper. To show the effectiveness and efficiency of this proposed method, we introduce two different applications in two-term fractional differential equations. The three-dimensional and two-dimensional plots for different values of the fractional derivative are depicted to compare our results with the exact solutions.

Parameterized Differential Transform Method and Its Application to Boundary Value Transmission Problems

In this study, we developed a new modification of the well-known differential transform method (DTM) that differs from the classical one by the algorithm for calculating the coefficients of an approximate solution given as a series. The proposed new algorithm we will call α-parameterized DTM (α-p DTM,for short). By using the proposed α-p DTM, we solve the boundary value transmission problem for the third-order differential equation. For the same problem, an approximate solution is also found using also the classical DTM. The solutions obtained were compared graphically.

A Thermal Analysis of a Convective–Radiative Porous Annular Fin Wetted in a Ternary Nanofluid Exposed to Heat Generation under the Influence of a Magnetic Field

Fins are utilized to considerably increase the surface area available for heat emission between a heat source and the surrounding fluid. In this study, radial annular fins are considered to investigate the rate of heat emission from the surface to the surroundings. The effects of a ternary nanofluid, magnetic field, permeable medium and thermal radiation are considered to formulate the nonlinear ordinary differential equation. The differential transformation method, one of the most efficient approaches, has been used to arrive at the analytical answer. Graphical analysis has been performed to show how nondimensional characteristics dominate the thermal gradient of the fin. The thickness and inner radius of a fin are crucial factors that impact the heat transmission rate. Based on the analysis, it can be concluded that a cost-effective annular rectangular fin can be achieved by maintaining a thickness of 0.1 cm and an inner radius of 0.2 cm.

Approximate Analytical Solutions for Non-linear Telegraph Equations with Source Term

This study proposed a new version of Multistep Modified Reduced Differential Transform Method (MMRDTM) which is applicable for obtaining semi-analytical approximation on Non-linear Telegraph Equations (NLTEs) with source term. Through the experimental approach, we unveiled various advantages of the proposed method, especially its ability to analytically approximate high-speed converging series. Moreover, a significant reduction in the number of calculated terms can also be observed. Before deploying the multistep technique, the substitution of non-linear term in NLTEs with the corresponding Adomian polynomial takes place. Consequently, a simpler technique for approximating NLTEs with source term was discovered. Besides that, the solutions can be estimated more precisely over a longer time. To justify our findings, solutions of NLTEs with source term are obtained using the MMRDTM. The accuracy of the proposed method for solving these equations was tested by comparing the absolute errors between the solutions obtained by the proposed method and the Modified Reduced Differential Transform Method (MRDTM) with the exact solution. The obtained solutions evidenced that the proposed MMRDTM can deliver highly precise approximations for the NLTEs with source term.

Approximation of the Time-Fractional Klein-Gordon Equation using the Integral and Projected Differential Transform Methods

In the present investigation, a new integral transform method (NITM) and the projected differential transform method (PDTM) are used to give an analytical solution to the time-fractional Klein-Gordon (TFKG) equation. The time-fractional derivative is used in the Caputo sense. The huge advantage of the suggested approach is the ease with which the nonlinear term can be effortlessly treated by projected differential transform without using Adomian's and He's polynomials. The solution of fractional partial differential equations using the aforementioned method is very simple and straightforward. The efficiency and accuracy of the proposed method are demonstrated by three examples, and the effects of various fractional Brownian motions are demonstrated graphically.

Analysis of Dynamic Characteristics of Attached High Rise Risers

The exhaust chimney of third-generation nuclear power units is a typical attached high-rise riser structure. In this paper, the simplified mechanical model and dynamic model of China’s third-generation nuclear power Hualong-1 VNA system, including multiple nonlinear factors, are established for the first time. The DTM (differential transformation method) was first applied to solve the natural vibration characteristics of a multi-point constrained variable cross-section riser structure, and the effects of variable cross-section, variable mass, variable axial force, and different elastic constraint parameters on the natural vibration characteristics of the system were studied. The dynamic behavior of the VNA system under the combined action of internal flow velocity, vortex excitation, and foundation excitation was studied. The results show that the outer diameter function of the VNA system pipeline should be designed as a quadratic function or a near quadratic multi-segment constant value function. The “limiting” effect of constraining large stiffness can force low-order vibration modes with high constraint stiffness to jump to high-order vibration modes with low constraint stiffness. The elastic constraint arrangement scheme with near center symmetry can make the system vibration mode present a half stable and half-curved form. A new optimization design scheme has been proposed regarding the layout and stiffness parameters of the VNA system guide bracket. This can enable the VNA system pipeline to avoid severe oscillations near the response extreme values caused by multiple frequency excitations of seismic loads under design and accident conditions and ensure the service life of the equipment.

Influence of induced magnetic field and surface roughness of Casson nanofluid flow over an exponentially stretching sheet

In the present analysis, we have discussed the mixed convective of flow over an exponential sheet in presence of induced magnetic field and surface roughness. The effects of hall current and chemical reaction on flow are also considered. The non-Newtonian model behavior is characterized by Casson nanofluid. The similarity transformations are employed for the transformation of partial differential equations into non-dimensional form and obtained equations are solved by employing the Differential transformation method (DTM). Further, the importance of various parameters on various profiles and gradients are explored graphically with some suitable discussion. At the wall, the small (α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha$$\\end{document}) and frequency (n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n$$\\end{document}) parameters influence on gradients are illustrated.

