Two-dimensional Multi-term Time Research Articles

Spectral Collocation Technique for Solving Two-Dimensional Multi-Term Time Fractional Viscoelastic Non-Newtonian Fluid Model

Applications of non-Newtonian fluids have been widespread across industries, accompanied by theoretical developments in engineering and mathematics. This paper studies a two-dimensional multi-term time fractional viscoelastic non-Newtonian fluid model by using two autonomous consecutive spectral collocation strategies. A modification of the spectral approach is implemented, leading to an algebraic system of equations able to obtain an approximate symmetric solution for the model. Numerical examples illustrate the effectiveness of the technique in terms of accuracy and convergence.

Error estimate of L1-ADI scheme for two-dimensional multi-term time fractional diffusion equation

<abstract><p>A two-dimensional multi-term time fractional diffusion equation $ {D}_{t}^{\mathit{\boldsymbol{\alpha}}} u(x, y, t)- \Delta u(x, y, t) = f(x, y, t) $ is considered in this paper, where $ {D}_{t}^{\mathit{\boldsymbol{\alpha}}} $ is the multi-term time Caputo fractional derivative. To solve the equation numerically, L1 discretisation to each fractional derivative is used on a graded temporal mesh, together with a standard finite difference method for the spatial derivatives on a uniform spatial mesh. We provide a rigorous stability and convergence analysis of a fully discrete L1-ADI scheme for solving the multi-term time fractional diffusion problem. Numerical results show that the error estimate is sharp.</p></abstract>

Finite difference/finite element method for two-dimensional time–space fractional Bloch–Torrey equations with variable coefficients on irregular convex domains

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 are known to provide insights into 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 involve the Caputo time fractional derivative and the Riemann–Liouville space fractional derivative. An unstructured-mesh Galerkin finite element method is used to discretize the spatial fractional derivative, while for each time fractional derivative we use the 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.

Approximate solution of the multi-term time fractional diffusion and diffusion-wave equations

We develop a numerical scheme for finding the approximate solution for one- and two-dimensional multi-term time fractional diffusion and diffusion-wave equations considering smooth and nonsmooth solutions. The concept of multi-term time fractional derivatives is conventionally defined in the Caputo view point. In the current research, the convergence analysis of Legendre collocation spectral method was carried out. Spectral collocation method is consequently tested on several benchmark examples, to verify the accuracy and to confirm effectiveness of proposed method. The main advantage of the method is that only a small number of shifted Legendre polynomials are required to obtain accurate and efficient results. The numerical results are provided to demonstrate the reliability of our method and also to compare with other previously reported methods in the literature survey.

Coupling of the improved singular boundary method and dual reciprocity method for multi-term time-fractional mixed diffusion-wave equations

The main purpose of this work is to present an impressive numerical scheme to solve two-dimensional multi-term time fractional mixed diffusion-wave differential equations (TFMDWE). The proposed method is based on the compact dual reciprocity method and the meshless improved singular boundary method (ISBM). The most significant privilege of the proposed method is mathematically simple, efficient, quite accurate which requires relatively less computational cost. Singular boundary method is one of truly meshless methods and it does not need any element or mesh for both field interpolation and background integration. For the considered problems, time derivatives are approximated via the time stepping method. Several numerical results are provided to show the accuracy of the present method for multi-term time-fractional mixed diffusion-wave equations.

Unstructured-mesh Galerkin finite element method for the two-dimensional multi-term time–space fractional Bloch–Torrey equations on irregular convex domains

Models based on partial differential equations containing time–space fractional derivatives have attracted considerable interest in the past decade because of their ability to model anomalous transport phenomena. These phenomena are strongly connected to the interactions within complex and non-homogeneous media exhibiting spatial heterogeneity. The class of equations with multi-term time–space derivatives of fractional orders has been found to be very useful in the description of such interactions. This motivates the extension of the classical Bloch–Torrey equation through the application of the operators of fractional calculus to new multi-term time–space fractional Bloch–Torrey equations with Riesz fractional operators.In this paper, we firstly propose an unstructured-mesh Galerkin finite element method for the two-dimensional multi-term time–space fractional diffusion equation with Riesz fractional operators on irregular convex domains. Secondly, we rigorously establish the stability and convergence of the numerical scheme. Thirdly, we extend the computational model to solve a system of coupled two-dimensional multi-term time–space fractional Bloch–Torrey equations. Finally, some numerical results are given to demonstrate the versatility and application of the models.

