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In this paper, we propose a high-accuracy linear B-spline finite-difference approximation for variable-order (VO) derivative. We consider VO fractional differentiation as a control parameter for improving the stability in systems exhibiting vibrations. The method is applied to nonlinear feedback with VO fractional derivative. The results demonstrate the efficiency and high accuracy of the novel algorithm.

In this work, a second-order accuracy in both space and time Crank–Nicolson (C-N)-type scheme, a fourth-order accuracy in space and second-order accuracy in time compact scheme and a sixth-order accuracy in space and second-order accuracy in time compact scheme are proposed for the derivative nonlinear Schrödinger equation. The C-N-type scheme is tested to satisfy the conservation of discrete mass. For the two compact schemes, the iterative algorithm and the Thomas algorithm in block matrix form are adopted to enhance the computational efficiency. Numerical experiment is given to test the mass conservation for the C-N-type scheme as well as the accuracy order of the three schemes. In addition, the numerical simulation of binary collision and the influence on the solitary solution by adding a small random perturbation to the initial condition are also discussed.

In this paper, a variable-coefficient and nonisospectral Ablowitz–Kaup–Newell–Segur (vcniAKNS) hierarchy with Lax integrability is constructed by embedding a finite number of differentiable and time-dependent functions into the well-known AKNS spectral problem and its time evolution equation. In the framework of inverse scattering transform method with time-varying spectral parameter, the constructed vcniAKNS hierarchy is solved exactly. As a result, exact solutions and their reduced n -soliton solutions of the vcniAKNS hierarchy are obtained. It is graphically shown that the parity of an embedded time-dependent function has connection with the symmetrical characteristics of the spatial structures and singular points of the obtained one-soliton solutions.

During the process of aerodynamic shape design of supersonic and hypersonic space planes, laminar flow design and boundary layer transition prediction play important roles in aero-thermal numerical simulations and aero-thermal protection design. Therefore, in this study, a computational fluid dynamics compatible transition closure model for high speed laminar-to-turbulent transitional flows is formulated with consideration of the analysis results from stability theory. The proposed model contains two transport equations to describe the transition mechanism using local variables. Specifically, the eddy viscosity of laminar fluctuations and intermittency factor are chosen to be the characteristic parameters and modeled by transport equations. Accounting for the dominant instability modes at supersonic/hypersonic conditions, the first- and second- modes are modeled using local variables through the analysis of laminar self-similar boundary layers. Then, the present transition model is applied with compressibility corrected k$k$-ω$\omega$ shear stress transport turbulence model. Thus, as the main significance of the current work, the present model is enabled to capture the overshoot phenomena as well as predict the transition onset position. Finally, comparisons between the predictions using the present model and the wind tunnel experimental results of several well-documented flow cases are provided to validate the proposed transition turbulence model.

A novel macromodel based on Krylov subspace projection method for micromixers with serpentine channels is presented. The physical equations are discretized using Galerkin method. The orthogonal basis is obtained and the discrete matrix is assembled with Arnoldi procedure based on Krylov subspace projection. The obtained macromodel can be used to calculate the concentration of the sample at arbitrary location of serpentine micromixers. The maximal relative deviation is 2 % between macromodel and only numerical simulation. The computational efficiency of the macromodel will be improved significantly with the numbers of serpentine channels increasing. Simulation results demonstrated that the macromodel is flexible, effective and easily operated for rapid design and computation of serpentine micromixers.

Evaluating the order of accuracy (order) is an integral part of the development and application of numerical algorithms. Apart from theoretical functional analysis to place bounds on error estimates, numerical experiments are often essential for nonlinear problems to validate the estimates in a reliable answer. The common workflow is to apply the algorithm using successively finer temporal/spatial grid resolutions δi ${\delta _i}$, measure the error \isini ${\isin _i}$ in each solution against the exact solution, the order is then obtained as the slope of the line that fits (log\isini,logδi) $(\log {\isin _i}, \log {\delta _i})$. We show that if the problem has singularities like divergence to infinity or discontinuous jump, this common workflow underestimates the order if solution at regions around the singularity is used. Several numerical examples with different levels of complexity are explored. A simple one-dimensional theoretical model shows it is impossible to numerically evaluate the order close to singularity on uniform grids.

