Modelling infiltration by means of a nonlinear fractional diffusion model

In this paper we study linear and nonlinear fractional diffusion equations with the Caputo fractional derivative of non-singular kernel that has been launched recently (Caputo and Fabrizio in Prog. Fract. Differ. Appl. 1(2):73-85, 2015). We first derive simple and strong maximum principles for the linear fractional equation. We then implement these principles to establish uniqueness and stability results for the linear and nonlinear fractional diffusion problems and to obtain a norm estimate of the solution. In contrast with the previous results of the fractional diffusion equations, the obtained maximum principles are analogous to the ones with the Caputo fractional derivative; however, extra necessary conditions for the existence of a solution of the linear and nonlinear fractional diffusion models are imposed. These conditions affect the norm estimate of the solution as well.

Purpose The objectives of this study are threefold: (1) to introduce the tempered Chebyshev wavelet (TCW), (2) to propose the TCW method for solving linear tempered Caputo fractional diffusion-type equations and (3) to develop a fast tempered Chebyshev wavelet (fTCW) method for solving one- and two-dimensional nonlinear tempered Caputo fractional diffusion-type equations. Design/methodology/approach The fTCW method integrates the TCW method with L2−1σ and sum-of-exponentials approximations of the tempered Caputo fractional time derivative. For this purpose, we introduce the TCW and derive its new operational matrices for tempered fractional integration by using the hypergeometric function. For nonlinear problems, we employ the interpolation technique in conjunction with operational matrices and fast approximations. The efficiency of the fTCW method is demonstrated through comparisons with the exact solution and the solution obtained using the TCW method. Findings We have derived the TCW operational matrix of tempered fractional integration and the TCW operational matrix of tempered fractional integration for boundary value problems. These matrices, in conjunction with the fast approximations of the tempered Caputo fractional time derivative and the interpolation technique, form the basis for the construction of the fTCW method. We present a detailed methodology for solving linear tempered Caputo fractional equations using the TCW method. Additionally, we provide a comprehensive methodology for solving both one- and two-dimensional nonlinear tempered Caputo fractional diffusion-type equations using the fTCW method. The convergence and error analysis of both methods are thoroughly discussed. Numerical simulations are presented to validate and illustrate the theoretical results. These simulations demonstrate the effectiveness and accuracy of the proposed methods by solving three test problems, including one linear and two nonlinear tempered Caputo fractional diffusion-type equations. The results are compared with analytical solutions or with each other to highlight the efficiency and precision of the TCW and fTCW methods. Originality/value Many engineers and scientists can utilize the presented methods for solving their linear and nonlinear tempered Caputo fractional diffusion-type models.

In this article, the fractional derivatives in the sense of modified Riemann–Liouville and the exp-function method are used to construct exact solutions for some nonlinear partial fractional differential equations via the nonlinear fractional Liouville equation and nonlinear fractional Zoomeron equation. These nonlinear fractional equations can be turned into another nonlinear ordinary differential equation by complex transform method. This method is efficient and powerful in solving wide classes of nonlinear fractional order equations. The exp-function method appears to be easier and more convenient by means of a symbolic computation system.

The pivotal aim of the present work is to obtain analytical and approximate solution for nonlinear time-fractional Swift-Hohenberg equations (FS-HEs) using the conformable residual series method (CRSM). The fractional derivative is proposed within a conformable concept. The proposed method is graceful amalgamations of the conformable residual error functions and generalized Talyor series in the sense of conformable operator. The truncated approximate solution is substituted in the nonlinear fractional model where the conformable derivative to the residual function is equal to zero. The convergence analysis is discussed to show the accuracy and efficiency of the CRSM to obtain approximate solutions for the FS-HEs. Numerical simulation with graphical representation is also given to validate and illustrate the proposed method. The obtained results indicate that the CRS technique is effective, simple, and systematic for analyzing the behavior of nonlinear partial differential equations of fractional order arisen in many areas of physics and science.

Homotopy analysis method for solving linear and nonlinear fractional diffusion-wave equation

Stability analysis of traveling wave solution for gravity-driven flow

In this work, a modified residual power series method is implemented for providing efficient analytical and approximate solutions for a class of coupled system of nonlinear fractional integrodifferential equations. The proposed algorithm is based on the concept of residual error functions and generalized power series formula. The fractional derivative is described under the Caputo concept. To illustrate the potential, accuracy, and efficiency of the proposed method, two numerical applications of the coupled system of nonlinear fractional integrodifferential equations are tested. The numerical results confirm the theoretical predictions and depict that the suggested scheme is highly convenient, is quite effective, and practically simplifies computational time. Consequently, the proposed method is simple, accurate, and convenient in handling different types of fractional models arising in the engineering and physical systems.

