Class Of Initial-boundary Value Problems Research Articles

Global well-posedness of a new class of initial-boundary value problem on incompressible MHD/MHD-Boussinesq equations

The global well-posedness of a new class of initial-boundary value problem on incompressible MHD equations in the bounded domain with the smooth boundary is studied. The existence of a class of global weak solution to the initial boundary value problem for two/three-dimensional incompressible MHD equation with the given pressure-velocity's relation boundary condition for the fluid field at the boundary and with one perfectly insulating boundary condition for the magnetic field at the boundary is obtained, and the global existence and uniqueness of the smooth solution to the corresponding problem in two-dimensional case for the smooth initial data is also established. The corresponding results are also extended to the two/three-dimensional incompressible MHD-Boussinesq equations with the density-velocity's relation boundary condition for the density.

NONEXISTENCE OF GLOBAL SOLUTIONS FOR TIME FRACTIONAL WAVE EQUATIONS IN AN EXTERIOR DOMAIN

This paper concerns two classes of initial-boundary value problems of semilinear time fractional wave equations in an exterior domain. Based on a special technique of test functions, we determine some critical exponents in the sense of Fujita for a corresponding problem. We also obtain the existence and nonexistence of global weak solutions with certain appropriate conditions.

Two theorems on estimates for solutions of one class of nonlinear equations in a finite-dimensional space

The need to study boundary value problems for elliptic parabolic equations is dictated by numerous practical applications in the theoretical study of the processes of hydrodynamics, electrostatics, mechanics, heat conduction, elasticity theory and quantum physics. In this paper, we obtain two theorems on a priori estimates for solutions of nonlinear equations in a finite-dimensional Hilbert space. The work consists of four items. In the first subsection, the notation used and the statement of the main results are given. In the second subsection, the main lemmas are given. The third section is devoted to the proof of Theorem 1. In the fourth section, Theorem 2 is proved. The conditions of the theorems are such that they can be used in studying a certain class of initial-boundary value problems to obtain strong a priori estimates in the presence of weak a priori estimates. This is the meaning of these theorems.

On a Weak Solution of a Fractional-order Temporal Equation

Several real-world phenomena emerging in engineering and science fields can be described successfully by developing certain models using fractional-order partial differential equations. The exact, analytical, semi-analytical or even numerical solutions for these models should be examined and investigated by distinguishing between their solvablities and non-solvabilities. In this paper, we aim to establish some sufficient conditions for exploring the existence and uniqueness of solution for a class of initial-boundary value problems with Dirichlet condition. The gained results from this research paper are established for the class of fractional-order partial differential equations by a method based on Lax Milgram theorem, which relies in its construction on properties of the symmetric part of the bilinear form. Lax Milgram theorem is deemed as a mathematical scheme that can be used to examine the existence and uniqueness of weak solutions for fractional-order partial differential equations. These equations are formulated here in view of the Caputo fractional-order derivative operator, which its inverse operator is the Riemann-Louville fractional-order integral one. The results of this paper will be supportive for mathematical analyzers and researchers when a fractional-order partial differential equation is handled in terms of finding its exact, analytical, semi-analytical or numerical solution.

A Class of Quasilinear Equations with Riemann–Liouville Derivatives and Bounded Operators

The existence and uniqueness of a local solution is proved for the incomplete Cauchy type problem to multi-term quasilinear fractional differential equations in Banach spaces with Riemann–Liouville derivatives and bounded operators at them. Nonlinearity in the equation is assumed to be Lipschitz continuous and dependent on lower order fractional derivatives, which orders have the same fractional part as the order of the highest fractional derivative. The obtained abstract result is applied to study a class of initial-boundary value problems to time-fractional order equations with polynomials of an elliptic self-adjoint differential operator with respect to spatial variables as linear operators at the time-fractional derivatives. The nonlinear operator in the considered partial differential equations is assumed to be smooth with respect to phase variables.

