Classical solution of a mixed problem for the wave equation with discontinuous initial conditions in a curvilinear half-strip

A singularly perturbed parabolic problem of convection-diffusion type with a discontinuous initial condition is examined. An analytic function is identified which matches the discontinuity in the initial condition and also satisfies the homogenous parabolic differential equation associated with the problem. The difference between this analytical function and the solution of the parabolic problem is approximated numerically, using an upwind finite difference operator combined with an appropriate layer-adapted mesh. The numerical method is shown to be parameter-uniform. Numerical results are presented to illustrate the theoretical error bounds established in the paper.

A singularly perturbed parabolic problem of convection-diffusion type with a discontinuous initial condition is examined. A particular complimentary error function is identified which matches the discontinuity in the initial condition. The difference between this analytical function and the solution of the parabolic problem is approximated numerically. A coordinate transformation is used so that a layer-adapted mesh can be aligned to the interior layer present in the solution. Numerical analysis is presented for the associated numerical method, which establishes that the numerical method is a parameter-uniform numerical method. Numerical results are presented to illustrate the pointwise error bounds established in the paper.

Moving interfaces in peridynamic diffusion models and the influence of discontinuous initial conditions: Numerical stability and convergence

Variation formulae are proved for solutions of non-linear differential equations with variable delays and discontinuous initial conditions. The discontinuity of the initial condition means that at the initial moment of time the values of the initial function and the trajectory, generally speaking, do not coincide. The formulae obtained contain a new summand connected with the discontinuity of the initial condition and the variation of the initial moment.

We solve the one dimensional heat equation with discontinuous initial condition, the discontinuity being atx = 0, using the heat kernel and the Sine collocation method for convolution integrals. In [7], it is shown that the method is converging at an exponential rate. We compare the following formulations: the first one is based on the fact that fort > 0, the solution of the forced heat equation, the forcing term being a smooth function, is infinitely many time differentiable; the other one ignores the smoothing properties of the heat kernel. With the first approach, we apply the Sine collocation algorithm presented in [7] to the double integral for the forcing term ignoring the discontinuity in the initial condition. With the other one, we break the double integral for the forcing term into two integrals, one over (−∞,0) × [0,T] and the other over (0,∞) × [0,T], then we make the change of variable ξ = −η in the integral over (−∞,0) × [0,T] to only compute integrals over (0,∞) × [0,T]. In the first case, the conformal map is the identity; in the other one it is log(sinh(z)). Our numerical results show that for the problem considered here, the first approach is more efficient and more accurate than the second one. The method presented here is easily generalized to the case of finitely many jumps in the initial condition.

For the perturbed controlled nonlinear delay differential equation with the discontinuous initial condition, a formula of the analytic representation of solution is proved in the left neighborhood of the endpoint of the main interval. In the formula, the effects of perturbations of the delay parameter, the initial vector, the initial and control functions are detected.

Purpose The purpose of this paper is to determine numerically the existence of blow-up in finite time of a one-dimensional, bidirectional equation that has been proposed for the study of wave propagation in shallow waters and the deformation of viscoelastic materials, subject to smooth initial conditions. Design/methodology/approach An implicit, time-linearized, finite difference method is used to solve the nonlinear wave equation in a truncated domain subject to homogeneous Dirichlet boundary conditions, and smooth initial conditions of both the Gaussian type and those corresponding to the exact solution of the inviscid, generalized regularized-long wave equation (RLW). Findings A large set of numerical experiments on the effects of the linear and nonlinear drift, dispersion and viscosity coefficients and relaxation time, and the amplitude and width of the Gaussian conditions and the amplitude of those corresponding to the initial conditions of the inviscid RLW, on blow-up are reported. The results of these experiments indicate that, for the same initial mass, if there exists blow-up, it occurs at longer times for Gaussian conditions due to the larger width of these conditions and its corresponding smaller initial kinetic energy. Blow-up has also been found to depend on the power of the nonlinear drift, and the viscosity coefficient and relaxation time. Comparisons with other previously published results that made used of nonsmooth or even discontinuous initial conditions indicate that the blow-up time decreases as the smoothness of the initial conditions is decreased, in accord with the fact that the stretching energy increases as the gradient of the solution increases. Originality/value The blow-up of the solution to a one-dimensional, bidirectional, nonlinear wave equation that models wave propagation in shallow waters and the deformation of viscoelastic materials, subject to two types of smooth initial conditions has been studied as a function of the parameters that characterize the linear and nonlinear drift, dispersion, viscosity and relaxation time, and the amplitude and width of the initial Gaussian conditions. Two conditions based on the growth of the local solution and the positiveness of a time-dependent function that depends on the kinetic and stretching energies have been used to determine the blow-up time.