Finite difference scheme for simulating a generalized two-dimensional multi-term time fractional non-Newtonian fluid model

A finite difference scheme, based upon the Crank–Nicolson scheme, is applied to the numerical approximation of a two-dimensional time fractional non-Newtonian fluid model. This model not only possesses a multi-term time derivative, but also contains a special time fractional operator on the spatial derivative. And a very important lemma is proposed and also proved, which plays a vital role in the proof of the unconditional stability. The stability and convergence of the finite difference scheme are discussed and theoretically proved by the energy method. Numerical experiments are given to validate the accuracy and efficiency of the scheme, and the results indicate that this Crank–Nicolson difference scheme is very effective for simulating the generalized non-Newtonian fluid diffusion model.

Finite Element Method for Two-Dimensional Unsteady MHD Couette Flow of a Generalized Oldroyd-B Fluid

In recent years, non-Newtonian fluids have been widely applied in engineering and industry. Due to the complexity of these fluids, there are several constitutive equations of non- Newtonian fluids. One particular subclass of non-Newtonian fluids is the class of a generalized Oldroyd-B fluids with fractional derivative constitutive equations, which has been found to approximate the response of many dilute polymeric liquids. Although the analytic solution of these equations can be derived, it contains a multi-level sum of an infinite series and Fox H-functions, which is extremely complex and difficult to evaluate. Therefore, seeking the numerical solution of the equation is of great importance. Different from the general time fractional diffusion equation, the constitutive equation is a novel multi-term time fractional diffusion equation and has a special time fractional operator on the spatial derivative, which is challenging to approximate. In this paper, we will consider the finite element method for the two-dimensional multi-term time fractional diffusion equation. Firstly, we utilise the L1 and L2 schemes to approximate the two different time fractional derivatives, respectively. Secondly, we apply the finite element method to the equation and establish the fully variational formulation. Then we adopt linear polynomials on triangular elements to derive the matrix form of the numerical scheme. Furthermore, we present the stability and convergence of the scheme. Finally, two numerical examples are investigated to show the effectiveness of our method, in which a two-dimensional unsteady MHD Couette flow of a generalized Oldroyd-B fluid is considered.

The analytical solution and numerical solutions for a two-dimensional multi-term time fractional diffusion and diffusion-wave equation

In this paper we consider the analytical and numerical solutions for a two-dimensional multi-term time-fractional diffusion and diffusion-wave equation. We derive the analytical solution for the equation using the method of separation of variables and properties of the multivariate Mittag-Leffler function. An implicit difference approximation is constructed. Stability and convergence analysis of the numerical scheme are proved by the energy method. Numerical examples are constructed to evaluate the working of the numerical scheme as compared to theoretical analysis. • A new two-dimensional multi-term time-fractional diffusion and diffusion. • Wave equation (2D-MT-TFD-DWE) is considered. • The analytical solution for the 2D-MT-TFD-DWE is derived. • A novel implicit difference method (IDM) for 2D-MT-TFD-DWE is proposed. • The stability and convergence of the IDM are proved by the energy method. • Numerical examples are given.

The Unstructured Mesh Finite Element Method for the Two-Dimensional Multi-term Time–Space Fractional Diffusion-Wave Equation on an Irregular Convex Domain

In this paper, the two-dimensional multi-term time-space fractional diffusion-wave equation on an irregular convex domain is considered as a much more general case for wider applications in fluid mechanics. A novel unstructured mesh finite element method is proposed for the considered equation. In most existing works, the finite element method is applied on regular domains using uniform meshes. The case of irregular convex domains, which would require subdivision using unstructured meshes, is mostly still open. Furthermore, the orders of the multi-term time-fractional derivatives have been considered to belong to (0, 1] or (1, 2] separately in existing models. In this paper, we consider two-dimensional multi-term time-space fractional diffusion-wave equations with the time fractional orders belonging to the whole interval (0, 2) on an irregular convex domain. We propose to use a mixed difference scheme in time and an unstructured mesh finite element method in space. Detailed implementation and the stability and convergence analyses of the proposed numerical scheme are given. Numerical examples are conducted to evaluate the theoretical analysis.

