Approximate analytic solution of fractional heat-like and wave-like equations with variable coefficients using the differential transforms method

Variational iteration method for fractional heat- and wave-like equations

The basic motivation of the present study is to apply the local fractional Sumudu variational iteration method (LFSVIM) for solving nonlinear wave-like equations with variable coefficients and within local fractional derivatives. The derivatives operators are taken in the local fractional sense. The results show that the LFSVIM is an appropriate method to find non-differentiable solutions for similar problems. The results of the solved examples showed the flexibility of applying this method and its ability to reach accurate results even with these new differential equations.

The application of differential transformation method to nonlinear equations arising in heat transfer

A novel approach to solving nonlinear fractional wave-like differential equations

In this paper, a compact difference scheme is established for nonlinear delayed fractional sub-diffusion equations. By the discrete energy method, it is shown that it achieves the convergence rate of in L∞- norm, and is unconditionally stable. Here, α is the order of fractional derivative. Besides, the extension of the solver to multi-term time fractional sub-diffusion equations with delays is discussed. Finally, numerical results demonstrate the effectiveness of our solvers.

In this work, we applied the fractional reduced differential transform method (FRDTM) to find the exact solutions of the three-dimensional fractional Helmholtz equation (FHE) and compared our outcomes with the tenth-order approximate solutions for diverse fractional orders. Different values of fractional derivatives are signified explicitly in three dimensions. Outcomes are denoted in the tables for nonfractional exact and approximate solutions and fractional approximate solutions for different fractional orders. Two examples are given to prove the proficiency of the suggested method, to resolve diverse categories of fractional partial differential equations.

In magnetic resonance imaging of the human brain, the diffusion process of tissue water is considered in the complex tissue environment of cells, membranes and connective tissue. Models based on fractional order Bloch-Torrey equations can provide a new insights into further investigations of tissue structures and the microenvironment. In this paper, we consider new two-dimensional multi-term time and space fractional Bloch-Torrey equations with variable coefficients on irregular convex domains, which involves the Caputo time fractional derivative and the Riemann-Liouville space fractional derivative. An unstructured-mesh Galerkin finite element method is used to discretise the spatial fractional derivative, while for each time fractional derivative we use a novel $L1$ scheme on a temporal graded mesh. The stability and convergence of the fully discrete scheme are proved. Numerical examples are given to verify the efficiency of our method.

Solution of free vibration equation of elastically supported Timoshenko columns with a tip mass by differential transform method

In this work, we proposed a hybrid algorithm to approximate the solution of Conformable Fractional Fokker-Planck Equation (CFFPE). This algorithm comprises of unification of two methods named Fractional Wave Transformation Method (FWTM) and Differential Transform Method (DTM). The method is based on two steps. The first step is to reduce the given CFPDEs to corresponding Partial Differential Equations (PDEs). Then, the second step is to solve obtained PDEs iteratively by using DTM. Moreover, the algorithm’s efficiency is shown by employing the method successfully to conformable time-fractional Fokker-Planck equation arising in surface physics, plasma physics, polymer physics, laser physics, biophysics, engineering, neurosciences, nonlinear hydrodynamics, population dynamics, pattern formation and marketing. As a result, the obtained data demonstrate that the algorithm is reliable and applicable.

This paper proposed a new analytic method to calculate the Lyapunov exponent based on the Differential Transform Method (DTM). In this work, the DTM method is applied for finding approximate variable coefficients. In this method, first the time series of the dynamical system is estimated based on DTM method, then the Lyapunov exponent is calculated analytically. The Lorenz system and the Colpitts oscillator are used as a practical case study of this research to validate the results. Simulation results are used to show the main points of this paper. We have calculated of the Lyapunov exponent based on DTM method for the first time.

The main purpose of this paper is to present a new computational for approximate analytical solutions of nonlinear time-fractional wave-like equations with variable coefficients using the fractional residual power series method (FRPSM). The fractional derivative is considered in the Caputo sense. This method is based on the generalized Taylor series formula and residual error function. Unlike other analytical methods, FRPSM has a special advantage, that it solves the nonlinear problems without using linearization, discretization, perturbation or any other restrictions. By numerical examples, it is shown that the FRPSM is a simple, effective, and powerful method for finding approximate analytical solutions of nonlinear fractional partial differential equations.

