The present paper aims to compare Harrington's direct method of moment (MoM) with the conjugate gradient method (CGM) by evaluating the total current solving the electric field integral equation (EFIE). Based on their performances, the number of iterations needed for convergence, storage, and the level of precision, it is found that the direct MoM is more efficient than other iterative CGM.

By using the bifurcation theory of dynamical systems, we study the coupled Higgs field equation and the existence of new solitary wave solutions, and uncountably infinite many periodic wave solutions are obtained. Under different parametric conditions, various sufficient conditions to guarantee the existence of the above solutions are given. All exact explicit parametric representations of the above waves are determined.

Based on Lyapunov-Krasovskii functional method and stochastic analysis theory, we obtain some new delay-dependent criteria ensuring mean square stability of a class of impulsive stochastic equations. Numerical examples are given to illustrate the effectiveness of the theoretical results.

We use homotopy perturbation method (HPM) to handle the foam drainage equation. Foaming occurs in many distillation and absorption processes. The drainage of liquid foams involves the interplay of gravity, surface tension, and viscous forces. The concept of He's homotopy perturbation method is introduced briefly for applying this method for problem solving. The results of HPM as an analytical solution are then compared with those derived from Adomian's decomposition method (ADM) and the variational iteration method (VIM). The results reveal that the HPM is very effective and convenient in predicting the solution of such problems, and it is predicted that HPM can find a wide application in new engineering problems.

A fractional boundary value problem is considered. By means of Banach contraction principle, Leray-Schauder nonlinear alternative, properties of the Green function, and Guo-Krasnosel'skii fixed point theorem on cone, some results on the existence, uniqueness, and positivity of solutions are obtained.

Due to the well-known Srivastava-Attiya operator, we investigate here some results relating the p-valent of the operator with differential subordination and subordination. Further, we obtain some interesting results on sandwich-type theorem for the same.

We obtain new result of the existence of positive solutions of a class of singular impulse periodic boundary value problem via a nonlinear alternative principle of Leray-Schauder. We do not require the monotonicity of functions in paper (Zhang and Wang, 2003). An example is also given to illustrate our result.

The author discusses the multiple positive solutions for an infinite boundary value problem of first-order impulsive superlinear integrodifferential equations on the half line in a Banach space by means of the fixed point theorem of cone expansion and compression with norm type.

We study a nonlinear parabolic system governing the biological dynamic in the soil. We prove global existence (in time) and uniqueness of weak and positive solution for this reaction-diffusion semilinear system in a bounded domain, completed with homogeneous Neumann boundary conditions and positive initial conditions.

This paper is concerned with linear thermoelastic systems defined in domains with moving boundary. The uniform rate of decay of the energy associated is proved.

The extended Jacobi elliptic function expansion method is applied for Zakharov-Kuznetsov-modified equal-width (ZK-MEW) equation. With the aid of symbolic computation, we construct some new Jacobi elliptic doubly periodic wave solutions and the corresponding solitary wave solutions and triangular functional (singly periodic) solutions.

We obtain a global existence result for the higher-dimensional thermoviscoelastic equations. Using semigroup approach, we will establish the global existence of homogeneous, nonhomogeneous, linear, semilinear, and nonlinear, thermoviscoelastic systems.

By using the fixed point theorem on cone, some sufficient conditions are obtained on the existence of positive periodic solutions for a class of $n$-species competition systems with impulses. Meanwhile, we point out that the conclusion of (Yan, 2009) is incorrect.

We consider the following class of nonlinear singular differential equation subject to the Neumann boundary condition $y\text{'}\left(0\right)=y\text{'}\left(1\right)=0$. Conditions on $p\left(x\right)$ and $q\left(x\right)$ ensure that $x=0$ is a singular point of limit circle type. A simple approximation scheme which is iterative in nature is considered. The initial iterates are upper and lower solutions which can be ordered in one way $(v_0\le u_0)$ or the other $(u_0\le v_0)$.

We establish the existence of ground states on ${ℝ}^N$ for the Laplace operator involving the Hardy-type potential. This gives rise to the existence of the principal eigenfunctions for the Laplace operator involving weighted Hardy potentials. We also obtain a higher integrability property for the principal eigenfunction. This is used to examine the behaviour of the principal eigenfunction around 0.

We consider a thermoelastic diffusion problem in one space dimension with second sound. The thermal and diffusion disturbances are modeled by Cattaneo's law for heat and diffusion equations to remove the physical paradox of infinite propagation speed in the classical theory within Fourier's law. The system of equations in this case is a coupling of three hyperbolic equations. It poses some new analytical and mathematical difficulties. The exponential stability of the slightly damped and totally hyperbolic system is proved. Comparison with classical theory is given.

