Multi-step Differential Transformation Method Research Articles

The APPROXIMATE SOLUTIONS OF MALARIA DISEASE TRANSMISSION MODEL: USING MUTI-STEP DIFFERENTIAL TRANSFORMATION METHOD

In this paper, numerical solutions to the SPEIR-SEI Malaria disease model were obtained using the Multi-Step Differential Transformation Method (MS-DTM). MS-DTM is a semi-analytical method for solving a system of a non-linear differential equation where its exact solution is difficult to obtain. The analytical solution of MS-DTM was compared with the solution of Maple21’s in-built classical fourth-order Runge-Kutta method. The results demonstrate the reliability and efficiency of the method and the graphs show that the solutions from both methods agreed well with each other.

Dynamical behaviour of a tumour-immune model focusing on the dosage of targeted chemotherapeutic drug

In this paper, we have considered a tumour-immune model with the dosage effect of the targeted chemotherapeutic drug and its efficacy. The chemotherapeutic drug efficacy is characterized via way of means of its impact on the abnormal size of beginning tumours. In pharmacology, the effectiveness of a drug is a very important factor in developing a dosage for any disease. With these points of view regarding drug efficacy, we present a four compartmental mathematical model, viz, the population of tumour cells, , CD8 T-killer cells, , CD4 T-helper cells, , and the amount of targeted chemotherapeutic drug, , in the bloodstream, at any time t. A qualitative analysis of the proposed model, including the existence and boundedness of the model has been described here. The dynamics of the system are presented by examining the admissibility and stability of the system at trivial, dead, tumour-free, and co-existing equilibrium points with their respective biological significance. The conditions for the local stability of all equilibrium points are derived by using the Jacobian matrix and Routh–Hurwitz's criterion. Numerical calculations have been performed using the Multi-step Differential Transformation Method (Ms-DTM), and are presented and discussed briefly in terms of graphs. We have expressed that the model is suitable for the tumour removal with the passage of time, regardless of the initial population or tumour size. Also, the amount the dosage of the drug with an assigned numerical value for its efficacy is one of the important factors in the treatment. The efficacy term is included to see the effect of drug dosages in CD8 T-killer cells and CD4 T-helper cells compartment.

Effect of Thermal Radiation and Double-Diffusion Convective Peristaltic Flow of a Magneto-Jeffrey Nanofluid through a Flexible Channel

The noteworthiness of double-diffusive convection with magneto-Jeffrey nanofluid on a peristaltic motion under the effect of MHD and porous medium through a flexible channel with the permeable wall has been theoretically examined. A non-linearized Rosseland approximation is utilized to show the thermal radiation effect. The governing equations are converted to standard non-linear partial differential equations by using suitable non-dimensional parameters. Solutions of emerging equations are obtained by using the multi-step differential transformation method (Ms-DTM). The differential transformation method (DTM) can be applied directly to nonlinear differential equations without requiring linearization and discretization; therefore, it is not affected by errors associated with discretization. The role of influential factors on concentration, temperature, volume fraction, and velocity are determined using graphs. A significant outcome of the present article is that the presence of double-diffusive convection can change the nature of convection in the system. The present results have a wide biological applicability, including for biomicrofluidic devices that regulate the fluid flow through a flexible endoscope and other medical pumping systems.

Peristaltic Transport of Carreau Nanofluid in Presence of Triple Diffusion in an Asymmetric Channel by Multi-Step Differential Transformation Method

The present work investigates the influence of triple diffusion on Carreau nanoliquid in peristaltic flow through an asymmetric channel. By using appropriate non-dimensional parameters, governing equations are transformed to conventional non-linear partial differential equations. The Ms-DTM is used to find solutions to developing equations. Because of the buoyancy force that prevails inside the boundary layer, velocity is impacted by the buoyancy ratio. The current investigation found that as the varied values of the modified Dufour parameter were increased, the temperature profile increased. The thermal conductivity increases as thermal diffusivity increases. It has also been discovered that the existence of triple-diffusing components with low diffusivity might alter the type of convection in the system. Graphs depict the influence of several parameters on velocity, salt1 and salt2 concentrations, solute concentration, and temperature.

