On Time Fractional Partial Differential Equations and Their Solution by Certain Formable Transform Decomposition Method

This paper aims to investigate a new efficient method for solving time fractional partial differential equations.In this orientation,a reliable formable transform decomposition method has been designed and developed,which is a novel combination of the formable integral transform and the decomposition method.Basically,certain accurate solutions for time-fractional partial differential equations have been presented.Themethod under concern demandsmore simple calculations and fewer efforts compared to the existingmethods.Besides,the posed formable transformdecompositionmethod has been utilized to yield a series solution for given fractional partial differential equations.Moreover,several interesting formulas relevant to the formable integral transform are applied to fractional operators which are performed as an excellent application to the existing theory.Furthermore,the formable transform decomposition method has been employed for finding a series solution to a time-fractional Klein-Gordon equation.Over and above,some numerical simulations are also provided to ensure reliability and accuracy of the new approach.

On the Approximation of Fractal-Fractional Differential Equations Using Numerical Inverse Laplace Transform Methods

Laplace transform is one of the powerful tools for solving differential equations in engineering and other science subjects.Using the Laplace transform for solving differential equations,however,sometimes leads to solutions in the Laplace domain that are not readily invertible to the real domain by analyticalmeans.Thus,we need numerical inversionmethods to convert the obtained solution fromLaplace domain to a real domain.In this paper,we propose a numerical scheme based on Laplace transform and numerical inverse Laplace transform for the approximate solution of fractal-fractional differential equations with orderα,β.Our proposed numerical scheme is based on three main steps.First,we convert the given fractal-fractional differential equation to fractional-differential equation in Riemann-Liouville sense,and then into Caputo sense.Secondly,we transformthe fractional differential equation in Caputo sense to an equivalent equation in Laplace space.Then the solution of the transformed equation is obtained in Laplace domain.Finally,the solution is converted into the real domain using numerical inversion of Laplace transform.Three inversion methods are evaluated in this paper,and their convergence is also discussed.Three test problems are used to validate the inversion methods.We demonstrate our results with the help of tables and figures.The obtained results show that Euler’s and Talbot’s methods performed better than Stehfest’s method.

Solving shock wave with discontinuity by enhanced differential transform method(EDTM)

An enhanced differential transform method （EDTM）, which introduces the Pad@ technique into the standard differential transform method （DTM）, is proposed. The enhanced method is applied to the analytic treatment of the shock wave. It accelerates the convergence of the series solution and provides an exact Dower series solution.

Modified Differential Transform Method (DTM) Simulation of Hydromagnetic Multi-Physical Flow Phenomena from a Rotating Disk

A similarity solution for the steady hydromagnetic convective heat and mass transfer with slip flow from a spinning disk with viscous dissipation and Ohmic heating yields a system of non-linear, coupled, ordinary differential equations. These equations are analytically solved by applying a newly developed method namely the DTM-Padé technique which is a combination of the Differential Transform Method (DTM) and the Padé approximation. A full analytical solution is presented, as a benchmark for alternative numerical solutions. DTM-Padé is implemented without requiring linearization, discretization, or perturbation, and holds significant potential for solving strongly nonlinear differential equations which arise frequently in fluid dynamics. The regime studied is shown to be controlled by the slip parameter (γ), magnetohydrodynamic body force parameter (M), Eckert (viscous heating) number (Ec), Schmidt number (Sc), Soret number (Sr), Dufour number (Du) and Prandtl number (Pr). The influence of selected parameters on the evolution of dimensionless velocity, temperature and concentration distributions is studied graphically. Increasing magnetic field (M) is found to significantly inhibit the radial (f) and tangential (g) velocities, but to accentuate the axial velocity field (h);furthermore temperature (θ) and concentration (φ) are both enhanced with increasing M. Increasing Soret number (Sr) acts to boost the dimensionless concentration (φ). Temperatures are significantly elevated in the boundary layer regime with a rise in Eckert number (Ec). Excellent correlation between the DTM-Padé technique and numerical (shooting) solutions is achieved. The model has important applications in industrial energy systems, process mechanical engineering, electromagnetic materials processing and electro-conductive chemical transport processes.