Solvability of Dirichlet Problem For a Fractional Partial Differential equation by using energy inequality and Faedo-Galerkin method

In this paper, we establish sufficient conditions for the existence and uniqueness of the solution for a class of initial-boundary value problems with Dirichlet condition for a class of fractional partial differential equations. The results are established by a method based on a priori estimate "energy inequality" and the Faedo-Galerkin method.

On the Unique Solvability of Incomplete Cauchy Type Problems for a Class of Multi-Term Equations with the Riemann–Liouville Derivatives

Incomplete Cauchy-type problems are considered for linear multi-term equations solved with respect to the highest derivative in Banach spaces with fractional Riemann–Liouville derivatives and with linear closed operators at them. Some new existence and uniqueness theorems for solutions are presented explicitly and the analyticity of the solutions of the homogeneous equations are also shown. The asymmetry of the Cauchy-type problem under study is expressed in the presence of a so-called defect, which shows the number of lower-order initial conditions that should not be set when setting the problem. As applications, our abstract results are used in the study of a class of initial-boundary value problems for multi-term equations with Riemann–Liouville derivatives in time and with polynomials of a self-adjoint elliptic differential operator with respect to spatial variables.

Unique solvability of a Dirichlet problem for a fractional parabolic equation using energy-inequality method

In this paper, we establish sufficient conditions for the existence and uniqueness of the solution in fractional functional space for a class of initial boundary-value problems for a class of partial fractional parabolic differential equations that include a fractional derivative of Caputo. The results are established by the application of the method based on a priori estimate 'energy inequality' and the density of the range of the operator generated by the problem considered. <br><br> Встановлені достатні умови існування та єдиності розв'язку з дробового функціонального простору для одного класу початково-крайових задач для деяких дробово-параболічних диференціальних рівнянь із дробовою похідною Капуто. Результати отримано шляхом застосування методу енергетичних нерівностей. Доведена щільність образу оператора, що відповідає задачі.

Continuous Dependence on Data for Solutions of Fractional Extended Fisher–Kolmogorov Equation

Abstract This paper is concerned with the continuous dependence of mild solutions on initial values and orders for a general class of initial boundary-value problem to fractional extended Fisher–Kolmogorov equation. The results obtained in this paper can be considered as a contribution to this emerging field.

A priori estimates for weak solution for a time-fractional nonlinear reaction-diffusion equations with an integral condition

In this paper, we establish sufficient conditions for the existence and uniqueness of the solution in functional weighted Sobolev space for a class of initial-boundary value problems with integral condition for a class of nonlinear partial fractional reaction-diffusion (RD) equations. The results are established by using a priori estimate in Bouziani fractional spaces and applying an iterative process based on results obtained for the linear problem, we prove the existence, uniqueness of the weak generalized solution of the nonlinear problem.

Invariance and existence analysis for semilinear hyperbolic equations with damping and conical singularity

In this paper, we will discuss about the invariance of solution set and present the existence and non-existence of the global solutions a class of initial-boundary value problems with dissipative terms is considered for a class of semilinear degenerate hyperbolic equations on the cone Sobolev spaces. First, we will discuss the invariance of some sets corresponding to the problem (1.1) and then, by using a family of potential wells and concavity methods, we obtain existence and non-existence results of global solutions with exponential decay and show the blow-up in finite time of solutions on a manifold with conical singularities.

Preconditioned iterative methods for space-time fractional advection-diffusion equations

In this paper, we propose practical numerical methods for solving a class of initial-boundary value problems of space-time fractional advection-diffusion equations. First, we propose an implicit method based on two-sided Grünwald formulae and discuss its stability and consistency. Then, we develop the preconditioned generalized minimal residual (preconditioned GMRES) method and preconditioned conjugate gradient normal residual (preconditioned CGNR) method with easily constructed preconditioners. Importantly, because resulting systems are Toeplitz-like, fast Fourier transform can be applied to significantly reduce the computational cost. We perform numerical experiments to demonstrate the efficiency of our preconditioners, even in cases with variable coefficients.

