Solution Of Initial-boundary Value Problem Research Articles

Existence Local and Global of Solution of Initial-Boundary Value Problem for Class of Fractional p−Laplacian Equation with Logarithmic Nonlinearity

Existence Local and Global of Solution of Initial-Boundary Value Problem for Class of Fractional p−Laplacian Equation with Logarithmic Nonlinearity

Визначення двовимірних нестаціонарних температурних полів у плитах та панелях з плоскопаралельними межами за наявності джерел тепла

Non-stationary two-dimensional problems of thermal conductivity for plates and panels with plane-parallel boundaries in the presence of volume-distributed non-stationary heat sources are formulated. A method of constructing a solution to the formulated heat conduction problems for the considered bodies is proposed. The technique uses the approximation of the temperature distribution in both elements by the thickness variable by a cubic polynomial. The coefficients of the approximation polynomial are given through the integral over the thickness variable temperature characteristics and the conditions for the boundary values of the temperature on the outer surfaces of the plate and panel. As a result, the initial two-dimensional initial boundary value problems for the temperature for the plate and panel are reduced to one-dimensional initial boundary value problems for the integral temperature characteristics. To construct the solution of the initial-boundary value problem for the integral characteristics of the temperature in the case of a plate infinite in longitudinal and transverse coordinates, the integral Laplace transform in time and the integral Fourier transform in the longitudinal coordinate were used. The solution of the problem on the integral temperature characteristics in the case of the panel is found using the integral Laplace transform in time and the finite integral transform in the transverse coordinate. Expressions of integral temperature characteristics for the plate and panel are obtained in the form of convolutions of functions corresponding to homogeneous solutions of initial-boundary value problems for integral temperature characteristics and functions describing the available non-stationary heat sources in these bodies and given surface temperature values. The general solutions of the two-dimensional initial boundary value problems of thermal conductivity for slabs and panels are recorded for the presence of arbitrarily variable spatial coordinates of non-stationary heat sources and the conditions of convective heat exchange with the external environment on the surfaces of the considered bodies.

A combined BEM and Laplace transform for unsteady modified-Helmholtz equation of time–space variable coefficients for anisotropic media

PurposeTwo-dimensional (2D) problems are governed by unsteady anisotropic modified-Helmholtz equation of time–space dependent coefficients are considered. The problems are transformed into a boundary-only integral equation which can be solved numerically using a standard boundary element method (BEM). Some examples are solved to show the validity of the analysis and examine the accuracy of the numerical method.Design/methodology/approach The 2D problems which are governed by unsteady anisotropic modified-Helmholtz equation of time–space dependent coefficients are solved using a combined BEM and Laplace transform. The time–space dependent coefficient equation is reduced to a time-dependent coefficient equation using an analytical transformation. Then, the time-dependent coefficient equation is Laplace transformed to get a constant coefficient equation, which can be written as a boundary-only integral equation. By utilizing a BEM, this integral equation is solved to find numerical solutions to the problems in the frame of the Laplace transform. These solutions are then inversely transformed numerically to obtain solutions in the original time–space frame.FindingsThe main finding of this research is the derivation of a boundary-only integral equation for the solutions of initial-boundary value problems governed by a modified-Helmholtz equation of time–space dependent coefficients for anisotropic functionally graded materials with time-dependent properties.Originality/valueThe originality of the research lies on the time dependency of properties of the functionally graded material under consideration.

Numerical solution of initial-boundary value problems for a multi-dimensional pseudoparabolic equation

Numerical solution of initial-boundary value problems for a multi-dimensional pseudoparabolic equation

Application of a Robust Multigrid Technique for the Parallel Solution of Initial-Boundary Value Problems

Application of a Robust Multigrid Technique for the Parallel Solution of Initial-Boundary Value Problems

Asymptotic behavior of solutions of initial-boundary value problems for 1D viscous and heat-conducting ionized gas

