We consider the free boundary problem for current-vortex sheets in ideal incompressible magnetohydrodynamics. The problem of current-vortex sheets arises naturally, for instance, in geophysics and astrophysics. We prove the existence of a unique solution to the constant-coefficient linearized problem and an a priori estimate with no loss of derivatives. This is a preliminary result to the study of linearized variable-coefficient current-vortex sheets, a first step to prove the existence of solutions to the nonlinear problem.

We prove the existence and uniqueness of entropy solution for nonlinear anisotropic elliptic equations with Neumann homogeneous boundary value condition for $L^1$-data. We prove first, by using minimization techniques, the existence and uniqueness of weak solution when the data is bounded, and by approximation methods, we prove a result of existence and uniqueness of entropy solution.

We review recent results on the homogenization in Sobolev spaces with variable exponents. In particular, we are dealing with the Γ-convergence of variational functionals with rapidly oscillating coefficients, the homogenization of the Dirichlet and Neumann variational problems in strongly perforated domains, as well as double porosity type problems. The growth functions also depend on the small parameter characterizing the scale of the microstructure. The homogenization results are obtained by the method of local energy characteristics. We also consider a parabolic double porosity type problem, which is studied by combining the variational homogenization approach and the two-scale convergence method. Results are illustrated with periodic examples, and the problem of stability in homogenization is discussed.

The main feature of the boundary layer flow problems of nanofluids or classical fluids is the inclusion of the boundary conditions at infinity. Such boundary conditions cause difficulties for any of the series methods when applied to solve such a kind of problems. In order to solve these difficulties, the authors usually resort to either Padé approximants or the commercial numerical codes. However, an intensive work is needed to perform the calculations using Padé technique. Due to the importance of the nanofluids flow as a growing field of research and the difficulties caused by using Padé approximants to solve such problems, a suggestion is proposed in this paper to map the semi-infinite domain into a finite one by the help of a transformation. Accordingly, the differential equations governing the fluid flow are transformed into singular differential equations with classical boundary conditions which can be directly solved by using the differential transformation method. The numerical results obtained by using the proposed technique are compared with the available exact solutions, where excellent accuracy is found. The main advantage of the present technique is the complete avoidance of using Padé approximants to treat the infinity boundary conditions.

We discuss the continuity of analytic resolvent in the uniform operator topology and then obtain the compactness of Cauchy operator by means of the analytic resolvent method. Based on this result, we derive the existence of mild solutions for nonlocal fractional differential equations when the nonlocal item is assumed to be Lipschitz continuous and neither Lipschitz nor compact, respectively. An example is also given to illustrate our theory.

We investigate the properties of a general class of differential equations described by $dy(t)/dt=f_{k+\mathrm{1}}(t)y(t{)}^{k+\mathrm{1}}+f_k(t)y(t{)}^k+\cdots +f_{\mathrm{2}}(t)y(t{)}^{\mathrm{2}}+f_{\mathrm{1}}(t)y(t)+f_{\mathrm{0}}(t),$ with $k>\mathrm{1}$ a positive integer and $f_i(t),\mathrm{\hspace{0.17em}0}\le i\le k+\mathrm{1}$, with $f_i(t)$, real functions of $t$. For $k=\mathrm{2}$, these equations reduce to the class of Abel differential equations of the first kind, for which a standard solution procedure is available. However, for $k>\mathrm{2}$ no general solution methodology exists, to the best of our knowledge, that can lead to their solution. We develop a general solution methodology that for odd values of $k$ connects the closed form solution of the differential equations with the existence of closed-form expressions for the roots of the polynomial that appears on the right-hand side of the differential equation. Moreover, the closed-form expression (when it exists) for the polynomial roots enables the expression of the solution of the differential equation in closed form, based on the class of Hyper-Lambert functions. However, for certain even values of $k$, we prove that such closed form does not exist in general, and consequently there is no closed-form expression for the solution of the differential equation through this methodology.

We prove that a discrete semigroup $𝕋=\{T(n):n\in {\mathbb{Z}}_{+}\}$ of bounded linear operators acting on a complex Banach space $X$ is uniformly exponentially stable if and only if, for each $x\in A{P}_0({\mathbb{Z}}_{+},X)$, the sequence $n\mapsto {\sum }_{k=0}^n\mathrm{‍}T(n-k)x(k):{\mathbb{Z}}_{+}\to X$ belongs to $A{P}_0({\mathbb{Z}}_{+},X)$. Similar results for periodic discrete evolution families are also stated.