In this work, the electrohydrodynamic flow equation of a fluid in an ion drag configuration in a circular cylindrical conduit is considered. The differential equation of the problem is a highly nonlinear and singular boundary value problem (BVP). An analytical solution of the problem is presented using the differential transform method (DTM). In this method, the differential equation and related boundary conditions transformed into a recurrence set of equations and at the end, the coefficients of Taylor series are calculated based on the solution of this set of equations. DTM results are compared with the numerical solution of the problem and an excellent accuracy is observed. It showed that the proposed method overcame the nonlinearity and singularity of the problem without any need to restrict assumptions or linearization. Most of the scientific problems and engineering phenomena are usually defined based on the nonlinear differential equations of both initial value problems (IVP) and boundary value problems (BVP). It is important to remember that except a limited number of these problems, finding exact analytical solution is difficult. Therefore, researchers use semi-analytical techniques and numerical methods. The semi-analytical methods such as the perturbation method (PM), Adomian decomposition method (ADM), variational iteration method (VIM), homotopy perturbation method (HPM), homotopy analysis method (HAM), and differential transform method (DTM) are useful mathematical ones, which present the analytical solutions for nonlinear differential equations. In this paper, solution of electrohydrodynamic flow equation of a fluid in an ion drag configuration in a circular cylindrical conduit is presented using DTM. Already, the electrohydrodynamic flow of a fluid and its governing equations is considered (1). The electrohydrodynamic flow is important in analysis of the flow meters, accelerators, pumps and magnetohydrodynamic generators. The differential equation of the problem is the nonlinear singular boundary value problem. 2 2

This article discusses an approximate scheme for solving one-dimensional heat-like and wave-like equations in fuzzy environment based on the homotopy perturbation method (HPM). The concept of topology in homotopy is used to create a convergent series solution of the fuzzy equations. The objective of the study is to formulate the double parametric fuzzy HPM to obtain approximate solutions of fuzzy heat-like and fuzzy wave-like equations. The fuzzification and the defuzzification analysis for the double parametric form of fuzzy numbers of the fuzzy heat-like and the fuzzy wave-like equations is carried out. The proof of convergence of the solution under the developed approximate scheme is provided. The effectiveness of the proposed method is tested by numerically solving examples of fuzzy heat-like and wave-like equations where results indicate that the approach is efficient not only in terms of accuracy but also with respect to CPU time consumption.

Black–Scholes partial differential equation is a generally acceptable model in financial markets for option pricing. However, without variable transformations, the provision of symbolic solutions to the variable coefficient partial differential equation is not a straight-forward task. Moreover, the coefficients of the Black–Scholes can depend on the time and the asset price which makes the analytical solution of the Black–Scholes model very difficult to develop. In this paper, analytical solutions of the model of valuations of financial options are presented using Laplace and differential transform methods. The results of the solutions of the Laplace and differential transformation methods are compared with the results of the exact analytical solutions. Moreover, numerical examples for different options pricing are presented to establish the applications, speed and accuracy of the hybrid methods.

An infinite horizontally extended sparcely packed chemically inert porous channel flow of a Newtonian liquid is considered. The walls of the channels are assumed to be at different temperatures so that the viscosity of the fluid varies across the channel. A dimensionless variable viscosity coefficient is introduced in the Darcy–Forchheimer- Brinkman model, along with the Darcy number, Forchheimer number and the Brinkman number. A series solution is obtained for the Darcy-Forchheimer-Brinkman equation using the differential transform method (DTM). Using this solution for the fully developed flow velocity and the convective diffusion equation, the influence of the variable viscosity coefficient and the Darcy, Brinkman and Forchheimer numbers on the all-time valid dispersion coefficient is analyzed. An increase the variable viscosity parameter is to increase the dispersion coefficient while an increase in all the other parameters will decrease the dispersion coefficient. The Light hill, Taylor-Aris and Taylor dispersion coefficients are obtained as the limiting cases of the generalized dispersion coefficient.