Dynamics of the macroscopic motion of a compressible, viscous and heat-conducting, one-dimensional monatomic ionized gas can be described by a compressible Navier-Stokes type system. Compared with the corresponding compressible Navier-Stokes type systems modeling a viscous and heat conducting ideal polytropic gas [7,10,13–16,18,19,23,24,26,28,31,32], the real gases [17], nonlinear thermoviscoelasticity [5,6], and a viscous radiative and reactive gas [11,12,20,21,30], the dependence of the state of an ionized gas on the degree of ionization leads to the loss of concavity of the physical entropy in some small bounded domain. Such a property of the ionized gas makes it impossible to deduce the dissipative estimates on the first-order spatial derivative of the bulk velocity and the absolute temperature from the basic energy estimates obtained through the normalized entropy and, as a consequence, makes the construction of its global large solutions and the study of their large time behavior more difficult.By making full use of the intrinsic structure of equations, the existence of global large smooth solutions to two types of initial boundary value problems of one-dimensional, compressible Navier-Stokes system for a viscous and heat-conducting ionized gas is established in [22] for a class of density and/or temperature dependent transport coefficients, but since the estimates obtained there depend on the time variable, the problem on their large time behavior remains unsolved and the main purpose of this paper is to give the precise description of the large time behavior of the global solutions constructed in [22]. The key ingredient in our analysis is to deduce the uniform-in-time positive lower and upper bounds on the specific volume and the absolute temperature.

Bifurcation Analysis and Numerical Study of Wave Solution for Initial-Boundary Value Problem of the KdV-BBM Equation

In this work, we study the bifurcation and the numerical analysis of the nonlinear Benjamin-Bona-Mahony-KdV equation. According to the bifurcation theory of a dynamic system, the various kinds of traveling wave profiles are obtained including the behavior of solitary and periodic waves. Additionally, a two-level linear implicit finite difference algorithm is implemented for investigating the Benjamin-Bona-Mahony-KdV model. The application of a priori estimation for the approximate solution also provides the convergence and stability analysis. It was demonstrated that the current approach is singularly solvable and that both time and space convergence are of second-order precision. To confirm the computational effectiveness, two numerical simulations are prepared. The findings show that the current technique performs admirably in terms of delivering second-order accuracy in both time and space with the maximum norm while outperforming prior schemes.

A Finite Difference Approximation for Numerical Simulation of 2D Viscous Coupled Burgers Equations

Many of the physical phenomena in nature are usually expressed in terms of algebraic, differential or integral equations.Several nonlinear phenomena playing a very important role in engineering sciences, physics and computational mathematics are usually modeled by those non-linear partial differential equations (PDEs). It is usually difficult and problematic to examine and find out nalytical solutions of initial-boundary value problems consisting of PDEs. In fact, there is no a certain method or technique working well for all these type equations. For this reason, their approximate solutions are usually preferred rather than analytical ones of such type equations. Thus, many researchers are concentrated on approximate methods and techniques to obtain numerical solutions of non-linear PDEs. In the present article, the numerical simulation of the two-dimensional coupled Burgers equation (2D-cBE) has been sought by finite difference method based on Crank-Nicolson type approximation. Widely used three test examples given with appropriate initial and boundary conditions are used for the simulation process. During the simulation process,the error norms $L_{2}$, $L_{\infty}$ are calculated if the exact solutions are already known, otherwise the pointwise values and graphics are provided for comparison. The newly obtained error norms $L_{2}$, $L_{\infty}$ by the presented schemes are compared with those of some of the numerical solutions in the literature. A good consistency and accuracy are observed both by numerical values and visual illustrations.

On Inner Regularity of Solutions of Two-Dimensional Zakharov–Kuznetsov Equation

In this paper, we consider questions of inner regularity of weak solutions of initial-boundary value problems for the Zakharov–Kuznetsov equation with two spatial variables. The initial function is assumed to be irregular, and the main parameter governing the regularity is the decay rate of the initial function at infinity. The main results of the paper are obtained for the problem on a half-strip. In this problem, different types of initial conditions (e.g., Dirichlet or Neumann conditions) influence the inner regularity. We also give a survey of earlier results for other types of domains: a plane, a half-plane, and a strip.

