Abstract: The main objective of this paper is to investigate an accurate numerical method for solving a biological fractional model via Atangana-Baleanu fractional derivative. We focused our attention on linear and nonlinear Fisher’s equations. We use the spectral collocation method based on the Chebyshev approximations. This method reduced the nonlinear equations to a system of ordinary differential equations by using the properties of Chebyshev polynomials and then solved them by using the finite difference method. This is the first time that this method is used to solve nonlinear equations in Atangana-Baleanu sense. We present the effectiveness and accuracy of the proposed method by computing the absolute error and the residual error functions. The results show that the given procedure is an easy and efficient tool to investigate the solution of nonlinear equations with local and non-local singular kernels.

Abstract: We present in this paper construction and analysis of a block-centered finite difference method for the spatial discretization of the scalar auxiliary variable Crank-Nicolson scheme (SAV/CN-BCFD) for gradient flows, and show rigorously that scheme is second-order in both time and space in various discrete norms. When equipped with an adaptive time strategy, the SAV/CN-BCFD scheme is accurate and extremely efficient. Numerical experiments on typical Allen-Cahn and Cahn-Hilliard equations are presented to verify our theoretical results and to show the robustness and accuracy of the SAV/CN-BCFD scheme.

Abstract: In this paper, we propose a novel meshfree Generalized Finite Difference Method (GFDM) approach to discretize PDEs defined on manifolds. Derivative approximations for the same are done directly on the tangent space, in a manner that mimics the procedure followed in volume-based meshfree GFDMs. As a result, the proposed method not only does not require a mesh, it also does not require an explicit reconstruction of the manifold. In contrast to some existing methods, it avoids the complexities of dealing with a manifold metric, while also avoiding the need to solve a PDE in the embedding space. A major advantage of this method is that all developments in usual volume-based numerical methods can be directly ported over to surfaces using this framework. We propose discretizations of the surface gradient operator, the surface Laplacian and surface Diffusion operators. Possibilities to deal with anisotropic and discontinuous surface properties with large jumps are also introduced, and a few practical applications are presented.

Abstract: This study numerically examined unsteady double stratified EMHD mixed convection flow of nanofluid via permeable stretching sheet. It also looked at the convective heat and mass boundary conditions as well as the Navier velocity slip. In the thermal field, the effects of radiative heat transfer, heat generation/absorption, viscous dissipation, together with Ohmic heating (both magnetic and electric fields) were considered. The concentration field accounts for the chemical reaction. These show the physical behavior of electromagnetohydrodynamic flow associated with the problem formulation. The characteristics in regard to convective heat and mass, Navier slips conditions, as well as double stratification, were imposed. Such structure arises in energy efficiency and performance, which is achievable without higher pumping power, serves in the extrusion manufacturing process involving the thermal system for efficient devices particularly in polymeric, paper production, and food processing. The governing equations, which are nonlinear partial differential equations, were modelled by ordinary differential equations using suitable transformations. The ODEs were solved numerically, using implicit finite difference method (Keller box method). The physical implications deliberated on the behavior via the velocity, thermal energy, and concentration fields as well as the skin friction coefficient; the Nusselt and Sherwood numbers were scrutinized in relation to several parameters via mathematical model. The analysis shows that thermal and concentration stratifications decrease the distributions adjacent to the sheet surface, indicating decrease in the concentration nanoparticles and reduction in thermal energy. Augmentation occurs with convective heat and mass Biot numbers with the fields. The electric and magnetic parameters exhibit opposite flow behavior to the velocity and temperature. Chemical reaction and viscous dissipation weaken the concentration profile. Numerical results were compared with the published data available in the literature for limiting cases, and good agreement was noticed.

Abstract: In this paper the implicit-explicit (IMEX) two-step backward differentiation formula (BDF2) method with variable step-size, due to the nonsmoothness of the initial data, is developed for solving pa...

Abstract: This paper reports multiple slip effects on MHD unsteady flow heat and mass transfer over a stretching sheet with Soret effect; suction/injection and thermal radiation are numerically analyzed. We consider a time-dependent applied magnetic field and stretching sheet which moves with nonuniform velocity. Suitable similarity variables are used to transform governing partial differential equations into a system of coupled nonlinear ordinary differential equations. The transformed equations are then solved numerically by applying an implicit finite difference method with quasi-linearization technique. The influences of the various parameters on the velocity temperature and concentration profiles as well as on the skin friction coefficient and Sherwood and Nusselt numbers are discussed by the aid of graphs and tables.