The estimate of the upper bounds of eigenvalues for a class of systems of ordinary differential equations with higher order is considered by using the calculus theory. Several results about the upper bound inequalities of the ($n+1$)th eigenvalue are obtained by the first $n$ eigenvalues. The estimate coefficients do not have any relation to the geometric measure of the domain. This kind of problem is interesting and significant both in theory of systems of differential equations and in applications to mechanics and physics.

This paper is concerned with the oscillatory behavior of the second-order half-linear advanced dynamic equation ${(r(t){(x^{\mathrm{\Delta }}(t))}^{\gamma })}^{\mathrm{\Delta }}+p(t)x^{\gamma }(g(t))=0$ on an arbitrary time scale $𝕋$ with sup $𝕋=\infty$, where $g(t)\ge t$ and . Some sufficient conditions for oscillation of the studied equation are established. Our results not only improve and complement those results in the literature but also unify the oscillation of the second-order half-linear advanced differential equation and the second-order half-linear advanced difference equation. Three examples are included to illustrate the main results.

We study the Camassa-Holm (CH) equation and recently introduced μCH equation from the geometric point of view. We show that Kupershmidt deformations of these equations describe pseudospherical surfaces and hence are geometrically integrable.

The present work deals with applying the homotopy perturbation method to the problem of the nonlinear oscillations of multiwalled carbon nanotubes embedded in an elastic medium under various boundary conditions. A multiple-beam model is utilized in which the governing equations of each layer are coupled with those of its adjacent ones via the van der Waals interlayer forces. The amplitude-frequency curves for large-amplitude vibrations of single-walled, double-walled, and triple-walled carbon nanotubes are obtained. The influences of some commonly used boundary conditions, changes in material constant of the surrounding elastic medium, and variations of the nanotubes geometrical parameters on the vibration characteristics of multiwalled carbon nanotubes are discussed. The comparison of the generated results with those from the open literature illustrates that the solutions obtained are of very high accuracy and clarifies the capability and the simplicity of the present method. It is worthwhile to say that the results generated are new and can be served as a benchmark for future works.

We consider the existence of positive solutions for the Neumann boundary value problem $x^{\mathrm{\prime \prime }}\left(t\right)+m^2\left(t\right)x\left(t\right)=f\left(t,x\left(t\right)\right)+e\left(t\right),t\in \left(0,1\right),x^{\prime }\left(0\right)=0,x^{\prime }\left(1\right)=0$, where $m\in C\left(\right[\mathrm{0,1}],(0,+\infty \left)\right),e\in C[\mathrm{0,1}],$ and $f:\left[\mathrm{0,1}\right]\times (0,+\infty )\to [0,+\infty )$ is continuous. The theorem obtained is very general and complements previous known results.

A mathematical model is proposed to study the role of distributed delay on plankton ecosystem in the presence of a toxic producing phytoplankton. The model includes three state variables, namely, nutrient concentration, phytoplankton biomass, and zooplankton biomass. The release of toxic substance by phytoplankton species reduces the growth of zooplankton and this plays an important role in plankton dynamics. In this paper, we introduce a delay (time-lag) in the digestion of nutrient by phytoplankton. The stability analysis of all the feasible equilibria are studied and the existence of Hopf-bifurcation for the interior equilibrium of the system is explored. From the above analysis, we observe that the supply rate of nutrient and delay parameter play important role in changing the dynamical behaviour of the underlying system. Further, we have derived the explicit algorithm which determines the direction and the stability of Hopf-bifurcation solution. Finally, numerical simulation is carried out to support the theoretical result.

We use generalized differential transform method (GDTM) to derive the solution of space-time fractional telegraph equation in closed form. The space and time fractional derivatives are considered in Caputo sense and the solution is obtained in terms of Mittag-Leffler functions.

We obtain some $W^{2,2}$ a priori bounds for a class of uniformly elliptic second-order differential operators, both in a no-weighted and in a weighted case. We deduce a uniqueness and existence theorem for the related Dirichlet problem in some weighted Sobolev spaces on unbounded domains.

We study the existence of periodic solutions for n-th order functional differential equations $x^{(n)}(t)={\displaystyle {\sum }_{i=0}^{n-1}{b}_i{[x^{(i)}(t)]}^k+f(x(t-\tau (t)))+p(t)}$. Some new results on the existence of periodic solutions of the equations are obtained. Our approach is based on the coincidence degree theory of Mawhin.

The object of investigations is a system of impulsive differential equations with “supremum.” These equations are not widely studied yet, and at the same time they are adequate mathematical model of many real world processes in which the present state depends significantly on its maximal value on a past time interval. Practical stability for a nonlinear system of impulsive differential equations with “supremum” is defined and studied. It is applied Razumikhin method with piecewise continuous scalar Lyapunov functions and comparison results for scalar impulsive differential equations. To unify a variety of stability concepts and to offer a general framework for the investigation of the stability theory, the notion of stability in terms of two measures has been applied to both the given system and the comparison scalar equation. An example illustrates the usefulness of the obtained sufficient conditions.