In this article, the numerical solutions of stochastic Volterra–Fredholm integral equations have been obtained by hybrid Legendre block-pulse functions (BPFs) and stochastic operational matrix. The hybrid Legendre BPFs are orthonormal and have compact support on [0,1)$[0, 1)$. The numerical results obtained by the above functions have been compared with those obtained by second kind Chebyshev wavelets. Furthermore, the results of the proposed computational method establish its accuracy and efficiency.

The positivity and stability of standard and fractional descriptor continuous-time linear and nonlinear systems are addressed. Necessary and sufficient conditions for the positivity of descriptor linear and sufficient conditions for nonlinear systems are established. Using an extension of Lyapunov method sufficient conditions for the stability of positive nonlinear systems are given. The considerations are extended to fractional nonlinear systems.

By using some new analytical techniques, modified inequalities and Mawhin’s continuation theorem of coincidence degree theory, some sufficient conditions are obtained for the boundedness of the solution and the existence of at least one positive almost periodic solution of a kind of two-species model of facultative mutualism with time delays. Further, the global asymptotic stability of the positive almost periodic solution of this model is also considered. Some examples and numerical simulations are also given to illustrate the main results of this paper.

This paper is concerned with the controllability issue of fractional semilinear evolution inclusions with noninstantaneous impulses. Using weak sequentially closed graph operators, we establish sufficient conditions to guarantee controllability results. We do not assume that the semigroup is compact or we do not assume a compactness-type condition on the multivalued function. Finally, two examples are given to illustrate our theory.

This article is concerned with a Leslie–Gower predator–prey system with the predator being harvested and the prey having a delay due to the gestation of prey species. By regarding the gestation delay as a bifurcation parameter, we first derive some sufficient conditions on the stability of positive equilibrium point and the existence of Hopf bifurcations basing on the local parametrization method for differential-algebra system. In succession, we also investigate the direction of Hopf bifurcations and the stability of bifurcating periodic solutions on the center manifold by employing the center manifold reduction for functional differential equations. Finally, to verify our theoretical predictions, several numerical simulations are given.

With employing the von Karman plate theory, and considering the linearly thickness variation in one direction, the bending problem of a rectangular magnetoelectroelastic plates with linear variable thickness is investigated. According to the Maxwell’s equations, when applying the magnetoelectric load on the plate’s surfaces and neglecting the in-plane electric and magnetic fields in thin plates, the electric and magnetic potentials varying along the thickness direction for the magnetoelectroelastic plates are determined. The nonlinear differential equations for magnetoelectroelastic plates with linear variable thickness are established based on the Hamilton’s principle. The Galerkin procedure is taken to translate a set of differential equations into algebraic equations. The numerical examples are presented to discuss the influences of the aspect ratio and span–thickness ratio on the nonlinear load–deflection curves for magnetoelectroelastic plates with linear variable thickness. In addition, the induced electric and magnetic potentials are also presented with the various values of the taper constants.

In this paper, we study the numerical method for a class of nonlinear singularly perturbed delay differential equations using parametric cubic spline. Quasilinearization process is applied to convert the nonlinear singularly perturbed delay differential equations into a sequence of linear singularly perturbed delay differential equations. When the delay is not sufficiently smaller order of the singular perturbation parameter, the approach of expanding the delay term in Taylor’s series may lead to bad approximation. To handle the delay term, we construct a special type of mesh in such a way that the term containing delay lies on nodal points after discretization. The parametric cubic spline is presented for solving sequence of linear singularly perturbed delay differential equations. The error analysis of the method is presented and shows second-order convergence. The effect of delay parameter on the boundary layer behavior of the solution is discussed with two test examples.