Differential order analysis and sensitivity analysis of a CoVID-19 infection system with memory effect

<abstract><p>The paper deals with numerical analysis of solutions for state variables of a CoVID-19 model in integer and fractional order. The solution analysis for the fractional order model is done by the new generalized Caputo-type fractional derivative and Predictor-Corrector methodology, and that for the integer order model is carried out by Multi-step Differential Transformation Method. We have performed sensitivity analysis of the basic reproduction number with the help of a normalized forward sensitivity index. The Arzelá-Ascoli theorem and Fixed point theorems with other important properties are used to establish a mathematical analysis of the existence and uniqueness criteria for the solution of the fractional order. The obtained outcomes are depicted with the help of diagrams, narrating the nature of the state variables. According to the results, the Predictor-Corrector methodology is favorably unequivocal for the fractional model and very simple in administration for the system of equations that are non-linear. The research done in this manuscript can assure the execution and relevance of the new generalized Caputo-type fractional operator for mathematical physics.</p></abstract>

The kinematics behaviour of coupled pendulum using differential transformation method

The topic of coupled oscillations is rich in physical content which is both interesting and complex. In this research work we aimed to study the kinematics behaviour of an important physical system known as coupled pendulum, in which two identical pendulums were coupled together by a spring. The Lagrangian of the system was first constructed, then the equations of motion have been derived. Analytical solutions were obtained for these equations, and in addition we apply a reliable algorithm based on an adaptation of the standard differential transformation method is presented, which is the multi-step differential transformation method to obtain numerical solutions for the considered system for some selected initial conditions. The solutions of the equations of motion were obtained by the multi-step differential transform method. Of course, this method is needed if it predicts the motion of the considered system. For this reason figurative comparisons between the multi-step differential transformation method, the standard differential transformation method and the classical fourth-order Runge–Kutta method are given. The results reveal that the proposed technique is a promising tool to predict the behaviours of the considered system in long time interval. Moreover, the motion of curves represents the initial conditions that sensitive depend and how much oscillatory motion. In conclusion, the proposed technique is a reliable method to solve the considered double pendulum problem in long time interval. We believe that the system considered in this work along with the method used is interesting for physicists, mathematicians and engineers.

Applications of the Differential Transformation Method and Multi-Step Differential Transformation Method to Solve a Rotavirus Epidemic Model

Epidemic models are essential in understanding the transmission dynamics of diseases. These models are often formulated using differential equations. A variety of methods, which includes approximate, exact and purely numerical, are often used to find the solutions of the differential equations. However, most of these methods are computationally intensive or require symbolic computations. This article presents the Differential Transformation Method (DTM) and Multi-Step Differential Transformation Method (MSDTM) to find the approximate series solutions of an SVIR rotavirus epidemic model. The SVIR model is formulated using the nonlinear first-order ordinary differential equations, where S; V; I and R are the susceptible, vaccinated, infected and recovered compartments. We begin by discussing the theoretical background and the mathematical operations of the DTM and MSDTM. Next, the DTM and MSDTM are applied to compute the solutions of the SVIR rotavirus epidemic model. Lastly, to investigate the efficiency and reliability of both methods, solutions obtained from the DTM and MSDTM are compared with the solutions from the Runge-Kutta Order 4 (RK4) method. The solutions from the DTM and MSDTM are in good agreement with the solutions from the RK4 method. However, the comparison results show that the MSDTM is more efficient and converges to the RK4 method than the DTM. The advantage of the DTM and MSDTM over other methods is that it does not require a perturbation parameter to work and does not generate secular terms. Therefore the application of both methods

A numerical approach of tumor‐immune model with B cells and monoclonal antibody drug by multi‐step differential transformation method

Monoclonal antibody (mAb) drugs are used to kill malignant tumors by making them able to see into the immune system. In this article, we developed a mathematical model with the help of tumor‐immune system with antibodies effect and efficacy of mAb drug. We defined the existence of solutions of the model, and by the Lyapunov method, we showed that the solutions are bounded; thereafter, we illustrated the local and global stability phenomenon, which is described by the Lyapunov method and localization of compact invariant sets (LCIS) method for disease‐free equilibrium points including the biological significance. In actual fact, this model suggested some ideas about the malignant tumors reciprocate to mAb drug dosage and its efficacy, eventually to overcome malignancy. It is likely to control the steadiness of the tumor, which is very important for any kind of treatment. The numerical calculations for the proposed model have been carried out by multi‐step differential transformation method (Ms‐DTM). The simulation of the model is depicted through Figures 1–5, which shows the effect of interaction rate of tumor cells and antibodies and dose of the drug for various values of efficacy.

