Fractional Quasilinear Hyperbolic Equations with Variable Sources

we propose a new high‐order approximation for the solution of two‐space‐dimensional quasilinear hyperbolic partial differential equation of the formutt=A(x,y,t,u)uxx+B(x,y,t,u)uyy+g(x,y,t,u,ux,uy,ut), 0 <x,y< 1,t> 0 subject to appropriate initial and Dirichlet boundary conditions , wherek> 0 andh> 0 are mesh sizes in time and space directions, respectively. We use only five evaluations of the functiongas compared to seven evaluations of the same function discussed by (Mohanty et al., 1996 and 2001). We describe the derivation procedure in details and also discuss how our formulation is able to handle the wave equation in polar coordinates. The proposed method when applied to a linear hyperbolic equation is also shown to be unconditionally stable. Some examples and their numerical results are provided to justify the usefulness of the proposed method.

Asymptotic stability of solutions to quasilinear hyperbolic equations with variable sources

In this paper, we consider the following quasilinear damped hyperbolic equation involving variable exponents: $$ u_{tt}-\operatorname{div}( |\nabla u|^{r(x)-2}\nabla u)+|u_t|^{m(x)-2} u_t-\Delta u_t=|u|^{q(x)-2}u, $$ with homogenous Dirichlet initial boundary value condition. An energy estimate and Komornik's inequality are used to prove uniform estimate of decay rates of the solution. We also show that $u(x, t)=0$ is asymptotic stable in terms of natural energy associated with the solution of the above equation. As we know, such results are seldom seen for the variable exponent case. At last, we give some numerical examples to illustrate our results.

In this paper, we consider the following quasilinear damped hyperbolic equation involving variable exponents: $$ u_{tt}-\operatorname{div}( |\nabla u|^{r(x)-2}\nabla u)+|u_t|^{m(x)-2} u_t-\Delta u_t=|u|^{q(x)-2}u, $$ with homogenous Dirichlet initial boundary value condition. An energy estimate and Komornik's inequality are used to prove uniform estimate of decay rates of the solution. We also show that $u(x, t)=0$ is asymptotic stable in terms of natural energy associated with the solution of the above equation. As we know, such results are seldom seen for the variable exponent case. At last, we give some numerical examples to illustrate our results. Bibliography: 16 titles.

Lower and upper bounds for the blow-up time for nonlinear wave equation with variable sources

In this paper, we propose a new three-level implicit method based on a half-step spline in compression method of order two in time and order four in space for the solution of one-space dimensional quasi-linear hyperbolic partial differential equation of the form u_{tt} =A(x,t,u)u_{xx} +f(x,t,u,u_{x},u_{t}). We describe spline in compression approximations and their properties using two half-step grid points. The new method for one-dimensional quasi-linear hyperbolic equation is obtained directly from the consistency condition. In this method we use three grid points for the unknown function u(x,t) and two half-step points for the known variable ‘x’ in x-direction. The proposed method, when applied to a linear test equation, is shown to be unconditionally stable. We have also established the stability condition to solve a linear fourth-order hyperbolic partial differential equation. Our method is directly applicable to solve hyperbolic equations irrespective of the coordinate system, which is the main advantage of our work. The proposed method for a scalar equation is extended to solve the system of quasi-linear hyperbolic equations. To assess the validity and accuracy, the proposed method is applied to solve several benchmark problems, and numerical results are provided to demonstrate the usefulness of the proposed method.

In this paper, we propose a new three level implicit method based on half-step spline in tension method of order two in time and four in space for the solution of one-space dimensional quasi-linear hyperbolic partial differential equation of the form $$w_{tt}=K(x,t,w)w_{xx} + {\varphi }(x,t,w,w_{x},w_{t})$$ . We describe spline in tension approximations and its properties using two half-step grid points. The new method for one dimensional quasi-linear hyperbolic equation is obtained directly from the consistency condition. In this method we use three grid points for the unknown function w(x, t) and two half-step points for the known variable ‘x’ in x-direction. The proposed method when applied to Telegraphic equation is shown to be unconditionally stable. Further, the stability condition for 1-D linear hyperbolic equation with variable coefficients is established. Our method is directly applicable to hyperbolic equations irrespective of the coordinate system which is the main advantage of our work. The proposed method for scalar equation is extended to solve the system of quasi-linear hyperbolic equations. To assess the validity and accuracy, the proposed method is applied to solve several benchmark problems and numerical computations are provided to demonstrate the effectualness of the method.

We report a new 3-level implicit compact numerical method of order four in time and four in space based on off-step discretization for the solution of two-space-dimensional quasilinear hyperbolic equation [Formula: see text], [Formula: see text], [Formula: see text] defined in the region [Formula: see text], [Formula: see text], [Formula: see text]. We require only 19 grid points for the unknown variable [Formula: see text] and two extra off-step points each in [Formula: see text]-, [Formula: see text]- and [Formula: see text]-directions. The proposed method is directly applicable to two-dimensional hyperbolic equations with singular coefficients, which is the main attraction of our work. We do not require any fictitious points for computation. The proposed method when applied to a two-dimensional damped wave equation is shown to be unconditionally stable. Operator splitting method is used to solve damped wave equation. Many benchmark problems are solved to confirm the fourth-order convergence of the proposed method.