This paper is concerned with the existence and uniqueness of a mild solution of a semilinear fractional-order functional evolution differential equation with the infinite delay and impulsive effects. The existence and uniqueness of a mild solution is established using a solution operator and the classical fixed-point theorems.

This paper is concerned with a time-varying fishing model with delay. By means of the continuation theorem of coincidence degree theory, we prove that it has at least one positive almost periodic solution.

This paper is concerned with time-periodic solution of the weakly dissipative Camassa-Holm equation with a periodic boundary condition. The existence and uniqueness of a time periodic solution is presented.

We consider a kind of Sturm-Liouville boundary value problems. Using variational techniques combined with the methods of upper-lower solutions, the existence of at least one positive solution is established. Moreover, the upper solution and the lower solution are presented.

A fractional order time-independent form of the wave equation or diffusion equation in two dimensions is obtained from the standard time-independent form of the wave equation or diffusion equation in two-dimensions by replacing the integer order partial derivatives by fractional Riesz-Feller derivative and Caputo derivative of order and respectively. In this paper, we derive an analytic solution for the fractional time-independent form of the wave equation or diffusion equation in two dimensions in terms of the Mittag-Leffler function. The solutions to the fractional Poisson and the Laplace equations of the same kind are obtained, again represented by means of the Mittag-Leffler function. In all three cases, the solutions are represented also in terms of Fox's $H$-function.

We employ the bifurcation method of planar dynamical systems and qualitative theory of polynomial differential systems to derive new bounded traveling-wave solutions for a variant of the $K(3,2)$ equation. For the focusing branch, we obtain hump-shaped and valley-shaped solitary-wave solutions and some periodic solutions. For the defocusing branch, the nonexistence of solitary traveling wave solutions is shown. Meanwhile, some periodic solutions are also obtained. The results presented in this paper supplement the previous results.

The stationary Boussinesq equations describing the heat transfer in the viscous heat-conducting fluid under inhomogeneous Dirichlet boundary conditions for velocity and mixed boundary conditions for temperature are considered. The optimal control problems for these equations with tracking-type functionals are formulated. A local stability of the concrete control problem solutions with respect to some disturbances of both cost functionals and state equation is proved.

We study the Navier-Stokes-Nernst-Planck-Poisson system modeling the flow of electrohydrodynamics. For small initial data, the global existence, uniqueness, and asymptotic stability as time goes to infinity of self-similar solutions to the Cauchy problem of this system posed in the whole three dimensional space are proved in the function spaces of pseudomeasure type.

We improve some results of Pan and Xing (2008) and extend the exponent range in Liouville-type theorems for some parabolic systems of inequalities with the time variable on $\mathbb{R}$. As an immediate application of the parabolic Liouville-type theorems, the range of the exponent in blow-up rates for the corresponding systems is also improved.

Carlson's type theorem on removable sets for $\alpha$-Holder continuous solutions is investigated for the quasilinear elliptic equations $\text{div}\hspace{0.17em}A(x,u,\nabla u)=0,$ having degeneration $\omega$ in the Muckenhoupt class. In partial, when $\alpha$ is sufficiently small and the operator is weighted $p$-Laplacian, we show that the compact set $E$ is removable if and only if the Hausdorff measure ${\Lambda }_{\omega }^{-p+\left(p-1\right)\alpha }\left(E\right)=0$.

A SI-type ecoepidemiological model that incorporates reproduction delay of predator is studied. Considering delay as parameter, we investigate the effect of delay on the stability of the coexisting equilibrium. It is observed that there is stability switches, and Hopf bifurcation occurs when the delay crosses some critical value. By applying the normal form theory and the center manifold theorem, the explicit formulae which determine the stability and direction of the bifurcating periodic solutions are determined. Computer simulations have been carried out to illustrate different analytical findings. Results indicate that the Hopf bifurcation is supercritical and the bifurcating periodic solution is stable for the considered parameter values. It is also observed that the quantitative level of abundance of system populations depends crucially on the delay parameter if the reproduction period of predator exceeds the critical value.

A class of nonautonomous two-species competitive system with stage structure and impulse is considered. By using the continuation theorem of coincidence degree theory, we derive a set of easily verifiable sufficient conditions that guarantee the existence of at least a positive periodic solution, and, by constructing a suitable Lyapunov functional, the uniqueness and global attractivity of the positive periodic solution are presented. Finally, an illustrative example is given to demonstrate the correctness of the obtained results.