On -differential transform method

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

Free Vibrational Analysis of Non-uniform Double Eulerbernoulli Beams on a Winkler Foundation Using Laplace Differential Transform Method

In this study, the free vibrational analysis of non-uniform Euler-Bernoulli double beams on a Winkler foundation under simply supported and fixed-fixed boundary conditions is examined. The governing equation is solved using the Laplace Differential Transform Method, the combined form of the Laplace transform and differential transform technique (DTM). The accuracy of the method used is demonstrated by comparing the natural frequencies obtained using LDTM with previously published results available in the literature. It is discovered that for non-uniform double Euler-Bernoulli beams on a Winkler foundation with fixed-fixed end conditions, the natural frequencies are higher than those of simply supported end conditions. It is also observed that as the non-uniformity of the cross section of the beam increases, the natural frequencies reduce. Hence, it is suggested that the non-uniformity of the cross-section of the beam for a simply supported end condition be between 0 and less than 0.8. While the fixed-fixed end condition should have a value between 0 and 0.95. Journal of Engineering Science 14(1), 2023, 111-121

On the free convection of magneto-micropolar fluid in association with thermal radiation and chemical reaction and optimized heat transfer rate using response surface methodology

An analysis is carried out for the free convection of magneto-micropolar liquid via a stretching surface for the inclusion of thermal radiation and chemical reaction. The transverse magnetic field is employed on the normal direction of flow with the impact of Peclet number relating to thermal and solutal transfer profiles. Referring to the current applications in several engineering problems, industrial applications, and more importantly the peristaltic pumping processes, blood flow phenomena, etc. the role of micropolar fluid is significant. Therefore, the objective of thismodel is to develop by incorporating thermal radiation which has several applications in aforesaid areas. However, the proposed model is solved analytically using the differential transform method (DTM) and prior to that transformation to ordinary system is obtained by using similarity transformations. The characteristic of various physical components associated with the governing equations is deployed graphically. The analysis of these parameters is described briefly in the discussion section. Further, a statistical approach response surface methodology (RSM) is used to optimize the heat transfer rate for the factors such as magnetic parameter, thermal radiation, and Peclet number.

Comparison of Adomian Decomposition Method with Differential Transformation Method for Unsteady MHD Flow and Heat Transfer Over a Stretching/Shrinking Permeable Sheet with Ohmic Heating

In this paper, two semi-analytical techniques were implemented to solve a two-dimensional unsteady MHD fluid flow and heat transfer through a stretching/shrinking permeable sheet with ohmic heating and viscous dissipation. The governing flow equations in PDE form were reduced to ordinary differential equations using appropriate similarity transformation. We obtained approximate expressions for the velocity and temperature profiles. Comparative results obtained employing Adomian decomposition method and differential transformation method were benchmarked against a numerical solution using Keller box scheme. Our findings revealed that the approximate analytical solution obtained using DTM is more dependable with fast convergence, highly accurate with minimal calculations and computationally convenient. However, the requirement of Adomian polynomials to tackle the nonlinear terms in ADM makes its execution sometimes cumbersome and difficult.

Solution of Lane-Emden Type Differential Equations by Using Differential Transform Method

The temperature in a self-gravitating star has been widely described in astrophysics by the Lane-Emden differential equation. To solve the non-linear equations, the Lane-Emden differential equation is used as a prototype for testing new mathematical and numerical techniques. The problems of singular initial value of Lane-Emden type was solved by the Differential Transform Method in this study. This method is applicable to solve various linear and nonlinear problems which decrease size of computational work. Here, we introduced some numerical problems to describe the high accuracy and efficiency of our proposed method.

Generalized Differential Transform Method – Application to Integral Equations of Fractional Order

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.

Thermal performance of stretching/shrinking fully wet porous cylindrical and conical pin fin structures by differential transformation approach

In this paper, the thermal behavior of cylindrical and conical pin fin structures operating in a convective–radiative environment has been examined. The fully wet porous fin structures are assumed to be mounted with a stretching/shrinking mechanism similar to a conveyer belt. The conical pin fin has been comparatively analyzed with the cylindrical one for three distinct cases of stretching, stagnant, and shrinking. Further, the Darcy model has been employed to analyze the fluid–solid interactions. The differential transformation method (DTM) has been used to solve the resulting second-order nonlinear ordinary differential equations. On the thermal performance of conical and cylindrical pin fin structures, the effects of the stretching/shrinking parameter, Peclet number, wet porous parameter, convective parameter, and radiative parameter have been established graphically. The conical pin fin is found to aid the heat transmission process better than the cylindrical one. The work is also helpful in the realm of microelectronics, particularly in the development of micro-pin-fin structures.

Effect of temperature-dependent internal heat generation over exponential and dovetail convective-radiative porous fin wetted in hybrid nanofluid

The main objective of this research is to conduct a comparative study of the thermal variations in a dovetail and exponential fin, considering fully wet conditions, ternary nanofluid, and internal heat generation in the form of both linear and non-linear functions. Additionally, the influence of a magnetic field as a driving force for heat transfer systems is taken into account. The mathematical modelling involves the use of dimensionless transformations to convert the energy equation into an ordinary differential equation, which is then analytically solved using the Differential transformation method (DTM). The study focuses on exploring the effects of significant thermal parameters such as radiation conduction, wet factor, heat generation, and ambient temperature on the temperature profile. Graphical analysis is employed to investigate the thermal performance. The main findings indicate that ternary nanoliquid exhibits a higher thermal response compared to nano and hybrid nanoliquid. Moreover, the nonlinear forms of internal heat generation and ternary nanofluids demonstrate a higher rate of heat transfer.