Numerical solution of the modified equal width wave equation is obtained by using lumped Galerkin method based on cubic B-spline finite element method. Solitary wave motion and interaction of two solitary waves are studied using the proposed method. Accuracy of the proposed method is discussed by computing the numerical conserved laws $L_2$ and $L_{\infty }$ error norms. The numerical results are found in good agreement with exact solution. A linear stability analysis of the scheme is also investigated.

We consider the steady-state two-dimensional motion of an inviscid incompressible fluid which obeys a nonlinear Poisson equation. By seeking solutions of a specific form, we arrive at some interesting new nonlinear vortex solutions.

The theory of control analyzes the proprieties of commanded systems. Problems of optimal control (OC) have been intensively investigated in the world literature for over forty years. During this period, series of fundamental results have been obtained, among which should be noted the maximum principle (Pontryagin et al., 1962) and dynamic programming (Bellman, 1963). For many of the problems of the optimal control theory (OCT), adequate solutions are found (Bryson and Yu-chi, 1969, Lee and Markus, 1967, Gabasov and Kirillova, 1977, 1978, 1980). Results of the theory were taken up in various fields of science, engineering, and economics. The present paper aims at extending the constructive methods of Balashevich et al., (2000) that were developed for the problems of optimal control with the bounded initial state is not fixed are considered.

Unsteady flow in a collapsible tube is analyzed to simulate a diseased human coronary artery. The novelty of the approach is that the set of equations governing the fluid-structure interaction is reduced to a single integrodifferential equation in the transient state. The equation is then solved using the finite difference method to obtain the flow characteristics and compliant wall behavior. Three control parameters are investigated, namely, Reynolds number, inlet transmural pressure, and the wall thickness. The predicted wall deflection is quite large at low Reynolds numbers, suggesting possible approach to breakdown in equilibrium. The transmural pressure increases with wall deflection and bulges appear at the ends of the membrane indicating critical stage of stability, consistent with previous studies. Increase in wall thickness reduces the wall deflection and ultimately results in its collapse which may indicate another breakdown in equilibrium. An increase in internal pressure is required to maintain membrane stability.

This work is devoted to the study of uniqueness and existence of positive solutions for a second-order boundary value problem with integral condition. The arguments are based on Banach contraction principle, Leray Schauder nonlinear alternative, and Guo-Krasnosel’skii fixed point theorem in cone. Two examples are also given to illustrate the main results.

We prove that a transversely holomorphic foliation, which is transverse to the fibers of a fibration, is a Seifert fibration if the set of compact leaves is not a zero measure subset. Similarly, we prove that a finitely generated subgroup of holomorphic diffeomorphisms of a connected complex manifold is finite provided that the set of periodic orbits is not a zero measure subset.

We establish certain new sufficient conditions which guarantee the existence of periodic solutions for a nonlinear differential equation of the third order with multiple deviating arguments. Using the Lyapunov functional approach, we prove a specific theorem and provide an example to illustrate the theoretical analysis in this work and the effectiveness of the method utilized here.

We utilize the improved Riccati equation method to construct more general exact solutions to nonlinear equations. And we obtain the travelling wave solutions involving parameters, which are expressed by the hyperbolic functions, the trigonometric functions, and the rational functions. When the parameters are taken as special values, the method provides not only solitary wave solutions but also periodic waves solutions. The method appears to be easier and more convenient by means of a symbolic computation system. Of course, it is also effective to solve other nonlinear evolution equations in mathematical physics.

We study the existence of multiple nonnegative solutions for the doubly singular three-point boundary value problem with derivative dependent data function . Here, $p\in C[\mathrm{0,1}]\cap C^1(\mathrm{0,1}]$ with $p(t)>0$ on $(\mathrm{0,1}]$ and $q(t)$ is allowed to be discontinuous at $t=0$. The fixed point theory in a cone is applied to achieve new and more general results for existence of multiple nonnegative solutions of the problem. The results are illustrated through examples.

We prove a higher-order inequality of Hardy type for functions in anisotropic Sobolev spaces that vanish at the boundary of the space domain. This is an important calculus tool for the study of initial-boundary-value problems of symmetric hyperbolic systems with characteristic boundary.

