Nonlinear Diffusion-wave Equation Research Articles

Chelyshkov polynomials method for distributed-order time fractional nonlinear diffusion-wave equations

This work deals with the distributed-order time fractional nonlinear diffusion-wave equations. These equations are generated by replacing the first- and second-order time derivative terms with the distributed-order fractional derivative terms. The distributed-order fractional derivatives used in these problems are in the Caputo sense. The Chelyshkov polynomials as a well-known family of basis functions are used to develop a spectral collocation method for these problems. Through the way, some operational matrices regarding the classical and distribute-order fractional derivatives for these polynomials are extracted. In the proposed method, after approximating the solution of the problem in terms of the Chelyshkov polynomials and employing the expressed matrices, solving the primary equations transforms into solving algebraic systems which can be easily solved. Some numerical examples are studied to show the adequacy of the approach.

A linearized finite difference scheme for time–space fractional nonlinear diffusion-wave equations with initial singularity

Abstract 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.

A numerical combined algorithm in cubic B-spline method and finite difference technique for the time-fractional nonlinear diffusion wave equation with reaction and damping terms

The applications of the diffusion wave model of a time-fractional kind with damping and reaction terms can occur within classical physics. This quantification of the activity can measure the diagnosis of mechanical waves and light waves. The goal of this work is to predict and construct numerical solutions for such a diffusion model based on the uniform cubic B-spline functions. The Caputo time-fractional derivative has been estimated using the standard finite difference technique, whilst, the uniform cubic B-spline functions have been employed to achieve spatial discretization. The convergence of the suggested blueprint is discussed in detail. To assert the efficiency and authenticity of the study, we compute the approximate solutions for a couple of applications of the diffusion model in electromagnetics and fluid dynamics. To show the mathematical simulation, several tables and graphs are shown, and it was found that the graphical representations and their physical explanations describe the behavior of the solutions lucidly. The key benefit of the resultant scheme is that the algorithm is straightforward and makes it simple to implement as utilized in the highlight and conclusion part.

EFFICIENT NUMERICAL SOLUTION OF TWO-DIMENSIONAL TIME-SPACE FRACTIONAL NONLINEAR DIFFUSION-WAVE EQUATIONS WITH INITIAL SINGULARITY

<abstract> In this paper, we present an efficient linearized alternating direction implicit (ADI) scheme for two-dimensional time-space fractional nonlinear diffusion-wave equations with initial singularity. First, the original problem is equivalently transformed into its partial integro-differential form. Then, for the time discretization, the Crank-Nicolson technique combined with the midpoint formula and the second order convolution quadrature formula are used. Meanwhile, the classical central difference formula and fractional central difference formula are adopted to approximate the second order derivative and the Riesz derivative in space, respectively. The unconditional stability and convergence of the proposed scheme are proved by the energy method. Numerical experiments support the theoretical results. </abstract>

Analysis of a robust implicit scheme for space–time fractional stochastic nonlinear diffusion wave model

The current paper develops and analyzes a numerical scheme for the space–time fractional stochastic nonlinear diffusion wave equations. The implicit scheme is based on the matrix transform technique for discretizing the Riesz-space fractional derivative, -order approximation to the Caputo-fractional derivative in time and Taylor's series method to linearize the nonlinear source term, and has been successfully applied to solve a class of nonlinear fractional diffusion wave equation. We prove that the implicit scheme is convergent with β-order in space and order in time, respectively. The optimum error estimates in the temporal-spatial direction and unconditional stability of the implicit scheme have been theoretically investigated. Moreover, several specific numerical experiments confirm the consistency and high efficacy of the provided algorithms, which minimizes the computational costs.

Two linearized finite difference schemes for time fractional nonlinear diffusion-wave equations with fourth order derivative

<abstract> In this paper, we present a linearized finite difference scheme and a compact finite difference scheme for the time fractional nonlinear diffusion-wave equations with space fourth order derivative based on their equivalent partial integro-differential equations. The finite difference scheme is constructed by using the Crank-Nicolson method combined with the midpoint formula, the weighted and shifted Gr$ \ddot{\text{u}} $nwald difference formula and the second order convolution quadrature formula to deal with the temporal discretizations. Meanwhile, the classical central difference formula and fourth order Stephenson scheme are used in the spatial direction. Then, the compact finite difference scheme is developed by using the fourth order compact difference formula for the spatial direction. The proposed schemes can deal with the nonlinear terms in a flexible way while meeting weak smoothness requirements in time. Under the relatively weak smoothness conditions, the stability and convergence of the proposed schemes are strictly proved by using the discrete energy method. Finally, some numerical experiments are presented to support our theoretical results. </abstract>

A numerical method for two-dimensional multi-term time-space fractional nonlinear diffusion-wave equations