We consider the following class of three point boundary value problem $y\prime \prime (t)+f(t,y)=0$, $y\prime (0)=0,y(1)=\delta y(\eta )$, where $\delta >0$, , the source term $f(t,y)$ is Lipschitz and continuous. We use monotone iterative technique in the presence of upper and lower solutions for both well-order and reverse order cases. Under some sufficient conditions, we prove some new existence results. We use examples and figures to demonstrate that monotone iterative method can efficiently be used for computation of solutions of nonlinear BVPs.

A linear different operator $L$ is called weakly hypoelliptic if any local solution $u$ of $Lu=\mathrm{0}$ is smooth. We allow for systems, that is, the coefficients may be matrices, not necessarily of square size. This is a huge class of important operators which coverall elliptic, overdetermined elliptic, subelliptic, and parabolic equations. We extend several classical theorems from complex analysis to solutions of any weakly hypoelliptic equation: the Montel theorem providing convergent subsequences, the Vitali theorem ensuring convergence of a given sequence, and Riemann's first removable singularity theorem. In the case of constant coefficients, we show that Liouville's theorem holds, any bounded solution must be constant, and any $L^p$-solution must vanish.

This paper is concerned with the existence, uniqueness, and stability of the solution of some impulsive fractional problem in a Banach space subjected to a nonlocal condition. Meanwhile, we give a new concept of a solution to impulsive fractional equations of multiorders. The derived results are based on Banach's contraction theorem as well as Schaefer's fixed point theorem.

We derive sufficient conditions for the existence of positive solutions to higher order $(p,q)$-Laplacian two-point boundary value problem, $(-\mathrm{1}{)}^{m_{\mathrm{1}}+n_{\mathrm{1}}-\mathrm{1}}{[{\varphi }_p(u^{(\mathrm{2}{m}_{\mathrm{1}})}(t))]}^{(n_{\mathrm{1}})}=f_{\mathrm{1}}(t,u(t),v(t))$, $t\in [\mathrm{0,1}]$, $\mathrm{}(-\mathrm{1}{)}^{m_{\mathrm{2}}+n_{\mathrm{2}}-\mathrm{1}}{[{\varphi }_q(v^{(m_{\mathrm{2}})}(t))]}^{(\mathrm{2}{n}_{\mathrm{2}})}=f_{\mathrm{2}}(t,u(t),v(t))$, $t\in [\mathrm{0,1}]$, $\mathrm{}\mathrm{}{u}^{(\mathrm{2}i)}(\mathrm{0})=\mathrm{0}=u^{(\mathrm{2}i)}(\mathrm{1})$, $i=\mathrm{0,1},\mathrm{2},\dots ,m_{\mathrm{1}}-\mathrm{1}$, ${[{\varphi }_p(u^{(\mathrm{2}{m}_{\mathrm{1}})}(t))]}_{\text{at\hspace{0.17em}}t=\mathrm{0}}^{(j)}=\mathrm{0}$, $j=\mathrm{0,1},\dots ,n_{\mathrm{1}}-\mathrm{2}$; $[{\varphi }_p(u^{(\mathrm{2}{m}_{\mathrm{1}})}(\mathrm{1}))]=\mathrm{0}$, ${[{\varphi }_q(v^{(m_{\mathrm{2}})}(t))]}_{\text{at\hspace{0.17em}}t=\mathrm{0}}^{(\mathrm{2}i)}=\mathrm{0}={[{\varphi }_q(v^{(m_{\mathrm{2}})}(t))]}_{\text{at\hspace{0.17em}}t=\mathrm{1}}^{(\mathrm{2}i)}$, $i=\mathrm{0,1},\dots ,n_{\mathrm{2}}-\mathrm{1}$, $v^{(j)}(\mathrm{0})=\mathrm{0}$, $j=\mathrm{0,1},\mathrm{2},\dots ,m_{\mathrm{2}}-\mathrm{2}$, and $v(\mathrm{1})=\mathrm{0}$, where $f_{\mathrm{1}},f_{\mathrm{2}}$ are continuous functions from $[\mathrm{0,1}]\times \mathrm{}\mathrm{}\mathrm{}ℝ{\mathrm{}}^{\mathrm{2}}$ to $[\mathrm{0},\mathrm{\infty })$, $m_{\mathrm{1}},n_{\mathrm{1}},m_{\mathrm{2}},n_{\mathrm{2}}\in \mathrm{}\mathrm{}\mathrm{}\mathbb{N}\mathrm{}$ and $\mathrm{1}/p+\mathrm{1}/q=\mathrm{1}$. We establish the existence of at least three positive solutions for the two-point coupled system by utilizing five-functional fixed point theorem. And also, we demonstrate our result with an example.