Shifted Jacobi tau method for solving the space fractional diffusion equation

In this paper, approximation techniques based on the shifted Jacobi together with spectral tau technique are presented to solve a class of initial-boundary value problems for the fractional diffusion equations with variable coefficients on a finite domain. The fractional derivatives are described in the Caputo sense. The technique is derived by expanding the required approximate solution as the elements of shifted Jacobi polynomials. Using the operational matrix of the fractional derivative, the problem can be reduced to a set of linear algebraic equations. Numerical examples are included to demonstrate the validity and applicability of the technique and a comparison is made with the existing results to show that the proposed method is easy to implement and produce accurate results.

On the role of conservation laws in the problem on the occurrence of unstable solutions for quasilinear parabolic equations and their approximations

For a broad class of initial-boundary value problems for quasilinear parabolic equations with nonlinear source and their approximations, we show that if the initial energy is negative, then the solution always blows up in finite time. This is especially important for finding sufficiently simple and easy-to-verify conditions guaranteeing the presence of physical effects such as heat localization in peaking modes or thermal explosions and for deriving two-sided estimates of the solution lifespan. We construct the corresponding new classes of difference schemes for which grid analogs of integral conservation laws hold. We show that, to obtain efficient two-sided estimates for the blow-up time of the solution of the differential problem, in practice, one should use difference schemes with explicit as well as implicit approximation to the source.

Well-posedness of a class of non-homogeneous boundary value problems of the Korteweg-de Vries equation on a finite domain

In this paper, we study a class of Initial-Boundary Value Problems proposed by Colin and Ghidaglia for the Korteweg-de Vries equation posed on a bounded domain (0,L). We show that this class of Initial-Boundary Value Problems is locally well-posed in the classical Sobolev space H s (0,L) for s> � 3 , which provides a positive answer to one of the open questions of Colin and Ghidaglia (Adv.

The Degasperis–Procesi equation on the half-line

We analyze a class of initial-boundary value problems for the Degasperis–Procesi equation on the half-line. Assuming that the solution u(x,t) exists, we show that it can be recovered from its initial and boundary values via the solution of a Riemann–Hilbert problem formulated in the plane of the complex spectral parameter k.

On a class of initial-boundary value problems for equations of Korteweg-de Vries type

We consider a special class of initial-boundary value problems on the positive halfline x > 0 for the Korteweg-de Vries equation and its generalizations. For this class, we prove theorems on the nonexistence of global solutions for t > 0.

A homotopy perturbation technique for solving partial differential equations of fractional order in finite domains

In this paper, a homotopy perturbation technique is proposed to solve a class of initial-boundary value problems of partial differential equations of arbitrary (fractional) orders over finite domains. The basic idea of this technique is to utilize both the initial and boundary conditions in the recursive relation of the solution scheme so that we can obtain a good approximate solution. Numerical examples are presented to illustrate the validity of the proposed technique.

A tau approach for solution of the space fractional diffusion equation

Fractional differentials provide more accurate models of systems under consideration. In this paper, approximation techniques based on the shifted Legendre-tau idea are presented to solve a class of initial-boundary value problems for the fractional diffusion equations with variable coefficients on a finite domain. The fractional derivatives are described in the Caputo sense. The technique is derived by expanding the required approximate solution as the elements of shifted Legendre polynomials. Using the operational matrix of the fractional derivative the problem can be reduced to a set of linear algebraic equations. From the computational point of view, the solution obtained by this method is in excellent agreement with those obtained by previous work in the literature and also it is efficient to use.

Weighted finite difference methods for a class of space fractional partial differential equations with variable coefficients

A class of weighted finite difference methods (WFDMs) for solving a class of initial-boundary value problems of space fractional partial differential equations with variable coefficients is presented. Their stability and convergence properties are considered. It is proven that the WFDMs are unconditionally-stable for 0 ≤ r ≤ 1 2 , and conditionally-stable for 1 2 < r ≤ 1 , here r is the weighting parameter subjected to 0 ≤ r ≤ 1 . Some convergence results are given. These methods, problems and results generalize the corresponding methods, problems and results given in [7,8,10]. Some numerical examples are provided to show the effectiveness of the methods with different weighting parameters.