We present a constructive approach to establish existence and uniqueness of solution of singular boundary value problem $-(p(x)y^{\prime }(x){)}^{\prime }=q(x)f(x,y,p{y}^{\prime })$ for Here $p(x)>0$ on $(0,b)$ allowing $p(0)=0$. Further $q(x)$ may be allowed to have integrable discontinuity at $x=0$, so the problem may be doubly singular.

In this paper, we construct and verify the asymptotic expansion for the spectrum of a boundary-value problem in a unit circle periodically perforated along the boundary. It is assumed that the size of perforation and the distance to the boundary of the circle are of the same smallness. As an application of the obtained results, the asymptotic behavior of the best constant in a Friedrichs-type inequality is investigated.

We deal with fully nonlinear second-order equations assuming a superlinear growth in u with the aim to generalize previous existence and uniqueness results of viscosity solutions in the whole space without conditions at infinity. We also consider the solvability of the Dirichlet problem in bounded and unbounded domains and show a blow-up result.

A new iterative method introduced by Daftardar-Gejji and Jafari (2006) (DJ Method) is an efficient technique to solve nonlinear functional equations. In the present paper, sufficiency conditions for convergence of DJM have been presented. Further equivalence of DJM and Adomian decomposition method is established.

This paper contains a surprisingly large amount of material and indeed can serve as an introduction to some of ideas and methods of singular perturbation theory. The work done in this area during the periods 1984–2000 and 2000–2005 has already been surveyed in 2002 and 2007 but our main objective is to produce a collection of important research articles of physical significance. In this paper, the crux of research articles published by numerous researchers during 2006–2010 in referred journals has been presented, and this leads to conclusions and recommendations about what methods to use on singular perturbation problems.

We consider in this paper an abstract parabolic backward Cauchy problem associated with an unbounded linear operator in a Hilbert space $H$, where the coefficient operator in the equation is an unbounded self-adjoint positive operator which has a continuous spectrum and the data is given at the final time $t=T$ and a solution for is sought. It is well known that this problem is illposed in the sense that the solution (if it exists) does not depend continuously on the given data. The method of regularization used here consists of perturbing both the equation and the final condition to obtain an approximate nonlocal problem depending on two small parameters. We give some estimates for the solution of the regularized problem, and we also show that the modified problem is stable and its solution is an approximation of the exact solution of the original problem. Finally, some other convergence results including some explicit convergence rates are also provided.

We investigate a fractional diffusion/anti-diffusion equation proposed by Andrew C. Fowler to describe the dynamics of sand dunes sheared by a fluid flow. In this paper, we prove the global-in-time well-posedness in the neighbourhood of travelling-waves solutions of the Fowler equation.

We prove the existence of weak solution to a semilinear boundary value problem without the Landesman-Lazer condition.

We consider the dead-core problem for the fast diffusion equation with spatially dependent coefficient and obtain precise estimates on the single-point final dead-core profile. The proofs rely on maximum principle and require much delicate computation.

A high-order iterative scheme is established in order to get a convergent sequence at a rate of order $N$ ($N\ge 1$) to a local unique weak solution of a nonlinear Kirchhoff wave equation in the unit membrane. This extends a recent result in (EJDE, 2005, No. 138) where a recurrent sequence converges at a rate of order 2.

We are interested in the existence of solutions for Dirichlet problem associated to the degenerate quasilinear elliptic equations $-{\sum }_{j=1}^n{D}_j\left[{\omega }_2\right(x\left){\mathcal{A}}_j\right(x,u,\nabla u\left)\right]+{\omega }_1\left(x\right)g(x,u(x\left)\right)+H(x,u,\nabla u){\omega }_2\left(x\right)=$$f\left(x\right),\text{on}\Omega$ in the setting of the weighted Sobolev spaces ${\text{W}}_0^{1,p}(\Omega ,{\omega }_1,{\omega }_2)$.

Advection equations appear often in large-scale mathematical models arising in many fields of science and engineering. The Crank-Nicolson scheme can successfully be used in the numerical treatment of such equations. The accuracy of the numerical solution can sometimes be increased substantially by applying the Richardson Extrapolation. Two theorems related to the accuracy of the calculations will be formulated and proved in this paper. The usefulness of the combination consisting of the Crank-Nicolson scheme and the Richardson Extrapolation will be illustrated by numerical examples.

We study stochastic partly dissipative lattice systems with random coupled coefficients and multiplicative/additive white noise in a weighted space of infinite sequences. We first show that these stochastic partly dissipative lattice differential equations generate a random dynamical system. We then establish the existence of a tempered random bounded absorbing set and a global compact random attractor for the associated random dynamical system.

We study a system of infinitely many Riccati equations that arise from a cumulant control problem, which is a generalization of regulator problems, risk-sensitive controls, minimal cost variance controls, and $k$-cumulant controls. We obtain estimates for the existence intervals of solutions of the system. In particular, new existence conditions are derived for solutions on the horizon of the cumulant control problem.