Magnetohydrodynamics natural convection of nanofluid flow over a vertical circular cone embedded in a porous medium and subjected to thermal radiation

In this paper, magnetohydrodynamics natural convection of nanofluid flow over a vertical circular cone immersed in a porous medium under the influence of thermal radiation is investigated using multi-step differential transformation method. The accuracies of the analytical solutions are established through the verifications of the results of the present study with the results of the numerical solutions and the past studies. The approximate analytical solutions are used to examine the impacts of cone angle, flow medium porosity, magnetic field, nanoparticles volume-fraction and shape on the flow and heat transfer behaviours of the Copper (II) Oxide-water nanofluid. It is hoped that this study will enhance better understanding of flow process for the design of flow and heat transfer equipment.

Application of Multi-step Differential Transformation Method (Ms-DTM) for Solving Nonlinear Problem of Heat Transfer in Solar Air Collector

Application of Multi-step Differential Transformation Method (Ms-DTM) for Solving Nonlinear Problem of Heat Transfer in Solar Air Collector

Application of differential transformation finite element method in aperiodic vibration of non-prismatic beam

The paper presents the combination of multi-step differential transformation method (MsDTM) and finite element method (FEM) in vibration analysis of non-prismatic beam. A new algorithm, named differential transformation finite element method (DTFEM), combines the advantages of both methods. It enables to solve a wide class problems and obtain solutions in the form of piecewise functions that are very useful in further analyses, especially in time-consuming computations of dynamic systems. In contrary to standard DTM, with the proposed method, we can successfully calculate vibrations of non-prismatic structures governed by partial differential equations with variable coefficients. The application of FEM reduces partial differential equation to system of ordinary differential equations depending on time. After applying multi-step DTM the system is converting into a set of recursive algebraic equations which can be solved in a symbolic way. The final approximate solution is composed of a sequence of single solutions obtained in every time interval.The algorithm of DTFEM was described and illustrated with an example of non-prismatic simply supported beam subjected to uniformly distributed load increasing nonlinearly in time. The results in the form of beam displacements, velocities and accelerations were compared with numerical approach based on Newmark method. Two cases of damping model were taken into account: structural damping model and external damping model represented by two viscous dampers. A very good agreement was found in both cases. Presented analyses show that DTFEM is more accurate, with the same time step, than Newmark method. Therefore, in some cases, it may be possible to obtain sufficiently accurate results faster. The main advantage of the proposed method is the solution’s form which is a function of time and not a discrete set of results.

The Combination of Multi-Step Differential Transformation Method and Finite Element Method in Vibration Analysis of Non-Prismatic Beam

The paper presents a new algorithm being the combination of multi-step differential transformation method (MsDTM) and finite element method (FEM) as a powerful tool for solving variety dynamic problems. The proposed algorithm, named as differential transformation finite element method (DTFEM), transforms partial differential equation into a set of recursive algebraic equations. The final form of a solution is a piecewise function which in general case may be a symbolic function. High effectiveness and accuracy of DTFEM is demonstrated on the example of forced vibrations of non-prismatic Euler–Bernoulli beam. Computed time histories of displacements, velocities and accelerations are highly consistent with results obtained by Newmark method.

Implementation of multi-step differentialtransformation method for hyperchaotic Rossler system

In this work, the multi-step differential transformation method (MSDTM) is applied to approximate a solution of the hyperchaotic Rossler system. MSDTM is adapted from the differential transformation method (DTM). In this method, DTM is implemented in each subinterval. Results are compared with a fourth-order Runge Kutta method and a standard DTM. The results show that the MSDTM is an efficient and powerful technique for solving hyperchaotic Rossler systems and this method is more accurate than DTM.