Abstract: The phenomenon of natural convection in a square cavity filled with a copper-water nanofluid is investigated numerically. The studied domain is a square cavity with hot and cold isothermal walls at x = 0 and x = L, respectively, while the other walls are adiabatic. The fins are considered perfectly conductive with different lengths (Lf) and positioned at different locations. We examined the situation for Rayleigh numbers ranging between 104 and 106. The governing equations are expressed in the vorticity, stream function, and temperature formulation. The system of equations was solved by the finite difference method, using the upwind scheme. The computation code thus developed was used to analyze the effect of the different locations of the fins on the thermal performances. The obtained results were validated by comparing with those of a previously published work and with those obtained using COMSOL Multiphysics. It has been found that adding fins on the cold and adiabatic walls results in an increase in the average Nusselt number, while it decreases when the fin is located on the hot wall. That is to say, placing the fins on the cold and adiabatic walls increases the thermal performances of the transfer.

Abstract: This paper presents a hybrid element‐free Galerkin (HEFG) method for solving wave propagation problems. By introducing the dimension split method, the three‐dimensional wave propagation problems are transformed into a series of two‐dimensional ones in other one‐dimensional directions. The two‐dimensional problems are solved using the improved element‐free Galerkin (IEFG) method, and the finite difference method is used in the one‐dimensional splitting direction and the time space. Then, the formulas of the HEFG method for three‐dimensional wave propagation problems are obtained. Numerical examples are selected to show the effectiveness and the advantage of the HEFG method. The convergence and error analysis of the HEFG method are discussed according to the numerical results under different splitting directions, weight functions, node distributions, scale parameters of the influence domain, penalty factors, and time steps. The numerical results are given to show the convergence and advantages of the HEFG method over the IEFG method. Comparing with the IEFG method, the HEFG method has greater computational precision and speed for three‐dimensional wave propagation problems.

Abstract: In this paper, a meshless numerical wave flume, based on the generalized finite difference method (GFDM), is adopted to accurately and efficiently simulate the interactions of water waves and current. The GFDM, a newly-developed meshless method, is truly free from mesh generation and numerical quadrature. The proposed meshless numerical wave flume is the combination of the GFDM, the second-order Runge–Kutta method, the semi-Lagrangian approach, the sponge layer and the ramping function. The problems of wave-current interactions in flumes with horizontal and inclined bottoms are accurately and stably investigated by the proposed meshless scheme, respectively. The changes of waveform can be obviously found, while the cases of coplanar, opposing and no currents are stably simulated. Besides, the distribution of steady current in the flume with inclined bottom, which is governed by an inverse Cauchy problem, is acquired by the GFDM in a stable manner. Numerical results of wave-current interactions are compared with other solutions to verify the accuracy of the proposed meshless scheme. Additionally, different parameters of the proposed meshless numerical scheme are examined to validate the consistency and stability of the proposed numerical wave flume for solutions of wave-current interactions.

Abstract: By transforming a 3D problem into some related 2D problems, the dimension splitting element-free Galerkin (DSEFG) method is proposed to solve 3D transient heat conduction problems. The improved element-free Galerkin (IEFG) method is used for 2D transient heat conduction problems, and the finite difference method is applied in the splitting direction. The discretized system equation is obtained based on the Galerkin weak form of 2D problem; the essential boundary conditions are imposed with the penalty method; and the finite difference method is employed in the time domain. Four exemplary problems are chosen to verify the efficiency of the DSEFG method. The numerical solutions show that the efficiency and precision of the DSEFG method are greater than ones of the IEFG method for 3D problems.