A well-known mathematical model of radially symmetric tumour growth is revisited in the present work. Under this aim, a cancerous spherical mass lying in a finite concentric nutritive surrounding is considered. The host spherical shell provides the tumor with vital nutrients, receives the debris of the necrotic cancer cells, and also transmits to the tumour the pressure imposed on its exterior boundary. We focus on studying the type of inhomogeneity that the nutrient supply and the pressure field imposed on the host exterior boundary, can exhibit in order for the spherical structure to be supported. It turns out that, if the imposed fields depart from being homogeneous, only a special type of interrelated inhomogeneity between nutrient and pressure can secure the spherical growth. The work includes an analytic derivation of the related boundary value problems based on physical conservation laws and their analytical treatment. Implementations in cases of special physical interest are examined, and also existing homogeneous results from the literature are fully recovered.

We consider a nonlinear 4th-order degenerate parabolic partial differential equation that arises in modelling the dynamics of an incompressible thin liquid film on the outer surface of a rotating horizontal cylinder in the presence of gravity. The parameters involved determine a rich variety of qualitatively different flows. We obtain sufficient conditions for finite speed of support propagation and for waiting time phenomena by application of a new extension of Stampacchia's lemma for a system of functional equations.

A nonlinear differential equation for the polar angle of a point of an ellipse is derived. The solution of this differential equation can be expressed in terms of the Jacobi elliptic function dn(u,k). If the polar angle is extended to the complex plane, the Jacobi imaginary transformation properties and the dependence on the real and complex quarter periods can be described. From the differential equation of the polar angle, exact solutions of the Poisson Boltzmann and the sinh-Poisson equations are found in terms of the Jacobi elliptic functions.

We investigate solutions of $w^{\prime \prime }+\left(\left(N-1\right)/r\right)w^{\prime }+{\left|w\right|}^{p-1}w=0,\hspace{0.17em}r>0$ and focus on the regime $N>2$ and $p>N/(N-2)$. Our advance is to develop a technique to efficiently classify the behavior of solutions on $(r_{\text{min}},r_{\text{max}})$, their maximal positive interval of existence. Our approach is to transform the nonautonomous $w$ equation into an autonomous ODE. This reduces the problem to analyzing the phase plane of the autonomous equation. We prove the existence of new families of solutions of the $w$ equation and describe their asymptotic behavior. In the subcritical case there is a well-known closed-form singular solution, $w_1$, such that $w_1(r)\to \infty$ as $r\to 0^{+}$ and $w_1(r)\to 0$ as $r\to \infty$. Our advance is to prove the existence of a family of solutions of the subcritical case which satisfies$w(r_i)=w_1(r_i)$ for infinitely many values $r_i>0$. At the critical value $p=(N+2)/(N-2)$ there is a continuum of positive singular solutions, and a continuum of sign changing singular solutions. In the supercritical regime $p>(N+2)/(N-2)$ we prove the existence of a family of “super singular” sign changing singular solutions.

I study the geometric notion of a differential system describing surfaces of a constant negative curvature and describe a family of pseudospherical surfaces for the nonlinear partial differential equations with constant Gaussian curvature $-1$.

We have presented a numerical integration method to solve a class of singularly perturbed delay differential equations with small shift. First, we have replaced the second-order singularly perturbed delay differential equation by an asymptotically equivalent first-order delay differential equation. Then, Simpson’s rule and linear interpolation are employed to get the three-term recurrence relation which is solved easily by discrete invariant imbedding algorithm. The method is demonstrated by implementing it on several linear and nonlinear model examples by taking various values for the delay parameter $\delta$ and the perturbation parameter $\epsilon$.

We consider a delayed SIR epidemic model in which the susceptibles are assumed to satisfy the logistic equation and the incidence term is of saturated form with the susceptible. We investigate the qualitative behaviour of the model and find the conditions that guarantee the asymptotic stability of corresponding steady states. We present the conditions in the time lag $\tau$ in which the DDE model is stable. Hopf bifurcation analysis is also addressed. Numerical simulations are provided in order to illustrate the theoretical results and gain further insight into the behaviour of this system.