The idea proposed in this work is to extend the ZZ transform method to resolve the nonlinear fractional partial differential equations by combining them with the so-called homotopy perturbation method (HPM). We apply this technique to solve some nonlinear fractional equations as: nonlinear time-fractional Fokker-Planck equation, the cubic nonlinear time-fractional Schrodinger equation and the nonlinear timefractional KdV equation. The fractional derivative is described in the Caputo sense. The results show that this is the appropriate method to solve somme models of nonlinear partial differential equations with time-fractional derivative.

A method of soil moisture content prediction is presented using fractional differential generalization of one-dimensional Richards equation for the purposes of decision support in irrigation. We propose to use water head dynamics data gathered in the beginning of growth season to perform a calibration of the model by the meta-heuristic particle swarm algorithm. Van Genuchten model is used to describe water retention curve. The cases of the known values of its coefficients along with their identification from water head dynamics data are considered. The paper presents the results of model verification for both classical Richards equation and its fractional differential generalization. For the conditions of sprinkling irrigation in the south of Ukraine, we estimate the accuracy of modelling the water head dynamics measured by the Watermark sensors and the accuracy of seasonal irrigation rate assessment. The obtained results show that the use of water transport model that take soil fractality into consideration has no effect on assessing seasonal irrigation rate but gives up to 7% increase in accuracy while predicting the time and rate of next irrigation application.

The nonlinear advection-diffusion equation, also known as Richard’s equation, is one of the most famous equations expressing water content in unsaturated porous media with broad applications in hydrology, engineering, and soil sciences. Because of the inherent nonlinearity in the equation, its closed-form analytical solution is rare. The traveling wave solution (TWS) is one of the recent exact approaches used in solving nonlinear partial differential equations (PDEs) and seems to be a suitable and efficient approach to obtain analytical solutions for Richard's equation. This paper presents three major cases of Richard’s equation that are tackled with the TWS method using general and modified forms of tanh⁡ function scheme to obtain the exact solution for each equation. The typical forms of diffusivity and conductivity functions proposed by Brooks and Corey are considered, and the exact solution for Richard’s equation is presented. Solutions have a broad applicability in pertinent engineering and s...

Noise can create periodic behavior and stabilize nonlinear diffusions

In this paper, we examined a wide class of the variable order fractional problems such as linear and nonlin-ear fractional variable order differential equations, variable order fractional functional boundary value problems, variable order fractional pantograph differential equations. The proposed method is a collocation method based on the Bessel polynomials and the operational matrix of derivatives, which transformed equations into a system of non-linear algebraic equations to achieve the approximate solution. By using Caputo fractional derivative, the operational matrix of the variable-order fractional derivatives is constructed. The error analysis shows that the method is convergent. Several numerical results confirm the accuracy and efficiency of the proposed method. Keywords: Bessel collocation method, Variable-order fractional operational matrix, Variable order fractional differential equations, Variable order fractional functional boundary value problems, Variable order fractional panto-graph equations.

Solitons and other solutions of the nonlinear fractional Zoomeron equation

A new noise reduction and edge enhancement method, i.e., Laplacian pyramid-based nonlinear diffusion and shock filter (LPNDSF), is proposed for medical ultrasound imaging. In the proposed LPNDSF, a coupled nonlinear diffusion and shock filter process is applied in Laplacian pyramid domain of an image, to remove speckle and enhance edges simultaneously. The performance of the proposed method was evaluated on a phantom and a real ultrasound image. In the phantom study, we obtained an average gain of 0.55 and 1.11 in contrast-to-noise ratio compared to the speckle reducing anisotropic diffusion (SRAD) and nonlinear coherent diffusion (NCD), respectively. Also, the proposed LPNDSF showed clearer boundaries on the phantom and the real ultrasound image. These preliminary results indicate that the proposed LPNDSF can effectively reduce speckle noise while enhancing image edges for retaining subtle features.

This paper presents a linearized finite difference scheme for solving a kind of time-space fractional nonlinear diffusion-wave equations with initial singularity, where the Caputo fractional derivative in time and the Riesz fractional derivative in space are involved. First, the considered problem is equivalently transformed into its partial integro-differential form. Then, the fully discrete scheme is constructed by using the Crank–Nicolson technique, the L1 approximation, and the convolution quadrature formula to deal with the temporal discretizations. Meanwhile, the classical central difference formula and the fractional central difference formula are applied to approximate the second-order derivative and the Riesz fractional derivative in space, respectively. Moreover, the stability and convergence of the proposed scheme are strictly proved by using the discrete energy method. Finally, some numerical experiments are presented to illustrate the theoretical results.