Abstract: A numerical scheme for a class of singularly perturbed parabolic partial differential equation with the time delay on a rectangular domain in the x–t plane is constructed. The presence of the perturbation parameter in the second-order space derivative gives rise to parabolic boundary layer(s) on one (or both) of the lateral side(s) of the rectangle. Thus the classical numerical methods on the uniform mesh are inadequate and fail to give good accuracy and results in large oscillations as the perturbation parameter approaches zero. To overcome this drawback a numerical method comprising the Crank–Nicolson finite difference method consisting of a midpoint upwind finite difference scheme on a fitted piecewise-uniform mesh of $$N\times M$$ elements condensing in the boundary layer region is constructed. A priori explicit bounds on the solution of the problem and its derivatives which are useful for the error analysis of the numerical method are established. To establish the parameter-uniform convergence of the proposed method an extensive amount of analysis is carried out. It is shown that the proposed difference scheme is second-order accurate in the temporal direction and the first-order (up to a logarithmic factor) accurate in the spatial direction. To validate the theoretical results, the method is applied to two test problems. The performance of the method is demonstrated by calculating the maximum absolute errors and experimental orders of convergence. Since the exact solutions of the test problems are not known, the maximum absolute errors are obtained by using double mesh principle. The numerical results show that the proposed method is simply applicable, accurate, efficient and robust.

Abstract: Numerous geological processes are governed by thermal and mechanical interactions. In particular, tectonic processes such as ductile strain localization can be induced by the intrinsic coupling that exists between deformation, energy and rheology. To investigate this thermomechanical feedback, we have designed 2-D codes that are based on an implicit finite-difference discretization. The direct-iterative method relies on a classical Newton iteration cycle and requires assembly of sparse matrices, while the pseudo-transient method uses pseudo-time integration and is matrix-free. We show that both methods are able to capture thermomechanical instabilities when applied to model thermally activated shear localization; they exhibit similar temporal evolution and deliver coherent results both in terms of nonlinear accuracy and conservativeness. The pseudo-transient method is an attractive alternative, since it can deliver similar accuracy to a standard direct-iterative method but is based on a much simpler algorithm and enables high-resolution simulations in 3-D. We systematically investigate the dimensionless parameters controlling 2-D shear localization and model shear zone propagation in 3-D using the pseudo-transient method. Code examples based on the pseudo-transient and direct-iterative methods are part of the M2Di routines (R¨ass et al., 2017) and can be downloaded from Bitbucket and the Swiss Geocomputing Centre website.

Abstract: Bio-heat transfer is a branch of bio-medical engineering which has its foundation linked to engineering disciplines of heat transfer. The thermal properties and behaviour of various malfunctioning tissues in human body varies as compared with normal tissues. Among various cancer tissues one which is commonly diagnosed in women is breast cyst (cancer causing fluid). The aim of present work is to develop one, two and three-dimensional computational models to study bio-heat transfer problems using finite difference method. First of all, a numerical model based on finite difference method is developed to solve Pennes’s bio-heat transfer equation in one-dimension to get temperature profiles normal to skin surface and validated with existing analytical solutions. Secondly, the numerical model is extended to study the thermal behaviour of human breast section embedded with cyst using two-dimensional cylindrical coordinate systems and validated with previous researcher’s results. The effect of size, location and presence of multiple cysts on surface temperature is studied. Lastly, the work is extended for the case of three-dimensional breast section with cyst located at the centre. The numerical results obtained using one, two and three-dimensional computational models will be highly helpful in the early detection of breast cancer tissues and also the location of it inside the body.

Abstract: In this paper, the meshless Generalized Finite Difference Method (GFDM) in conjunction with the second-order explicit Runge-Kutta method (RK2 method) is presented to solve coupled unsteady nonlinear convection-diffusion equations (CDEs). Compared with the conventional Euler method, the RK2 method not only has higher accuracy but also reduces the possibility of numerical oscillation in time discretization, especially for the nonlinear and coupled cases. The generalized finite difference method, which is a localized collocation method, is famous for its simplicity and adaptability in the numerical solution of partial differential equations. Benefiting from Taylor series and moving least squares, its partial derivatives can be formed by a series of surrounding space points. In comparison with traditional finite difference methods, the proposed GFDM is free of mesh and available for irregular discretization nodes. In this study, the stencil selection algorithms are introduced to choose the stencil support of a certain node from the whole discretization nodes. Error analysis and numerical investigations are presented to demonstrate the effectiveness of the proposed GFDM for solving the coupled linear and nonlinear unsteady convection-diffusion equations. Then it is successfully applied to three benchmark examples of the coupled unsteady nonlinear CDEs encountered in the double-diffusive natural convection process, chemotaxis-haptotaxis model of cancer invasion, and thermo-hygro coupling model of concrete.