Volume 18, Issue 1 -

This paper proposes two schemes for a nonlinear Schrödinger equation with four-order spacial derivative by using compact scheme and discrete gradient methods. They are of fourth-order accuracy in space. We analyze two discrete invariants of the schemes. The numerical experiments are implemented to investigate the efficiency of the schemes.

The aim of the present analysis is to examine the effects of slip boundary conditions and mixed convection flow of Williamson fluid over a stretching cylinder. The boundary layer partial differential equations are transformed into ordinary differential equations by using group theory transformations. The required ordinary differential equations are solved numerically by using implicit finite difference method known as Keller box method. The influence of dimensionless physical parameters on velocity and temperature profile as well as skin friction coefficient and local Nusselt number are presented graphically. Comparison has been made to the previous literature in order to check the accuracy of the method.

Abstract: By means of Mawhin’s continuation theorem of coincidence degree theory and Lyapunov function, some simple sufficient conditions are obtained for the existence and stability of a unique positive almost periodic solution of a delayed Lotka–Volterra recurrent neural networks. To a certain extent, the work in this paper corrects the defect of a recent paper. Finally, an example and simulations are given to illustrate the feasibility and effectiveness of the main result.

In the current study, a numerical investigation of two-dimensional combined convection-radiation heat transfer of turbulent gas flow over a backward-facing step (BFS) in a horizontal rectangular duct is presented. The computational domain contains two different parts including gas flow and solid element that makes the problem as a conjugate one. The gas phase is considered to be a radiating media that can absorb, emit and scatter thermal radiation, where in solid phase, heat transfer takes place by conduction. The set of governing equations for gas flow is solved numerically using the CFD technique and the k−ε$$k - \varepsilon $$ model is employed for computation of turbulence fluctuations. To evaluate the radiative term in the gas energy equation, the radiative transfer equation (RTE) is solved by the discrete ordinates method (DOM). Inside the solid phase, the conduction equation is solved to obtain the temperature distribution. The effects of conduction ratio, optical thickness, radiation-conduction parameter and albedo coefficient on heat transfer behavior of the system are carried out.

In this paper, we propose a Monotone Schwarz Iterative Method (MSIM) under the framework of Domain Decomposition Strategy for solving semilinear parabolic singularly perturbed partial differential equations (SPPDEs). A three-step Taylor Galerkin Finite Element (3TGFE) approximation of semilinear parabolic SPPDE is carried out during each of the stages of the MSIM. Appropriate Interface Problems are introduced to update the subdomain boundary conditions in the Monotone Iterative Domain Decomposition (MIDD) method. The convergence of the MIDD method has been established. In addition, the stability and ϵ$\epsilon$-uniform convergence of 3TGFE based MIDD has been discussed. Further, by using maximum principle and induction hypothesis, the convergence of the proposed MSIM has been established. Also, the proposed 3TGFE based MIDD has been successfully implemented on a couple of test problems.

Traditional turbulence models are initially formulated and calibrated under incompressible conditions. Thus, these models are always of low fidelity when extended to high speed, complex and compressible flows. In this work, a compressible von Kármán length scale is proposed for compressible flows considering the variable densities. The length scale is the ratio between the new vorticity and its gradient. The new length scale is actually based on phenomenological theory, which is then integrated into the KDO (turbulence Kinetic energy Dependent Only) turbulence model, arriving at a compressible model called CKDO (Compressible KDO). In the CKDO turbulence model, all the extra terms produced by compressibility are modeled as dissipation. Compression corners of 8, 16, 20 and 24 angles are studied within SST, SA, KDO and CKDO. These test cases are known as the typical shock wave–boundary layer interactions. The results show that the new length scale in CKDO is able to well capture the surface pressure and skin friction distributions. Besides, compared with the standard von Kármán length scale, the new length scale in CKDO can better capture the size and position of the separation bubble. With the increase of the corner angle, CKDO shows more prominent potential for describing compressible flows.

Numerical solutions of nonlinear partial differential equations, such as the generalized and extended Burgers-Huxley equations which combine effects of advection, diffusion, dispersion and nonlinear transfer are considered in this paper. Such system can be divided into linear and nonlinear parts, which allow the use of two numerical approaches. Barycentric Jacobi spectral (BJS) method is employed for the spatial discretization, the resulting nonlinear system of ordinary differential equation is advanced with a fourth-order exponential time differencing predictor corrector. Comparative numerical results for the values of options are presented. The proposed method is very elegant from the computational point of view. Numerical computations for a wide variety of problems, show that the present method offers better accuracy and efficiency in comparison with other previous methods. Moreover the method can be applied to a wide class of nonlinear partial differential equations.

This paper deals with the existence and multiplicity of nontrivial solutions of fractional multi-point boundary value problems for higher-order nonlinear singular fractional differential equations with sign-changing nonlinear term. The main tool used in the proof is topological degree theory. Some examples are given to illustrate our main results.

Abstract: Evolution equations containing fractional derivatives can provide suitable mathematical models for describing important physical phenomena. In this paper, we propose an accurate method for numerical solutions of multi-dimensional time-fractional heat equations. The proposed method is based on a fractional exponential integrator scheme in time and the Lagrange regularized kernel method in space. Numerical experiments show the effectiveness of the proposed approach.