The sufficient conditions are obtained for the existence and uniqueness of continuous solution to the linear nonclassical Volterra equation that appears in the integral models of developing systems. The Volterra integral equations of the first kind with piecewise smooth kernels are considered. Illustrative examples are presented.

We propose a three-step block method of Adam’s type to solve nonlinear second-order two-point boundary value problems of Dirichlet type and Neumann type directly. We also extend this method to solve the system of second-order boundary value problems which have the same or different two boundary conditions. The method will be implemented in predictor corrector mode and obtain the approximate solutions at three points simultaneously using variable step size strategy. The proposed block method will be adapted with multiple shooting techniques via the three-step iterative method. The boundary value problem will be solved without reducing to first-order equations. The numerical results are presented to demonstrate the effectiveness of the proposed method.

Our effort is to develop a criterion on almost surely exponential stability of numerical solution to stochastic pantograph differential equations, with the help of the discrete semimartingale convergence theorem and the technique used in stable analysis of the exact solution. We will prove that the Euler-Maruyama (EM) method can preserve almost surely exponential stability of stochastic pantograph differential equations under the linear growth conditions. And the backward EM method can reproduce almost surely exponential stability for highly nonlinear stochastic pantograph differential equations. A highly nonlinear example is provided to illustrate the main theory.

We introduce positive linear operators which are combined with the Chlodowsky and Szász type operators and study some approximation properties of these operators in the space of continuous functions of two variables on a compact set. The convergence rate of these operators are obtained by means of the modulus of continuity. And we also obtain weighted approximation properties for these positive linear operators in a weighted space of functions of two variables and find the convergence rate for these operators by using the weighted modulus of continuity.

A predator-prey model with both cross diffusion and time delay is considered. We give the conditions for emerging Turing instability in detail. Furthermore, we illustrate the spatial patterns via numerical simulations, which show that the model dynamics exhibits a delay and diffusion controlled formation growth not only of spots and stripe-like patterns, but also of the two coexist. The obtained results show that this system has rich dynamics; these patterns show that it is useful for the diffusive predation model with a delay effect to reveal the spatial dynamics in the real model.

This paper deals with a new implementation of the Bernstein polynomials method to the numerical solution of a special kind of singular system. For this aim, first the truncated Bernstein series polynomials of the solution functions are substituted in the given problem. Using some properties of these polynomials, the solution of the problem is reduced to solve a linear system of algebraic equations. In order to confirm the reliability and accuracy of the proposed method, some weakly Abel integral equations systems with comparisons are solved in detail as numerical examples.

We consider the initial boundary value problem for the spherically symmetric isentropic compressible Navier-Stokes equations with density-dependent viscosity coefficients and discontinuous initial data in this paper. For piecewise regular initial density with bounded jump discontinuity, we show that there exists a unique global piecewise regular solution. In particular, the jump of density decays exponentially in time and the piecewise regular solution tends to the equilibrium state exponentially as $t\to +\infty$.

There are many published papers about fractional integrodifferential equations and system of fractional differential equations. The goal of this paper is to show that we can investigate more complicated ones by using an appropriate basic theory. In this way, we prove the existence and uniqueness of solution for a $k$-dimensional system of multiterm fractional integrodifferential equations with antiperiodic boundary conditions by applying some standard fixed point results. An illustrative example is also presented.

This paper proposes a kind of symplectic schemes for linear Schrödinger equations with variable coefficients and a stochastic perturbation term by using compact schemes in space. The numerical stability property of the schemes is analyzed. The schemes preserve a discrete charge conservation law. They also follow a discrete energy transforming formula. The numerical experiments verify our analysis.