Differential transform method for solving Richards' equation

An approximate solution to Richards＇ equation is presented, mathematically describing a sort of unsaturated single phase fluid flow in porous media. The approach is a differential transform method （DTM） with intermediate variables. Two examples are given to demonstrate the accuracy of the presented solution.

Differential transformation method for studying flow and heat transfer due to stretching sheet embedded in porous medium with variable thickness, variable thermal conductivity,and thermal radiation

This article presents a numerical solution for the flow of a Newtonian fluid over an impermeable stretching sheet embedded in a porous medium with the power law surface velocity and variable thickness in the presence of thermal radiation. The flow is caused by non-linear stretching of a sheet. Thermal conductivity of the fluid is assumed to vary linearly with temperature. The governing partial differential equations （PDEs） are transformed into a system of coupled non-linear ordinary differential equations （ODEs） with appropriate boundary conditions for various physical parameters. The remaining system of ODEs is solved numerically using a differential transformation method （DTM）. The effects of the porous parameter, the wall thickness parameter, the radiation parameter, the thermal conductivity parameter, and the Prandtl number on the flow and temperature profiles are presented. Moreover, the local skin-friction and the Nusselt numbers are presented. Comparison of the obtained numerical results is made with previously published results in some special cases, with good agreement. The results obtained in this paper confirm the idea that DTM is a powerful mathematical tool and can be applied to a large class of linear and non-linear problems in different fields of science and engineering.

Application of Multi-Step Differential Transform Method on Flow of a Second-Grade Fluid over a Stretching or Shrinking Sheet

In this study, a reliable algorithm to develop approximate solutions for the problem of fluid flow over a stretching or shrinking sheet is proposed. It is depicted that the differential transform method (DTM) solutions are only valid for small values of the independent variable. The DTM solutions diverge for some differential equations that extremely have nonlinear behaviors or have boundary-conditions at infinity. For this reason the governing boundary-layer equations are solved by the Multi-step Differential Transform Method (MDTM). The main advantage of this method is that it can be applied directly to nonlinear differential equations without requiring linearization, discretization, or perturbation. It is a semi analytical-numerical technique that formulizes Taylor series in a very different manner. By applying the MDTM the interval of convergence for the series solution is increased. The MDTM is treated as an algorithm in a sequence of intervals for finding accurate approximate solutions for systems of differential equations. It is predicted that the MDTM can be applied to a wide range of engineering applications.

Reduced Differential Transform Method for Solving Nonlinear Biomathematics Models

In this paper,we study the approximate solutions for some of nonlinear Biomathematics models via the e-epidemic SI1I2R model characterizing the spread of viruses in a computer network and SIR childhood disease model.The reduced differential transforms method(RDTM)is one of the interesting methods for finding the approximate solutions for nonlinear problems.We apply the RDTM to discuss the analytic approximate solutions to the SI1I2R model for the spread of virus HCV-subtype and SIR childhood disease model.We discuss the numerical results at some special values of parameters in the approximate solutions.We use the computer software package such as Mathematical to find more iteration when calculating the approximate solutions.Graphical results and discussed quantitatively are presented to illustrate behavior of the obtained approximate solutions.

New approximate solution for time-fractional coupled KdV equations by generalised differential transform method

In this paper, the genera]ised two-dimensiona] differentia] transform method （DTM） of solving the time-fractiona] coupled KdV equations is proposed. The fractional derivative is described in the Caputo sense. The presented method is a numerical method based on the generalised Taylor series expansion which constructs an analytical solution in the form of a polynomial. An illustrative example shows that the genera]ised two-dimensional DTM is effective for the coupled equations.

Mixture of a New Integral Transform and Homotopy Perturbation Method for Solving Nonlinear Partial Differential Equations

In this paper, we present a new method, a mixture of homotopy perturbation method and a new integral transform to solve some nonlinear partial differential equations. The proposed method introduces also He’s polynomials [1]. The analytical results of examples are calculated in terms of convergent series with easily computed components [2].