AN INITIAL-BOUNDARY VALUE PROBLEM FOR KELVIN-VOIGT EQUATIONS WITH (p(x), q(x),m(x))- STRUCTURE

The proof of the global in time existence of solutions of initial-boundary value problems for nonlinear equations in many The proof of the global in time existence of solutions of initial-boundary value problems for nonlinear equations in many cases,  is not easy, even in some cases it is impossible. However, by showing some qualitative properties of its solution, one can find answers to such questions. For example, by establishing the blow up in a finite time property of a solution, one can show that a solution does not exist globally in time. Such way, in last years, the investigating the quality properties of solutions as localization and/or blow up in a finite time, has been developing rapidly In this paper, we study the initial-boundary value problem for the Kelvin-Voigt equations with both of diffusion and relaxation term, modified by (p(x),q(x))-Laplacian, respectively, and with the variable exponent damping term. The damping term in the equation realizes as nonlinear source term. In this work, we study the nonlinear initial-boundary value problem for generalized Kelvin-Voigt equations describing the motion of incompressible viscoelastic non-Newtonian fluids. The equations generalized by replacing the diffusion and relaxation terms in equation with p(x)-Laplacian and q(x)-Laplacian, respectively, and adding a nonlinear absorption term with variable exponents and coefficients. The definition of a weak solution is given. Under suitable conditions for variable exponents and coefficients, and data of the problem, the blow up of weak solutions are established.

Uniqueness of Solutions of Initial-Boundary Value Problems for Parabolic Systems with Dini-Continuous Coefficients in Domains on the Plane

Uniqueness of Solutions of Initial-Boundary Value Problems for Parabolic Systems with Dini-Continuous Coefficients in Domains on the Plane

Initial-boundary value problem for the time-fractional degenerate diffusion equation

In this paper the initial-boundary value problems for the one-dimensional linear time-fractional diffusion equations with the time-fractional derivative ∂ α t of order α ∈ (0, 1) in the variable t and time-degenerate diffusive coefficients t β with β ≥ 1 − α are studied. The solutions of initialboundary value problems for the one-dimensional time-fractional degenerate diffusion equations with the time-fractional derivative ∂ α t of order α ∈ (0, 1) in the variable t, are shown. The second section present Dirichlet and Neumann boundary value problems, and in the third section has shown the solutions of the Dirichlet and Neumann boundary value problem for the one-dimensional linear time-fractional diffusion equation. The solutions of these fractional diffusive equations are presented using the Kilbas-Saigo function Eα,m,l(z). The solution of the problems is discovered by the method of separation of variables, through finding two problems with one variable. The existence and uniqueness to the solution of the problem are confirmed. In addition, the convergence of the solution has been proven using the estimate for the Kilbas-Saigo function Eα,m,l(z) from [13] and Parseval’s identity

Метод сеток приближенного решения начально-краевых задач для обобщенных уравнений конвекции-диффузии

В прямоугольной области исследуются начально-краевые задачи для одномерных по пространству обобщенных уравнений конвекции-диффузии с оператором Бесселя и дробными производными в смысле Римана~--- Лиувилля и Капуто порядка $\alpha$ (0<$\alpha$<1) и с граничные условия первого и третьего рода. Уравнение конвекции-диффузии дробного порядка с оператором Бесселя возникает при переходе от трехмерного уравнения конвекции-диффузии дробного порядка к цилиндрическим (сферическим) координатам, в случае, когда решение $u=u(r)$ не зависит ни от~$z$, ни от~$\varphi$. Для численного решения рассматриваемых задач строятся монотонные разностные схемы второго порядка точности по параметрам сетки, аппроксимирующие эти задачи на равномерных сетках. С~помощью метода энергетических неравенств для решения начально-краевых задач получены априорные оценки в дифференциальной и разностной трактовках при предположении существования регулярного решения исходной дифференциальной задачи. Из полученных априорных оценок следуют единственность и устойчивость решения по правой части и начальным данным, а также в~силу линейности разностных задач сходимость решения соответствующей разностной задачи к~решению исходной дифференциальной задачи со скоростью $O(h^2+\tau^2)$.

Soliton solutions of the nonlinear Schrödinger equation with defect conditions

Motivated by a recent development in the derivation of soliton solutions for initial-boundary value problems via the dressing method, we reconsider solutions to the nonlinear Schrödinger (NLS) equation on two half-lines connected via integrable defect conditions. The dressing method to construct soliton solutions is applied, while preserving the spectral boundary constraint with a time-dependent defect matrix. Moreover, N-soliton solutions are explicitly constructed and it is proven that the solitons are transmitted through the defect independently of one another.