We introduce a new approach called the enhanced multistage homotopy perturbation method (EMHPM) that is based on the homotopy perturbation method (HPM) and the usage of time subintervals to find the approximate solution of differential equations with strong nonlinearities. We also study the convergence of our proposed EMHPM approach based on the value of the control parameter h by following the homotopy analysis method (HAM). At the end of the paper, we compare the derived EMHPM approximate solutions of some nonlinear physical systems with their corresponding numerical integration solutions obtained by using the classical fourth order Runge-Kutta method via the amplitude-time response curves.

We study the analytic property of the (generalized) quadratic derivative Ginzburg-Landau equation $(1/2⩽\alpha ⩽1)$ in any spatial dimension $n⩾1$ with rough initial data. For , we prove the analyticity of local solutions to the (generalized) quadratic derivative Ginzburg-Landau equation with large rough initial data in modulation spaces $M_{p,1}^{1-2\alpha }(1⩽p⩽\infty )$. For $\alpha =1/2$, we obtain the analytic regularity of global solutions to the fractional quadratic derivative Ginzburg-Landau equation with small initial data in ${\dot{B}}_{\infty ,1}^0$ $({ℝ}^n)\cap M_{\infty ,1}^0({ℝ}^n)$. The strategy is to develop uniform and dyadic exponential decay estimates for the generalized Ginzburg-Landau semigroup $e^{-\left(a+i\right)t{\left(-\Delta \right)}^{\alpha }}$ to overcome the derivative in the nonlinear term.

This paper addresses some methods for interpretation of oil and gas well test data distorted by wellbore storage effects. Using these techniques, we can deconvolve pressure and rate data from drawdown and buildup tests dominated by wellbore storage. Some of these methods have the advantage of deconvolving the pressure data without rate measurement. The two important methods that are applied in this study are an explicit deconvolution method and a modification of material balance deconvolution method. In cases with no rate measurements, we use a blind deconvolution method to restore the pressure response free of wellbore storage effects. Our techniques detect the afterflow/unloading rate function with explicit deconvolution of the observed pressure data. The presented techniques can unveil the early time behavior of a reservoir system masked by wellbore storage effects and thus provide powerful tools to improve pressure transient test interpretation. Each method has been validated using both synthetic data and field cases and each method should be considered valid for practical applications.

We obtain the global existence of weak solutions for the Cauchy problem of the nonhomogeneous, resonant system. First, by using the technique given in Tsuge (2006), we obtain the uniformly bounded $L^{\mathrm{\infty }}$ estimates $z({\rho }^{\delta ,\epsilon },u^{\delta ,\epsilon })\le B(x)$ and $w({\rho }^{\delta ,\epsilon },u^{\delta ,\epsilon })\le \beta$ when $a(x)$ is increasing (similarly, $w({\rho }^{\delta ,\epsilon },u^{\delta ,\epsilon })\le B(x)$ and $z({\rho }^{\delta ,\epsilon },u^{\delta ,\epsilon })\le \beta$ when $a(x)$ is decreasing) for the $\epsilon$-viscosity and $\delta$-flux approximation solutions of nonhomogeneous, resonant system without the restriction $z_0(x)\le 0$ or $w_0(x)\le 0$ as given in Klingenberg and Lu (1997), where $z$ and $w$ are Riemann invariants of nonhomogeneous, resonant system; $B(x)>0$ is a uniformly bounded function of $x$ depending only on the function $a(x)$ given in nonhomogeneous, resonant system, and $\beta$ is the bound of $B(x)$. Second, we use the compensated compactness theory, Murat (1978) and Tartar (1979), to prove the convergence of the approximation solutions.

We study the blow-up and global solutions for a class of quasilinear parabolic problems with Robin boundary conditions. By constructing auxiliary functions and using maximum principles, the sufficient conditions for the existence of blow-up solution, an upper bound for the “blow-up time,” an upper estimate of the “blow-up rate,” the sufficient conditions for the existence of global solution, and an upper estimate of the global solution are specified.

The dynamics of a prey-predator system with a finite delay is investigated. We show that a sequence of Hopf bifurcations occurs at the positive equilibrium as the delay increases. By using the theory of normal form and center manifold, explicit expressions for determining the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions are derived.