Complex fracture systems including natural fractures and hydraulic fractures exist in shale gas reservoir with fractured horizontal well development. The flow of shale gas is a multi-scale flow process from microscopic nanometer pores to macroscopic large fractures. Due to the complexity of seepage mechanism and fracture parameters, it is difficult to realize fine numerical simulation for fractured horizontal wells in shale gas reservoirs. Mechanisms of adsorption–desorption on the surface of shale pores, slippage and Knudsen diffusion in the nanometer pores, Darcy and non-Darcy seepage in the matrix block and fractures are considered comprehensively in this paper. Through fine description of the complex fracture systems after horizontal well fracturing in shale gas reservoir, the problems of conventional corner point grids which are inflexible, directional, difficult to geometrically discretize arbitrarily oriented fractures are overcome. Discrete fracture network model based on unstructured perpendicular bisection grids is built in the numerical simulation. The results indicate that the discrete fracture network model can accurately describe fracture parameters including length, azimuth and density, and that the influences of fracture parameters on development effect of fractured horizontal well can be finely simulated. Cumulative production rate of shale gas is positively related to fracture half-length, fracture segments and fracture conductivity. When total fracture length is constant, fracturing effect is better if single fracture half-length or penetration ratio is relatively large and fracturing segments are moderate. Research results provide theoretical support for optimal design of fractured horizontal well in shale gas reservoir.

From the last five decades, inverted pendulum (IP) has been considered as a benchmark problem in the control literature due to its inherit nature of instability, non-linearity and underactuation. Its applicability in wide range of practical systems, demands the need of a robust controller. It is found in the literature that wide range of controllers had been tested on this problem, out of which the most robust being sliding mode controller while the most optimal being linear quadratic regulator (LQR) controller. The former has a problem of discontinuity and chattering, while the latter lacks the property of robustness. To address the robustness issue in LQR controller, this paper proposes a novel robust LQR-based adaptive neural based fuzzy inference system controller, which is a hybrid of LQR and fuzzy inference system. The proposed controller is designed and implemented on rotary inverted pendulum. Further, to validate the robustness of proposed controller to parametric uncertainties, pendulum mass is varied. Simulation and experimental results show that as compared to LQR controller, the proposed controller is robust to variations in pendulum mass and has shown satisfactory performance.

This paper aims to study the extended generalized (2+1)-dimensional Jaulent–Miodek equation (egJM), which arises in a number of significant nonlinear problems of physics and applied mathematics. We derive conservation laws using Noether theorem and find travelling wave solution of the egJM equation.

We discuss the numerical modeling of infiltration in variably-saturated porous media. Richards’ equation is used to describe the infiltration towards the water table. It is difficult to accurately approximate its solution especially when we deal with layered soil due to its highly nonlinear fact. In this work, the nonlinearity is handled by using Gardner model. In the case of homogeneous soil, the linearized equation is solved using radial basis function partition of unity method (RBFPUM) with the introduction of QR factorization of Gaussian in order to enhance the numerical solution for small values of the so-called “shape parameter.” In the case of layered soil, domain decomposition principle is introduced. It is based on decomposing the general problem into many subproblems. The latter are solved by RBFPUM-QR and patched by using the Steklov–Poincaré equation. Infiltration towards water table in homogeneous and layered soil is considered as a numerical test.

We study an integral expression that is encountered in some classical spin models of magnetism. The idea is to calculate the key integral that represents the building block for the expression of the partition function of these models. The general calculation allows one to have a better look at the internal structure of the quantity of interest which, in turn, may lead to potentially new useful insights. We find out that application of two different approaches to solve the problem in a general-case scenario leads to an interesting integral formula involving modified Bessel functions of the first kind which appears to be new. We performed Monte Carlo simulations to verify the correctness of the integral formula obtained. Additional numerical integration tests lead to the same result as well. The approach under consideration, when generalized, leads to a linear integral equation that might be of interest to numerical studies of classical spin models of magnetism that rely on the well-established transfer-matrix formalism.

A symmetry analysis of steady two-dimensional boundary layer stagnation-point flow and heat transfer of viscous incompressible non-Newtonian power-law fluids over a nonlinearly shrinking/stretching sheet with thermal radiation effect is presented. Lie group of continuous symmetry transformations is employed to the boundary layer flow and heat transfer equations, that gives scaling laws and self-similar equations for a special type of shrinking/stretching velocity (cx1/3 $c{x^{1/3}}$) and free-stream straining velocity (ax1/3 $a{x^{1/3}}$) along the axial direction to the sheet. The self-similar equations are solved numerically using very efficient shooting method. For the above nonlinear velocities, the unique self-similar solution is obtained for straining velocity being always less than the shrinking/stretching velocity for Newtonian and non-Newtonian power-law fluids. The thickness of velocity boundary layer becomes thinner with power-law index for shrinking as well as stretching sheet cases. Also, the thermal boundary layer thickness decreases with increasing values the Prandtl number and the radiation parameter.