Analytical solutions for a time-fractional axisymmetric diffusion–wave equation with a source term

In this paper, a time-fractional axisymmetric diffusion–wave equation with a source term is considered in cylindrical coordinates. The analytical solution is obtained with the help of an integral transform method and some properties of special functions. In addition, we discuss two kinds of different boundary conditions and different forms of the source term. Finally, we analyze the effects of the fractional derivative on the solutions by using numerical results and find that sub-diffusion phenomena and oscillations exist.     

On a nonlinear Kirchhoff–Carrier wave equation associated with Robin conditions

This paper is devoted to the study a nonlinear Kirchhoff–Carrier wave equation associated with Robin conditions. Existence and uniqueness of weak solutions are proved by using the Faedo–Galerkin method and the linearization method for nonlinear terms. An asymptotic expansion of high order in many small parameters of solutions is also discussed.     

On a problem for the Navier–Stokes equations with the infinite Dirichlet integral

We study existence of global in time solutions to the Navier–Stokes equations in a two dimensional domain with an unbounded boundary. The problem is considered with slip boundary conditions involving nonzero friction. The main result shows a new L-bound on the vorticity. A key element of the proof is the maximum principle for a reformulation of the problem. Under some restrictions on the curvature of the boundary and the friction the result for large data (including flux) with the infinite Dirichlet integral is obtained.     

On a problem for the Navier–Stokes equations with the infinite Dirichlet integral

We study existence of global in time solutions to the Navier–Stokes equations in a two dimensional domain with an unbounded boundary. The problem is considered with slip boundary conditions involving nonzero friction. The main result shows a new L-bound on the vorticity. A key element of the proof is the maximum principle for a reformulation of the problem. Under some restrictions on the curvature of the boundary and the friction the result for large data (including flux) with the infinite Dirichlet integral is obtained.Received: October 31, 2002; revised: September 17, 2003     

Dirichlet problem for the Boussinesq–Love equation

We study the cases of unique solvability of the Dirichlet problem for the Boussinesq–Love equation.     

Controllability for a parabolic equation with a nonlinear term involving the state and the gradient

In this paper, we prove the controllability of a quasi-linear heat equation involving gradient terms with Fourier boundary conditions in a bounded domain of ? ^N . The proofs of the main results in this paper involve such inequalities and rely on the study of these linear problems and appropriate fixed point arguments.     

Exact analytical solution of the Cauchy problem for a linear reaction–diffusion equation with time-dependent coefficients and space–time-dependent source term

The present work derives the exact analytical solution of the Cauchy problem for a linear reaction–diffusion equation with time-dependent coefficients and space–time-dependent source term. The work also emphasizes the role of reaction–diffusion models as important particular cases of much more general equations in the kinetic theory of active particles. The analytical expression derived shows the structure of the solution and the contributions of different terms of the model to it. The result obtained enables one to solve the Cauchy problem indicated by using the exact analytical representation rather than numerical methods, which are usually time-consuming, especially when the number of spatial dimensions is greater than 2.     

On the convergence of the Dirichlet grid problem with a singularity for a singularly perturbed convection–diffusion equation

The Dirichlet problem for a singulary perturbed convection–diffusion equation in a rectangle when a discontinuity at the flow exit the first derivative of the boundary condition gives rise to an inner layer for the solution. On piecewise-uniform Shishkin grids that condense near regular and characteristic layers, the solution obtained using the classical five-point difference scheme with a directed difference is shown to converge with respect to the small parameter to solve the original problem in the grid norm L _∞ ^h almost with the first order. This theoretical result is confirmed via numerical analysis.     

A new numerical scheme for the nonlinear Schrödinger equation with wave operator

In this paper, a new compact finite difference scheme is proposed for a periodic initial value problem of the nonlinear Schrödinger equation with wave operator. This is an explicit scheme of four levels with a discrete conservation law. The unconditional stability and convergence in maximum norm with order \(O(h^{4}+\tau ^{2})\) are verified by the energy method. Those theoretical results are proved by a numerical experiment and it is also verified that this scheme is better than the previous scheme via comparison.     