The conjugate gradient method is an efficient method for solving large-scale nonlinear optimization problems. In this paper, we propose a nonlinear conjugate gradient method which can be considered as a hybrid of DL and WYL conjugate gradient methods. The given method possesses the sufficient descent condition under the Wolfe-Powell line search and is globally convergent for general functions. Our numerical results show that the proposed method is very robust and efficient for the test problems.

An impulsive control method is proposed to eliminate the rotor-stator rubbing based on the phase characteristic. The relation between the vibration energy and the phase difference suggests the starting point for controlling the rotor-stator rubbing by implementing impulse. When the contact between the rotor and the stator occurs, the impulse is implemented in $x$-direction and $y$-direction several times to avoid the rotor-stator rubbing. The practical feasibility of this approach is investigated by numerical simulations.

By employing known Guo-Krasnoselskii fixed point theorem, we investigate the eigenvalue interval for the existence and nonexistence of at least one positive solution of nonlinear fractional differential equation with integral boundary conditions.

We present a new version of the reproducing kernel Hilbert space method (RKHSM) for the solution of systems of fractional integrodifferential equations. In this approach, the solution is obtained as a convergent series with easily computable components. Several illustrative examples are given to demonstrate the effectiveness of the present method. The method described in this paper is expected to be further employed to solve similar nonlinear problems in fractional calculus.

We demonstrate the efficiency of reproducing kernel Hilbert space method on the seventh-order boundary value problems satisfying boundary conditions. These results have been compared with the results that are obtained by variational iteration method (VIM), homotopy perturbation method (HPM), Adomian decomposition method (ADM), variation of parameters method (VPM), and homotopy analysis method (HAM). Obtained results show that our method is very effective.

The problem of peristaltic nanofluid flow in an asymmetric channel in the presence of the second-order slip boundary condition was investigated in this paper. To the best of the authors’ knowledge, this parameter was here incorporated for the first time in such field of a peristaltic flow. The system governing the current flow was found as a set of nonlinear partial differential equations in the stream function, pressure gradient, nanoparticle concentration, and temperature distribution. Therefore, this system has been successfully solved exactly via a very effective procedure. These exact solutions were then proved to reduce to well-known results in the absence of second slip which were published very recently in the literature. Effect of the second slip parameter on the present physical parameters was discussed through graphs and it was found that this type of slip is a very important one to predict the investigated physical model. Moreover, the variation of many physical parameters such as amplitudes of the lower and upper waves, phase difference on the temperature distribution, nanoparticle concentration, pressure rise, velocity, and pressure gradient were also discussed. Finally, the present results may be viewed as an optimal choice for their dependence on the exact solutions which are obtained due to the highly complex nonlinear system.

The Cauchy problem for a simplified shallow elastic fluids model, one $3\times 3$ system of Temple’s type, is studied and a global weak solution is obtained by using the compensated compactness theorem coupled with the total variation estimates on the first and third Riemann invariants, where the second Riemann invariant is singular near the zero layer depth $\left(\rho =0\right)$. This work extends in some sense the previous works, (Serre, 1987) and (Leveque and Temple, 1985), which provided the global existence of weak solutions for $2\times 2$ strictly hyperbolic system and (Heibig, 1994) for $n\times n$ strictly hyperbolic system with smooth Riemann invariants.

This paper presents a computational approach for solving a class of nonlinear Volterra integro-differential equations of fractional order which is based on the Bernoulli polynomials approximation. Our method consists of reducing the main problems to the solution of algebraic equations systems by expanding the required approximate solutions as the linear combination of the Bernoulli polynomials. Several examples are given and the numerical results are shown to demonstrate the efficiency of the proposed method.