A Comparison between the Reduced Differential Transform Method and Perturbation-Iteration Algorithm for Solving Two-Dimensional Unsteady Incompressible Navier-Stokes Equations

In this work, approximate analytical solutions to the lid-driven square cavity flow problem, which satisfied two-dimensional unsteady incompressible Navier-Stokes equations, are presented using the kinetically reduced local Navier-Stokes equations. Reduced differential transform method and perturbation-iteration algorithm are applied to solve this problem. The convergence analysis was discussed for both methods. The numerical results of both methods are given at some Reynolds numbers and low Mach numbers, and compared with results of earlier studies in the review of the literatures. These two methods are easy and fast to implement, and the results are close to each other and other numerical results, so it can be said that these methods are useful in finding approximate analytical solutions to the unsteady incompressible flow problems at low Mach numbers.

Reduced Differential Transform Method for Solving Linear and Nonlinear Goursat Problem

In this paper a new method for solving Goursat problem is introduced using Reduced Differential Transform Method (RDTM). The approximate analytical solution of the problem is calculated in the form of series with easily computable components. The comparison of the methodology presented in this paper with some other well known techniques demonstrates the effectiveness and power of the newly proposed methodology.

Double Elzaki Transform Decomposition Method for Solving Non-Linear Partial Differential Equations

In this paper, we discuss a new method employed to tackle non-linear partial differential equations, namely Double Elzaki Transform Decomposition Method (DETDM). This method is a combination of the Double ELzaki Transform and Adomian Decomposition Method. This technique is hereafter provided and supported with necessary illustrations, together with some attached examples. The results reveal that the new method is very efficient, simple and can be applied to other non-linear problems.

Reconstructed Elzaki Transform Method for Delay Differential Equations with Mamadu-Njoseh Polynomials

One of the solution techniques used for ordinary differential equations, partial and integral equations is the Elzaki Transform. This paper is an extension of Mamadu and Njoseh [1] numerical procedure (Elzaki transform method (ETM)) for computing delay differential equations (DDEs). Here, a reconstructed Elzaki transform method (RETM) is proposed for the solution of DDEs where Mamadu-Njoseh polynomials are applied as basis functions in the approximation of the analytic solution. Using this strategy, a numerical illustration as in Ref.[1] is provided to the RETM as a basis for comparison to guarantee accuracy and consistency of the method. All numerical computations were performed with MAPLE 18 software.

Solving <i>nth</i>-Order Integro-Differential Equations Using the Combined Laplace Transform-Adomian Decomposition Method

In this paper, the Combined Laplace Transform-Adomian Decomposition Method is used to solve nth-order integro-differential equations. The results show that the method is very simple and effective.

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 third- grade non-Newtonian fluid under the periodic body acceleration motion and the pulsatile pressure gradient. The hybrid multi-step differential transformation method （Hybrid-Ms- DTM） 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, 9, make higher velocity profiles on the center line of both arteries. Also, the maximum wall shear stress increases when Ag increases.

The Adomian Decomposition Method and the Differential Transform Method for Numerical Solution of Multi-Pantograph Delay Differential Equations

In this paper, the Adomian Decomposition Method (ADM) and the Differential Transform Method (DTM) are applied to solve the multi-pantograph delay equations. The sufficient conditions are given to assure the convergence of these methods. Several examples are presented to demonstrate the efficiency and reliability of the ADM and the DTM;numerical results are discussed, compared with exact solution. The results of the ADM and the DTM show its better performance than others. These methods give the desired accurate results only in a few terms and in a series form of the solution. The approach is simple and effective. These methods are used to solve many linear and nonlinear problems and reduce the size of computational work.

Differential Transform Method for Some Delay Differential Equations

This paper concentrates on the differential transform method (DTM) to solve some delay differential equations (DDEs). Based on the method of steps for DDEs and using the computer algebra system Mathematica, we successfully apply DTM to find the analytic solution to some DDEs, including a neural delay differential equation. The results confirm the feasibility and efficiency of DTM.

Exact Solution for a Class of Stiff Systems by Differential Transform Method

In this paper, Differential Transform Method (DTM) is proposed for the closed form solution of linear and non-linear stiff systems. First, we apply DTM to find the series solution which can be easily converted into exact solution. The method is described and illustrated with different examples and figures are plotted accordingly. The obtained result confirm that DTM is very easy, effective and convenient.