Recently, numerous numerical schemes have been developed for solving single-term time-space fractional diffusion-wave equations. Among them, some popular methods were constructed by using the graded meshes due to the solution with low temporal regularity. In this paper, we present an efficient alternating direction implicit (ADI) scheme for two-dimensional multi-term time-space fractional nonlinear diffusion-wave equations. Firstly, the considered problem is equivalently transformed into its partial integro-differential form with the Riemann-Liouville integral and multi-term Caputo derivatives. Secondly, the ADI scheme is constructed by using the first-order approximations and L1 approximations to approximate the terms in time, and using the fractional centered differences to discretize the multi-term Riesz fractional derivatives in space. Furthermore, the fast implement of the proposed ADI scheme is discussed by the sum-of-exponentials technique for both Caputo derivatives and Riemann-Liouville integrals. Then, the solvability, unconditional stability and convergence of the proposed ADI scheme are strictly established. Finally, two numerical examples are given to support our theoretical results, and demonstrate the computational performances of the fast ADI scheme.

A meshfree approach for solving 2D variable-order fractional nonlinear diffusion-wave equation

This paper is concerned with the moving least squares (MLS) meshless approach for the numerical solution of two-dimensional (2D) variable-order time fractional nonlinear diffusion-wave equation (V-OTFND-WE) on arbitrary domains. The variable-order fractional derivative of this equation is expressed in the Caputo type. The proposed method is based on the MLS approximation in conjunction with the finite difference technique. For spatial derivatives, the MLS approximation and for temporal derivative, the finite difference technique are utilized to discretize the equation. The aim of the present paper is to show that the meshfree method based on the MLS approximation and collocation scheme is suitable for solving the variable-order fractional partial differential equations (PDEs). The proposed method is validated, using some different nonlinear numerical examples with complex geometries. The results obtained demonstrate the accuracy and easy implementation of the method for nonlinear variable-order time fractional PDEs.

A computational method for solving variable-order fractional nonlinear diffusion-wave equation

In this paper, we generalize a one-dimensional fractional diffusion-wave equation to a one-dimensional variable-order space-time fractional nonlinear diffusion-wave equation (V-OS-TFND-WE) where the variable-order fractional derivatives are considered in the Caputo type. To solve this introduced equation, an easy-to-follow method is proposed which is based on the Chebyshev cardinal functions coupling with the tau and collocation methods. To carry out the method, an operational matrix of variable-order fractional derivative (OMV-OFD) is derived for the Chebyshev cardinal functions to be employed for expanding the unknown function. The proposed method can provide highly accurate approximate solutions by reducing the problem under study to a system of nonlinear algebraic equations which is technically simpler for handling. The experimental results confirm the applicability and effectiveness of the method.

Distributed modelling of urban runoff using a meta-channel concept

Hydraulic flood routing is advantageous for computational accuracy; however, it requires individual calculations of an extensive pipe network in the case of a two-dimensional analysis. In this study, a method for considering the characteristics of a two-dimensional pipe network has been developed using the meta-channel concept to simplify the detailed calculations involved in the estimation of runoff from urban catchments. In essence, the meta-channel concept turns a two-dimensional pipe network into a one-dimensional pipe with an effective hydraulic geometry. Once such geometry has been identified, the flood routing can then be performed for an urban drainage system. A nonlinear diffusion wave equation, derived from the Saint-Venant equation, was used for flood routing, with an explicit method used for the numerical solution. The celerity and diffusion coefficients, which are two parameters of the diffusion wave equation, were estimated for the Goonja drainage from a two-dimensional pipe network using the meta-channel concept. A comparison of the results of the meta-channel-based pipe routing with the distributed SWMM simulation and observed data, showed close similarities, and identified the applicability of the meta-channel concept in an urban drainage setting.

Homotopy analysis method for solving multi-term linear and nonlinear diffusion–wave equations of fractional order

In this paper we have used the homotopy analysis method (HAM) to obtain solutions of multi-term linear and nonlinear diffusion–wave equations of fractional order. The fractional derivative is described in the Caputo sense. Some illustrative examples have been presented.

Solving multi-term linear and non-linear diffusion–wave equations of fractional order by Adomian decomposition method

Multi-term linear and non-linear diffusion–wave equations of fractional order are solved using Adomian decomposition method. Some numerical examples are presented.

Flood routing based on diffusion wave equation using mixing cell method

A one-dimensional non-linear diffusion wave equation is derived from the Saint Venant equations with neglect of the inertia terms. This non-linear equation has no general analytical solution. Numerical schemes are therefore employed to discretize the space and time axes and convert the differential equation to difference form. In this study, the mixing cell method is used to convert the diffusion wave equation to difference form, in which the difference term can be eliminated by selecting an optimal space step size Δx when time step size Δt is given. When the time step size Δt → 0, the space step size Δx = Q/(2S 0 BC k ) where Q is discharge, S 0 is bed slope, B is channel width and C k is kinematic wave celerity, which is the same as the characteristic length proposed by Kalinin and Milyukov. The results of application to two cases show that the mixing cell and linear channel flow routing methods produce hydrographs that are in agreement with the observed flood hydrographs.