This paper considers two general concepts of dichotomy for noninvertible and nonautonomous linear discrete-time systems in Banach spaces. These concepts use two types of dichotomy projections sequences (invariant and strongly invariant) and generalize some well-known dichotomy concepts (uniform, nonuniform, exponential, and polynomial). In the particular case of strongly invariant dichotomy projections, we present characterizations of these sequences and connections with other dichotomy concepts existent in the literature. Some illustrative examples clarify the implications between these concepts.

A convenient computational approach for solving mathematical model related to diffusion dispersion during flow through packed bed is presented. The algorithm is based on the mixed collocation method. The method is particularly useful for solving stiff system arising in chemical and process engineering. The convergence of the method is found to be of order 2 using the roots of shifted Chebyshev polynomial. Model is verified using the literature data. This method has provided a convenient check on the accuracy of the results for wide range of parameters, namely, Peclet numbers. Breakthrough curves are plotted to check the effect of Peclet number on average and exit solute concentrations.

By using an integral inequality, we establish some sufficient conditions for the existence and p-exponential stability of periodic solutions for a class of impulsive stochastic BAM neural networks with time-varying delays in leakage terms. Moreover, we present an example to illustrate the feasibility of our results.

In this paper we discuss time integrators for nonlinear differential equations. In recent years, splitting approaches have become an important tool for reducing the computational time needed to solve differential equations. Moreover, nonlinearity is a challenge to splitting schemes, while one has to extend the exp-functions in terms of a nonlinear Magnus expansion. Here we discuss a novel extension of the so-called multiproduct expansion methods, which is used to improve the standard Strang splitting schemes as to their nonlinearity. We present an extension of linear splitting schemes and concentrate on nonlinear systems of differential equations and generalise in this respect the recent MPE method; see (Chin and Geiser, 2011). Some first numerical examples, of rigid body problems, are given as benchmarks.

We study the wave solutions for a degenerated reaction diffusion system arising from the invasion of cells. We show that there exists a family of waves for the wave speed larger than or equal a certain number and below which there are no monotonic wave solutions. We also investigate the monotonicity, uniqueness, and asymptotics of the waves.

We present some operator splitting methods improved by the use of the Zassenhaus product and designed for applications to multiphysics problems. We treat iterative splitting methods that can be improved by means of the Zassenhaus product formula, which is a sequential splitting scheme. The main idea for reducing the computation time needed by the iterative scheme is to embed fast and cheap Zassenhaus product schemes, since the computation of the commutators involved is very cheap, since we are dealing with nilpotent matrices. We discuss the coupling ideas of iterative and sequential splitting techniques and their convergence. While the iterative splitting schemes converge slowly in their first iterative steps, we improve the initial convergence rates by embedding the Zassenhaus product formula. The applications are to multiphysics problems in fluid dynamics. We consider phase models in computational fluid dynamics and analyse how to obtain higher order operator splitting methods based on the Zassenhaus product. The computational benefits derive from the use of sparse matrices, which arise from the spatial discretisation of the underlying partial differential equations. Since the Zassenhaus formula requires nearly constant CPU time due to its sparse commutators, we have accelerated the iterative splitting schemes.

In this paper, the exact solutions of space-time fractional telegraph equations are given in terms of Mittage-Leffler functions via a combination of Laplace transform and variational iteration method. New techniques are used to overcome the difficulties arising in identifying the general Lagrange multiplier. As a special case, the obtained solutions reduce to the solutions of standard telegraph equations of the integer orders.