Analytical and numerical solutions of the unsteady 2D flow of MHD fractional Maxwell fluid induced by variable pressure gradient

The nonlocal property of the fractional derivative can supply more precise mathematical models for depicting flow dynamics of complex fluid which cannot be modelled appropriately by normal integer order differential equations. This paper studies the analytical and numerical methods of unsteady 2D flow of Magnetohydrodynamic (MHD) fractional Maxwell fluid in a rectangular pipe driven by variable pressure gradient. The governing equation is formulated with Caputo time dependent fractional derivatives whose orders are distributed in interval (0, 2). A challenge is to firstly obtain the exact solution by combining modified separation of variables method with Mikusiński-type operational calculus. Meanwhile, the numerical solution is also obtained by the implicit finite difference method whose validity has been confirmed by the comparison with the exact solution constructed. Different to the most classical works, both the stability and convergence analysis of two-dimensional multi-term time fractional momentum equation are derived. Based on numerical analysis, the results show that the velocity increases with the rise of the fractional parameter and relaxation time. While an increase in the values of Hartmann number leads to a slower velocity in the rectangular pipe.

A 2D multi-term time and space fractional Bloch-Torrey model based on bilinear rectangular finite elements

The consideration of diffusion processes in magnetic resonance imaging (MRI) signal attenuation is classically described by the Bloch-Torrey equation. However, many recent works highlight the distinct deviation in MRI signal decay due to anomalous diffusion, which motivates the fractional order generalization of the Bloch-Torrey equation.In this work, we study the two-dimensional multi-term time and space fractional diffusion equation generalized from the time and space fractional Bloch-Torrey equation. By using the Galerkin finite element method with a structured mesh consisting of rectangular elements to discretize in space and the L1 approximation of the Caputo fractional derivative in time, a fully discrete numerical scheme is derived. A rigorous analysis of stability and error estimation is provided. Numerical experiments in the square and L-shaped domains are performed to give an insight into the efficiency and reliability of our method. Then the scheme is applied to solve the multi-term time and space fractional Bloch-Torrey equation, which shows that the extra time derivative terms impact the relaxation process.

Superconvergence analysis of an H1-Galerkin mixed finite element method for two-dimensional multi-term time fractional diffusion equations

ABSTRACTIn this paper, numerical approximation for two-dimensional (2D) multi-term time fractional diffusion equation is considered. By virtue of properties of bilinear element, Raviart–Thomas element and approximation, an -Galerkin mixed finite element fully discrete approximate scheme is established for 2D multi-term time fractional diffusion equation. And then, unconditionally stable of the approximate scheme is rigourously testified by dealing with fractional derivative skilfully. At the same time, superclose results for the original variable u in -norm and the flux in -norm are derived. Furthermore, the global superconvergence results for u in -norm are deduced by the interpolation postprocessing operator. Finally, numerical results demonstrate that the approximate scheme provides a valid and efficient way for solving 2D multi-term time fractional diffusion equation.

Convergence and superconvergence of a fully-discrete scheme for multi-term time fractional diffusion equations

Using finite element method in spatial direction and classical L1 approximation in temporal direction, a fully-discrete scheme is established for a class of two-dimensional multi-term time fractional diffusion equations with Caputo fractional derivatives. The stability analysis of the approximate scheme is proposed. The spatial global superconvergence and temporal convergence of order O(h2+τ2−α) for the original variable in H1-norm is presented by means of properties of bilinear element and interpolation postprocessing technique, where h and τ are the step sizes in space and time, respectively. Finally, several numerical examples are implemented to evaluate the efficiency of the theoretical results.

Efficient Numerical Solution of the Multi-Term Time Fractional Diffusion-Wave Equation

AbstractSome efficient numerical schemes are proposed to solve one-dimensional and two-dimensional multi-term time fractional diffusion-wave equation, by combining the compact difference approach for the spatial discretisation and an L1 approximation for the multi-term time Caputo fractional derivatives. The unconditional stability and global convergence of these schemes are proved rigorously, and several applications testify to their efficiency and confirm the orders of convergence.