This paper is concerned with the stability of non-Fickian reaction-diffusion equations with a variable delay. It is shown that the perturbation of the energy function of the continuous problems decays exponentially, which provides a more accurate and convenient way to express the rate of decay of energy. Then, we prove that the proposed numerical methods are sufficient to preserve energy stability of the continuous problems. We end the paper with some numerical experiments on a biological model to confirm the theoretical results.

We construct the conservation laws for a variable coefficient variant Boussinesq system, which is a third-order system of two partial differential equations. This system does not have a Lagrangian and so we transform it to a system of fourth-order, which admits a Lagrangian. Noether’s approach is then utilized to obtain the conservation laws. Lastly, the conservation laws are presented in terms of the original variables. Infinite numbers of both local and nonlocal conserved quantities are derived for the underlying system.

We study a fractional differential model of HIV infection of ${\text{CD4}}^{\text{+}}$ T-cell, in which the ${\text{CD4}}^{\text{+}}$ T-cell proliferation plays an important role in HIV infection under antiretroviral therapy. An appropriate method is given to ensure that both the equilibria are asymptotically stable for $\tau \ge 0$. We calculate the basic reproduction number $R_0$, the IFE $E_0$, two IPEs $E_1^{*}$ and $E_2^{*}$, and so on, and judge the stability of the equilibrium. In addition, we describe the dynamic behaviors of the fractional HIV model by using the Adams-type predictor-corrector method algorithm. At last, we extend the model to incorporate the term which we consider the loss of virion and a bilinear term during attacking the target cells.

A parallel characteristic curve method is applied in domain decomposition system to simulate the dispersion behavior of hydrogen in this work. The characteristic curve method is employed to approximate the Navier-Stokes equations and the convection diffusion equation, and the feasibility of solving complex multicomponent flow problems is demonstrated by the numerical simulation of hydrogen dispersion in a partially open space. An analogy of the Boussinesq approximation is applied and numerical results are validated by comparing them with the experimental data. The dilution effect of ventilation is investigated. The transient behavior of hydrogen and the process of accumulation in the partially open space are discussed.

We made use of groundwater flow and mass transport equations to investigate the crucial potential risk of water pollution from hydraulic fracturing especially in the case of the Karoo system in South Africa. This paper shows that the upward migration of fluids will depend on the apertures of the cement cracks and fractures in the rock formation. The greater the apertures, the quicker the movement of the fluid. We presented a novel sampling method, which is the combination of the Monte Carlo and the Latin hypercube sampling. The method was used for uncertainties analysis of the apertures in the groundwater and mass transport equations. The study reveals that, in the case of the Karoo, fracking will only be successful if and only if the upward methane and fracking fluid migration can be controlled, for example, by plugging the entire fracked reservoir with cement.

Common coupled fixed point theorems are examined in this paper for comparable mappings ensuring nonlinear contraction in ordered partial metric spaces. Given theorems enlarge and universalize some conclusions of Gnana Bhaskar and Lakshmikantham (2006).

We apply a lumped mass finite element to approximate Dirichlet problems for nonsmooth elliptic equations. It is proved that the lumped mass FEM approximation error in energy norm is the same as that of standard piecewise linear finite element approximation. Under the quasi-uniform mesh condition and the maximum angle condition, we show that the operator in the finite element problem is diagonally isotone and off-diagonally antitone. Therefore, some monotone convergent algorithms can be used. As an example, we prove that the nonsmooth Newton-like algorithm is convergent monotonically if Gauss-Seidel iteration is used to solve the Newton's equations iteratively. Some numerical experiments are presented.

The IVPs with local fractional derivative are considered in this paper. Analytical solutions for the homogeneous and nonhomogeneous local fractional differential equations are discussed by using the Yang-Laplace transform.

We propose the local fractional function decomposition method, which is derived from the coupling method of local fractional Fourier series and Yang-Laplace transform. The forms of solutions for local fractional differential equations are established. Some examples for inhomogeneous wave equations are given to show the accuracy and efficiency of the presented technique.