The aim of this paper is to study bounds for lifespan of solutions to the following equation:utt−Δu+∫0tg(t−τ)Δu(τ)dτ+|ut|m(x,t)−2ut=|u|p(x,t)−2u\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ u_{tt}-\\Delta u+ \\int _{0}^{t}g(t-\\tau )\\Delta u(\\tau )\\,d\\tau + \\vert u_{t} \\vert ^{m(x,t)-2}u _{t}= \\vert u \\vert ^{p(x,t)-2}u $$\\end{document} under homogeneous Dirichlet boundary conditions. It is worth pointing out that it is not a trivial generalization for constant-exponent problems because we have to face some essential difficulties in studying such problems. The first difficulty is that the monotonicity of the energy functional fails. Another one is that there exists a gap between the norm and the modular to the generalized function space, which leads to the failure of the Poincaré inequality for modular form. To overcome such difficulties, the authors construct control function and apply new energy estimates to establish the quantitative relationship between the source int _{varOmega }|u|^{p(x,t)},dx and the initial energy, and then obtain the finite-time blow-up of solutions for a positive initial energy, especially, the authors only assume that p_{t}(x,t) is integrable rather than uniformly bounded. Such weak conditions are seldom seen for the variable exponent case. At last, an estimate of lower bound for lifespan is established by applying differential inequality argument and energy inequalities.

The aim of this paper is to study bounds for blow-up time to the following viscoelastic hyperbolic equation of Kirchhoff type with initial-boundary value condition: $$ |u_{t}|^{\rho }u_{tt}-M(\|\nabla u\|_{2}^{2})\Delta u+\int _{0}^{t}g(t- \tau )\Delta u(\tau )d\tau +|u_{t}|^{m(x)-2}u_{t}=|u|^{p(x)-2}u. $$ Compared with constant exponents, it is difficult to discuss the above problem due to the existence of a gap between the modular and the norm. The authors construct suitable function spaces to discuss the upper bound for blow-up time with positive initial energy by means of a differential inequality technique. In addition, lower bounds for blow-up time in different range of exponent are obtained. These improve and generalize some recent results.

This paper is devoted to the summary of results about a quasilinear hyperbolic partial Differential equation of first order with a hysteresis operator v = ℱ[u]: ∂(u + v)/∂t + ∂u/∂x = 0. Hysteresis is represented by functional describing adsorption and desorption on the particles of the substance and is a possibly discontinuous generalized play operator. The results can be extended to possibly discontinuous generalized Prandtl-Ishlinskii operators of play type.

PurposeThis paper aims to develop a new 3-level implicit numerical method of order 2 in time and 4 in space based on half-step cubic polynomial approximations for the solution of 1D quasi-linear hyperbolic partial differential equations. The method is derived directly from the consistency condition of spline function which is fourth-order accurate. The method is directly applied to hyperbolic equations, irrespective of coordinate system, and fourth-order nonlinear hyperbolic equation, which is main advantage of the work.Design/methodology/approachIn this method, three grid points for the unknown function w(x,t) and two half-step points for the known variable x in spatial direction are used. The methodology followed in this paper is construction of a cubic spline polynomial and using its continuity properties to obtain fourth-order consistency condition. The proposed method, when applied to a linear equation is shown to be unconditionally stable. The technique is extended to solve system of quasi-linear hyperbolic equations. To assess the validity and accuracy, the method is applied to solve several benchmark problems, and numerical results are provided to demonstrate the usefulness of the method.FindingsThe paper provides a fourth-order numerical scheme obtained directly from fourth-order consistency condition. In earlier methods, consistency conditions were only second-order accurate. This brings an edge over other past methods. In addition, the method is directly applicable to physical problems involving singular coefficients. Therefore, no modification in the method is required at singular points. This saves CPU time, as well as computational costs.Research limitations/implicationsThere are no limitations. Obtaining a fourth-order method directly from consistency condition is a new work. In addition, being an implicit method, this method is unconditionally stable for a linear test equation.Practical implicationsPhysical problems with singular and nonsingular coefficients are directly solved by this method.Originality/valueThe paper develops a new fourth-order implicit method which is original and has substantial value because many benchmark problems of physical significance are solved in this method.

In this paper, we study a new numerical method of order 4 in space and time based on half-step discretization for the solution of three-space dimensional quasi-linear hyperbolic equation of the form [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] subject to prescribed appropriate initial and Dirichlet boundary conditions. The scheme is compact and requires nine evaluations of the function [Formula: see text] For the derivation of the method, we use two inner half-step points each in [Formula: see text]-, [Formula: see text]-, [Formula: see text]- and [Formula: see text]-directions. The proposed method is directly applicable to three-dimensional (3D) hyperbolic equations with singular coefficients, which is main attraction of our work. We do not require extra grid points for computation. The proposed method when applied to 3D damped wave equation is shown to be unconditionally stable. Operator splitting method is used to solve 3D linear hyperbolic equations. Few benchmark problems are solved and numerical results are provided to support the theory which is discussed in this paper.

Large Time Stability Control for a Class of Quasilinear Parabolic and Hyperbolic Equations

The aim of this chapter is to introduce the main notions for the construction of localized asymptotic solutions to quasilinear parabolic equations with a small parameter at the derivatives (i.e., solutions that satisfy the equations up to sufficiently smooth functions tending to zero as a sufficiently great power of a small parameter).