A second-order space–time accurate scheme for nonlinear diffusion equation with general capacity term

A fully implicit finite difference (FIFD) scheme with second-order space–time accuracy is studied for a nonlinear diffusion equation with general capacity term. A new reasoning procedure is introduced to overcome difficulties caused by the nonlinearity of the capacity term and the diffusion operator in the theoretical analysis. The existence of the FIFD solution is investigated at first which plays an important role in the analysis. It is established by choosing a new test function to bound the solution and its temporal and spatial difference quotients in suitable norms in the fixed point arguments, which is different from the traditional way. Based on these bounds, other fundamental properties of the scheme are rigorously analyzed consequently. It shows that the scheme is uniquely solvable, unconditionally stable, and convergent with second-order space–time accuracy in L^∞(L²) and L^∞(H¹) norms. The theoretical analysis adapts to both one- and multidimensional problems, and can be extended to schemes with first-order time accuracy. Numerical tests are provided to verify the theoretical results and highlight the high accuracy of the second-order space–time accurate scheme. The reasoning techniques can be extended to a broad family of discrete schemes for nonlinear problems with capacity terms.     

Correctness of the Dirichlet problem for the Lavrent’ev–Bitsadze equation

It is shown that the Dirichlet problem in a multidimensional domain for the Lavrent’ev–Bitsadze equation is uniquely solvable. A criterion of the uniqueness of the solution is obtained.     

The Cauchy problem for the wave equation with Lévy Laplacian

We present the solution of the Cauchy problem (the initial-value problem in the whole space) for the wave equation with infinite-dimensional Lévy Laplacian Δ _L , $$ \frac{{\partial ^2 U(t,x)}} {{\partial t^2 }} = \Delta _L U(t,x) $$ in two function classes, the Shilov class and the Gâteaux class.     

The Dirichlet problem on a strip for the α-translating soliton equation

In this paper, we investigate the Dirichlet problem associated with the α-translating equation. Using the Perron method and a family of grim reapers as barriers, we prove the existence of a solution on a strip of ${\mathbb{R}}^2$ and the boundary data is formed by two copies of a convex function.     

Entropy solution for a nonlinear elliptic problem with lower order term in Musielak–Orlicz spaces

In this paper, we shall be concerned with the existence result of the following problem,

$$\begin{aligned} \left\{ \begin{array}{l} -\text {div}\left( a(x,u,\nabla u)\right) -\text {div}(\Phi (x,u))= f \ \ \mathrm{in}\ \Omega ,\\ u=0 \text { on } \partial \Omega , \end{array} \right. \end{aligned}$$

(0.1)

with the second term f belongs to \(L^1(\Omega )\). The growth and the coercivity conditions on the monotone vector field a are prescribed by a generalized N-function M. We assume any restriction on M, therefore we work with Musielak–Orlicz spaces which are not necessarily reflexive. The lower order term \(\Phi \) is a Carathéodory function satisfying only a growth condition.

     

A Jacobi–Davidson type method with a correction equation tailored for integral operators

We propose two iterative numerical methods for eigenvalue computations of large dimensional problems arising from finite approximations of integral operators, and describe their parallel implementation. A matrix representation of the problem on a space of moderate dimension, defined from an infinite dimensional one, is computed along with its eigenpairs. These are taken as initial approximations and iteratively refined, by means of a correction equation based on the reduced resolvent operator and performed on the moderate size space, to enhance their quality. Each refinement step requires the prolongation of the correction equation solution back to a higher dimensional space, defined from the infinite dimensional one. This approach is particularly adapted for the computation of eigenpair approximations of integral operators, where prolongation and restriction matrices can be easily built making a bridge between coarser and finer discretizations. We propose two methods that apply a Jacobi–Davidson like correction: Multipower Defect-Correction (MPDC), which uses a single-vector scheme, if the eigenvalues to refine are simple, and Rayleigh–Ritz Defect-Correction (RRDC), which is based on a projection onto an expanding subspace. Their main advantage lies in the fact that the correction equation is performed on a smaller space while for general solvers it is done on the higher dimensional one. We discuss implementation and parallelization details, using the PETSc and SLEPc packages. Also, numerical results on an astrophysics application, whose mathematical model involves a weakly singular integral operator, are presented.     

Stability of solutions to extremum problems for the nonlinear convection–diffusion–reaction equation with the Dirichlet condition

The solvability of the boundary value and extremum problems for the convection–diffusion–reaction equation in which the reaction coefficient depends nonlinearly on the concentration of substances is proven. The role of the control in the extremum problem is played by the boundary function in the Dirichlet condition. For a particular reaction coefficient in the extremum problem, the optimality system and estimates of the local stability of its solution to small perturbations of the quality functional and one of specified functions is established.     

The initial value problem for the cubic nonlinear Klein–Gordon equation

We study the initial value problem for the cubic nonlinear Klein–Gordon equation

Integration of the nonlinear Korteweg–de Vries equation with an additional term

Theoretical and Mathematical Physics - We use the method of the inverse spectral problem to integrate the nonlinear Korteweg–de Vries equation with an additional term in the class of periodic...