This article presents a parameter-uniform hybrid numerical technique for singularly perturbed parabolic convection-diffusion problems (SPPCDP) with discontinuous initial conditions (DIC). It utilizes the classical backward-Euler technique for time discretization and a hybrid finite difference scheme ( which is a proper combination of the midpoint upwind scheme in the outer regions and the classical central difference scheme in the interior layer regions(generated by the DIC)) for spatial discretization. The scheme produces parameter-uniform numerical approximations on a piecewise- uniform Shishkin mesh. When the perturbation parameter ε(0 < ε ≪ 1) is small, it becomes difficult to solve these problems using the classical numerical methods (standard central difference or a standard upwind scheme) on uniform meshes because discontinuous initial conditions frequently appear in the solutions of this class of problems. The method is shown to converge uniformly in the discrete supremum with nearly second-order spatial accuracy. The suggested method is subjected to a stability study, and parameter-uniform error estimates are generated. In order to support the theoretical findings, numerical results are presented.

The partial differential equations which represent the one-dimensional heat and wave equations are generally solved by the techniques: exact, approximate, and purely numerical. Since the application of numerical method becomes more complex, computationally intensive, or needs complicated symbolic computations, there is a need to seek the help of integral transform methods for solving the partial differential equations representing the one-dimensional heat and wave equations. Heat and wave equations are some of the well-known partial differential equations applicable in basic sciences, and engineering. Integral transform methods provide effective ways for solving a variety of problems arising in basic sciences and engineering. The Gupta transform is a new integral transform applied for solving differential equations like other integral transforms. In this study, Gupta transform is introduced for solving the one-dimensional heat and wave equations formulated in terms of partial differential equations. The concept and properties of the Gupta transform are also discussed.

We consider dispersive shock waves of the focusing nonlinear Schrödinger equation generated by discontinuous initial conditions which are periodic or quasiperiodic on the left semiaxis and zero on the right semiaxis. As an initial function, we use a finite-gap potential of the Dirac operator given in an explicit form through hyperelliptic theta-functions. The aim of this paper is to study the long-time asymptotics of the solution of this problem in a vicinity of the leading edge, where a train of asymptotic solitons are generated. Such a problem was studied in the work of Kotlyarov and Khruslov [Teor. Mat. Fiz. 68(2), 751–761 (1986)] and Kotlyarov {Mat. Zametki 49(2), 84–94 (1991) [Math. Notes 49(1-2), 172–180 (1991)]} using Marchenko’s inverse scattering techniques. We investigate this problem exceptionally using the Riemann-Hilbert (RH) problem techniques that allow us to obtain explicit formulas for asymptotic solitons themselves in contrast with the cited papers where asymptotic formulas are obtained only for the square of the absolute value of solution. Using transformations of the main RH problems, we arrive at a model problem corresponding to the parametrix at the end points of the continuous spectrum of the Zakharov-Shabat spectral problem. The parametrix problem is effectively solved in terms of the generalized Laguerre polynomials, which naturally appeared after appropriate scaling of the Riemann-Hilbert problem in small neighborhoods of the end points of the continuous spectrum. Further asymptotic analysis gives an explicit formula for solitons at the edge of dispersive waves. Thus, we give the complete description of the train of asymptotic solitons: not only bearing the envelope of each asymptotic soliton, but its oscillating structure is found explicitly. Besides, the second term of asymptotics describing an interaction between these solitons and oscillating background is also found. This gives the fine structure of the edge of dispersive shock waves.

The analytic relation between solutions of the original Cauchy problem and a corresponding perturbed problem is established. In the representation formula of solution, the effects of the discontinuous initial condition and perturbation of the initial data are revealed.

The approach to optimization of finite-difference (FD) schemes for the linear advection equation (LAE) is proposed. The FD schemes dependent on the scalar dimensionless parameter are considered. The parameter is included in the expression, which approximates the term with spatial derivatives. The approach is based on the considering of the dispersive and dissipative characteristics of the schemes as the functions of the parameter. For the proper choice of the parameter, these functions are minimized. The approach is applied to the optimization of two-step schemes with an asymmetric approximation of time derivative and with various approximations of the spatial term. The cases of schemes from first to fourth approximation orders are considered. The optimal values of the parameter are obtained. Schemes with the optimal values are applied to the solution of test problems with smooth and discontinuous initial conditions. Also, schemes are used in the FD-based lattice Boltzmann method (LBM) for modeling of the compressible gas flow. The obtained numerical results demonstrate the convergence of the schemes and decaying of the numerical dispersion.

Truncation and accumulated errors in wave propagation

Homogenization of the variable-speed wave equation

A numerical method of solving the problem of the non-stationary potential flow of a fluid with movable boundaries