We describe the point spectrum of the generator of a $C_0$-semigroup associated with the M/M/1 queueing model that is governed by an infinite system of partial differential equations with integral boundary conditions. Our results imply that the essential growth bound of the $C_0$-semigroup is 0 and, therefore, that the semigroup is not quasi-compact. Moreover, our result also shows that it is impossible that the time-dependent solution of the M/M/1 queueing model exponentially converges to its steady-state solution.

For the four-dimensional nonhomogeneous wave equation boundary value problems that are multidimensional analogues of Darboux problems in the plane are studied. It is known that for smooth right-hand side functions the unique generalized solution may have a strong power-type singularity at only one point. This singularity is isolated at the vertex $O$ of the boundary light characteristic cone and does not propagate along the bicharacteristics. The present paper describes asymptotic expansions of the generalized solutions in negative powers of the distance to $O$. Some necessary and sufficient conditions for existence of bounded solutions are proven and additionally a priori estimates for the singular solutions are obtained.

Standard application of similarity method to find solutions of PDEs mostly results in reduction to ODEs which are not easily integrable in terms of elementary or tabulated functions. Such situations usually demand solving reduced ODEs numerically. However, there are no systematic procedures available to utilize these numerical solutions of reduced ODE to obtain the solution of original PDE. A practical and tractable approach is proposed to deal with such situations and is applied to obtain approximate similarity solutions to different cases of an initial-boundary value problem of unsteady gas flow through a semi-infinite porous medium.

Existence and uniqueness of solutions for $\alpha \in (\mathrm{2,3}]$ order fractional differential equations with three-point fractional boundary and integral conditions involving the nonlinearity depending on the fractional derivatives of the unknown function are discussed. The results are obtained by using fixed point theorems. Two examples are given to illustrate the results.

The existence and uniqueness of the $R_v$-generalized solution for the first boundary value problem and a second order elliptic equation with coordinated and uncoordinated degeneracy of input data and with strong singularity solution on all boundary of a two-dimensional domain are established.

We study the existence and uniqueness of solutions for a fractional boundary value problem involving Hadamard-type fractional differential equations and nonlocal fractional integral boundary conditions. Our results are based on some classical fixed point theorems. Some illustrative examples are also included.

The reaction diffusion system with anomalous diffusion and a balance law $u_t+{\left(-\mathrm{\Delta }\right)}^{\alpha /2}u=-f\left(u,v\right)$, $\hspace{0.17em}{v}_t+{\left(-∆\right)}^{\beta /2}v=f\left(u,v\right)$, , , is con sidered. The existence of global solutions is proved in two situations: (i) a polynomial growth condition is imposed on the reaction term $f$ when ; (ii) no growth condition is imposed on the reaction term $f$ when .

This paper deals with the computation of the eigenvalues of two-parameter Sturm-Liouville (SL) problems using the Regularized Sampling Method, a method which has been effective in computing the eigenvalues of broad classes of SL problems (singular, non-self-adjoint, nonlocal, impulsive, etc.). We have shown, in this work that it can tackle two-parameter SL problems with equal ease. An example was provided to illustrate the effectiveness of the method.

By choosing the trial function space to the immersed finite element space and the test function space to be piecewise constant function space, we develop a discontinuous Galerkin immersed finite volume element method to solve numerically a kind of anisotropic diffusion models governed by the elliptic interface problems with discontinuous tensor-conductivity. The existence and uniqueness of the discrete scheme are proved, and an optimal-order energy-norm estimate and $L^2$-norm estimate for the numerical solution are derived.

We establish Crouzeix-Raviart element adaptive algorithm based on Rayleigh quotient iteration and give its a priori/a posteriori error estimates. Our algorithm is performed under the package of Chen, and satisfactory numerical results are obtained.

The author considers the Neumann boundary value problem $-y^{\prime \prime }\left(t\right)+\mathbf{M}y\left(t\right)=\lambda \omega \left(t\right)f\left(t,y\left(t\right)\right),\mathrm{ }\mathrm{ }$ $t\in J,\mathrm{ }\mathrm{ }t\ne t_k,-\mathrm{\Delta }{y}^{\prime }{|}_{t=t_k}$ = $\lambda I_k\left(t_k,y\left(t_k\right)\right),\mathrm{ }k=\mathrm{1,2},\dots ,m,y^{\prime }(0)=y^{\prime }(1)=0$ and establishes the dependence results of the solution on the parameter $\lambda$, which cover equations without impulsive effects and are compared with some recent results by Nieto and O’Regan.

This paper investigates the existence, multiplicity, nonexistence, and uniqueness of positive solutions to a kind of two-point boundary value problem for nonlinear fractional differential equations with $p$-Laplacian operator. By using fixed point techniques combining with partially ordered structure of Banach space, we establish some criteria for existence and uniqueness of positive solution of fractional differential equations with $p$-Laplacian operator in terms of different value of parameter. In particular, the dependence of positive solution on the parameter was obtained. Finally, several illustrative examples are given to support the obtained new results. The study of illustrative examples shows that the obtained results are applicable.

We consider the existence of infinitely many classical solutions to a class of impulsive differential equations with Dirichlet boundary value condition. Our main tools are based on variant fountain theorems and variational method. We study the case in which the nonlinearity is sublinear. Some recent results are extended and improved.

We investigate the existence of solutions for a $k$-dimensional system of fractional finite difference equations by using the Kranoselskii’s fixed point theorem. We present an example in order to illustrate our results.

We study the existence and monotone iteration of solutions for a third-order four-point boundary value problem. An existence result of positive, concave, and pseudosymmetric solutions and its monotone iterative scheme are established by using the monotone iterative technique. Meanwhile, as an application of our results, an example is given.

We show the existence and uniqueness of solutions for an antiperiodic boundary value problem of nonlinear impulsive $q_k$-difference equations by applying some well-known fixed point theorems. An example is presented to illustrate the main results.