On the Numerical Solution of Initial-Boundary Value Problems for the Convection-Diffusion Equation with a Fractional Caputo Derivative and a Nonlocal Linear Source

In a rectangular domain the first and third initial-boundary value problems are studied for the one-dimensional with respect to the spatial variable diffusion convection equation with a fractional Caputo derivative and a nonlocal linear source of integral form. Using the method of energy inequalities, under the assumption of the existence of a regular solution, a priori estimates are obtained in differential form, which implies the uniqueness and continuous dependence of the solution on the input data of the problem. On a uniform grid, two difference schemes are constructed that approximate the first and third initial-boundary value problems, respectively. For the solution of the difference problems, a priori estimates are obtained in the difference interpretation. The obtained estimates in difference form imply uniqueness and stability, as well as convergence at a rate equal to the order of the approximation error. An algorithm for the approximate solution of the third boundary value problem is constructed, numerical calculations of test examples are carried out, illustrating the theoretical results obtained in this work.

A stable implementation measure of multi-transmitting formula in the numerical simulation of wave motion.

The Multi-Transmitting Formula (MTF) proposed by Liao et al. is a local artificial boundary condition widely used in numerical simulations of wave propagation in an infinite medium, while the drift instability is usually caused in its numerical implementation. In view of a physical interpretation of the Gustafsson, Kreiss and Sundström criterion on numerical solutions of initial-boundary value problems in the hyperbolic partial differential equations, the mechanism of the drift instability of MTF was discussed, and a simple measure for eliminating the drift instability was proposed by introducing a modified operator into the MTF. Based on the theory of spherical wave propagation and damping effect of medium, the physical implication on modified operator was interpreted. And the effect of the modified operator on the reflection coefficient of MTF was discussed. Finally, the validity of the proposed stable implementation measure was verified by numerical tests of wave source problem and scattering problem.

On the connection between solutions of initial boundary-value problems for a some class of integro-differential PDE and a linear hyperbolic equation

We consider the second initial boundary-value problem for a certain class of second-order integro-differential PDE with integral operator. The connection of its solution with the solution of the standard second linear initial boundary-value problem for the hyperbolic equation is shown. Thus, the nonlinear problem is reduced to a standard linear problem, whose numerical solution can be obtained, for example, by the Fourier method or Galerkin method. The article provides examples of five integro-differential equations for various integral operators as particular representatives of the class of integro-differential equations for a better understanding of the problem. The application of the main theorem to these examples is shown. Some simple natural requirement is imposed on the integral operator; so, in four cases out of five the problem’s solution satisfies some phase constraint. The form of these constraints is of particular interest for the further research.

On Inner Regularity of Solutions of Two-Dimensional Zakharov-Kuznetsov Equation

In this paper, we consider questions of inner regularity of weak solutions of initial-boundary value problems for the Zakharov-Kuznetsov equation with two spatial variables. The initial function is assumed to be irregular, and the main parameter governing the regularity is the decay rate of the initial function at infinity. The main results of the paper are obtained for the problem on a semistrip. In this problem, different types of initial conditions (e. g., Dirichlet or Neumann conditions) influence the inner regularity. We also give a survey of earlier results for other types of areas: a plane, a half-plane, and a strip.

Parabolic Minkowski convolutions and concavity properties of viscosity solutions to fully nonlinear equations

This paper is concerned with the Minkowski convolution of viscosity solutions of fully nonlinear parabolic equations. We adopt this convolution to compare viscosity solutions of initial-boundary value problems in different domains. As a consequence, we can for instance obtain parabolic power concavity of solutions to a general class of parabolic equations. Our results are applicable to the Pucci operator, the normalized q-Laplacians with 1<q≤∞, the Finsler Laplacian, and more general quasilinear operators.

Green\u2019s function homotopy perturbation method for the initial-boundary value problems

This article deals with the novel method for finding solutions for the initial-boundary value problems (IBVPs), which is called the Sawangtong’s Green function homotopy perturbation method, shortly called SGHPM. The SGHPM is a method which combines the homotopy perturbation method with Green’s function method. The convergence analysis for the SGHPM is shown. Furthermore, some examples are presented to illustrate the validity of the proposed method and to ensure that SGHPM is a technique which is powerful and efficient for finding approximate analytic solutions of IBVPs.