Investigation of third-grade non-Newtonian blood flow in arteries under periodic body acceleration using multi-step differential transformation method

In this paper, a non-Newtonian third-grade blood in coronary and femoral arteries is simulated analytically and numerically. The blood is considered as the thirdgrade non-Newtonian fluid under the periodic body acceleration motion and the pulsatile pressure gradient. The hybrid multi-step differential transformation method (Hybrid-MsDTM) and the Crank-Nicholson method (CNM) are used to solve the partial differential equation (PDE), and a good agreement between them is observed in the results. The effects of the some physical parameters such as the amplitude, the lead angle, and the body acceleration frequency on the velocity and shear stress profiles are considered. The results show that increasing the amplitude, Ag, and reducing the lead angle of body acceleration, φ, make higher velocity profiles on the center line of both arteries. Also, the maximum wall shear stress increases when Ag increases.

Motion of a spherical particle on a rotating parabola using Lagrangian and high accuracy Multi-step Differential Transformation Method

In this letter, the equation of a particle's motion on a rotating parabolic surface is introduced through Lagrange equations and is solved by Multi-step Differential Transformation Method (Ms-DTM). As a main outcome, it is shown that this method gives approximations of a high degree of accuracy and least computational effort for studying particle motion on rotating parabolic surfaces compared to previous analytical methods. Also, position trajectory of the particle, r(t), and its phase planes are depicted in the current study for different constant numbers.

High accuracy analysis for motion of a spherical particle in plane Couette fluid flow by Multi-step Differential Transformation Method

In this study, coupled equations of particle's motion in Couette fluid flow are solved by Multi-step Differential Transformation Method (Ms-DTM) considering the rotation and shear effects on lift force and neglecting gravity. The precious achievement of the present work is introducing a new, fast and efficient analytical technique for spherical particles in plane Couette fluid flow over the previous numerical and analytical results in the literature. Also, obtained values are compared with numerical solution and HPM–Pade which recently are done by Torabi et al. It is shown that presented method gives approximations with a high degree of accuracy and least computational effort for studying particle motion in Couette fluid flow without the need for any linearization, discretization or perturbation against the previous work. Also, the acceleration profiles of the particle are presented and positions of the particle in each 1s time step are depicted graphically in the current study.

Adaptive multi-step differential transformation method to solve ODE systems

In this paper, it is given a fast algorithm to solve chaotic differential systems using the multi-step differential transforms method (MsDTM). The approach is applied to a number of chaotic nonlinear differential equations and numerical results are given. Performance analyses reveal that the proposed approach is an efficiency tool to solve using fewer time step to the considered equation systems.

Thermal Analysis of Non-linear Convective–Radiative Hyperbolic Lumped Systems with Simultaneous Variation of Temperature-Dependent Specific Heat and Surface Emissivity by MsDTM and BPES

With the advent of temperatures near absolute zero, it is often claimed that at very low temperatures the effect of thermal wave propagation must be included by the hyperbolic heat conduction equation (HHCE). In this paper the non-linear convective–radiative HHCE is investigated. Opposite to common numerical analyses, analytical expressions are obtained for the temperature variations by the multi-step differential transformation method. Some conclusions about alteration of the specific heat of the material, temperature steeping, and Vernotte number have been formulated.

Numerical solution of the fractional-order Vallis systems using multi-step differential transformation method

In this paper, we examine a fractional order Vallis systems. We also fulfil a detailed analysis on the stability of equilibrium. Multi-step differential transform method (MsDTM) extends to give approximate and analytical solutions of a fractional order Vallis systems. Numerical simulations are submitted to verify the validity and reliability results obtained from these methods.

Chaotic systems via multi-step differential transformation method

In this article, the multi-step differential transform method is implemented to give approximate and analytical solutions of nonlinear fractional order ordinary differential equation systems, such as Chen, Lorenz, Arneodo–Coullet, Genesio, Lui, and Rikitake chaotic systems. The numerical solutions obtained from the proposed method indicate that the approach is easy, simple, and accurate when applied to systems of fractional differential equations. Numerical simulations are given to verify the reliability and effectiveness of this method.