6 research outputs found
A Handy Approximation Technique for Closedform and Approximate Solutions of Time- Fractional Heat and Heat-Like Equations with Variable Coefficients
In this paper, we propose a handy approximation
technique (HAT) for obtaining both closed-form and
approximate solutions of time-fractional heat and heat-like
equations with variable coefficients. The method is relatively
recent, proposed via the modification of the classical
Differential Transformation Method (DTM). It devises a
simple scheme for solving the illustrative examples, and some
similar PDEs. Besides being handy, the results obtained
converge faster to their exact forms. This shows that this
modified DTM (MDTM) is very efficient and reliable. It
involves less computational work, even without given up
accuracy. Therefore, we strongly recommend it for solving
both linear and nonlinear time-fractional partial differential
equations (PDEs) with applications in other aspects of pure
and applied sciences, management, and finance
The non-standard finite difference scheme for linear fractional PDEs in fluid mechanics
AbstractA non-standard finite difference scheme is developed to solve the linear partial differential equations with time- and space-fractional derivatives. The Grunwald–Letnikov method is used to approximate the fractional derivatives. Numerical illustrations that include the linear inhomogeneous time-fractional equation, linear space-fractional telegraph equation, linear inhomogeneous fractional Burgers equation and the fractional wave equation are investigated to show the pertinent features of the technique. Numerical results are presented graphically and reveal that the non-standard finite difference scheme is very effective and convenient for solving linear partial differential equations of fractional order
Application of Homotopy Perturbation Method to Biological Population Model
In this article, a well-known analytical approximation method, so-called the Homotopy perturbation method (HPM) is adopted for solving the nonlinear partial differential equations arising in the spatial diffusion of biological populations. The resulting solutions are compared with those of the existing solutions obtained by employing the Adomian’s decomposition method. The comparison reveals that our approximate solutions are in very good agreement with the solutions by Adomian’s method. Moreover, the results show that the proposed method is a more reliable, efficient and convenient one for solving the non-linear differential equations
Two Hybrid Methods for Solving Two-Dimensional Linear Time-Fractional Partial Differential Equations
A computationally efficient hybridization of the Laplace transform with two spatial discretization techniques is investigated for numerical solutions of time-fractional linear partial differential equations in two space variables. The Chebyshev collocation method is compared with the standard finite difference spatial discretization and the absolute error is obtained for several test problems. Accurate numerical solutions are achieved in the Chebyshev collocation method subject to both Dirichlet and Neumann boundary conditions. The solution obtained by these hybrid methods allows for the evaluation at any point in time without the need for time-marching to a particular point in time
On the application of partial differential equations and fractional partial differential equations to images and their methods of solution
This body of work examines the plausibility of applying partial di erential equations and
time-fractional partial di erential equations to images. The standard di usion equation
is coupled with a nonlinear cubic source term of the Fitzhugh-Nagumo type to obtain a
model with di usive properties and a binarizing e ect due to the source term. We examine
the e ects of applying this model to a class of images known as document images;
images that largely comprise text. The e ects of this model result in a binarization process
that is competitive with the state-of-the-art techniques. Further to this application,
we provide a stability analysis of the method as well as high-performance implementation
on general purpose graphical processing units. The model is extended to include
time derivatives to a fractional order which a ords us another degree of control over this
process and the nature of the fractionality is discussed indicating the change in dynamics
brought about by this generalization. We apply a semi-discrete method derived by
hybridizing the Laplace transform and two discretization methods: nite-di erences and
Chebyshev collocation. These hybrid techniques are coupled with a quasi-linearization
process to allow for the application of the Laplace transform, a linear operator, to a
nonlinear equation of fractional order in the temporal domain. A thorough analysis
of these methods is provided giving rise to conditions for solvability. The merits and
demerits of the methods are discussed indicating the appropriateness of each method