A boundary value problem with an integral condition for a certain fractional differential equation

We suggest and analyze a technique by combining the variational iteration method and the homotopy perturbation method. This method is called the variational homotopy perturbation method (VHPM). We use this method for solving higher dimensional initial boundary value problems with variable coefficients. The developed algorithm is quite efficient and is practically well suited for use in these problems. The proposed scheme finds the solution without any discritization, transformation, or restrictive assumptions and avoids the round‐off errors. Several examples are given to check the reliability and efficiency of the proposed technique.

In this paper, we discuss the higher dimensional initial boundary value problems of variable coefficients. Our approach rests mainly on Adomian decomposition method to construct the analytic solutions for these equations. The method is quite efficient and is practically well suited for use in these problems. The analysis is accompanied by numerical examples. The obtained results demonstrate the reliability and the efficiency of the method.

In this article, higher dimensional initial boundary value problems of variable coefficients are solved by means of an analytic technique, namely the Homotopy analysis method (HAM). Comparisons are made between the Adomian decomposition method (ADM), the exact solution and the homotopy analysis method. The results reveal that the proposed method is very effective and simple. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010

This paper deals with a singular two dimensional initial boundary value problem for a Caputo time fractional parabolic equation supplemented by Neumann and non-local boundary conditions. The well posedness of the posed problem is demonstrated in a fractional weighted Sobolev space. The used method based on some functional analysis tools has been successfully showed its efficiency in proving the existence, uniqueness and continuous dependence of the solution upon the given data of the considered problem. More precisely, for proving the uniqueness of the solution of the posed problem, we established an energy inequality for the solution from which we deduce the uniqueness. For the existence, we proved that the range of the operator generated by the considered problem is dense.

Part I: We consider the numerical solution of the Navier-Stokes equations governing the unsteady flow of a viscous incompressible fluid. The analysis of numerical approximations to smooth nonlinear problems reduces to the examination of related linearized problems. The well posedness of the linear Navier-Stokes equations and the stability of finite difference approximations are studied by making energy estimates for the initial boundary value problems. Flows with open boundaries (i.e., inflow and outflow) and with solid walls are considered. We analyse boundary conditions of several types involving the velocity components or a combination of the velocity components and the pressure. The properties of these different types of boundary conditions are compared with emphasis on the suppression of undesirable numerical boundary layers for high Reynolds number calculations. The formulation of the Navier-Stokes equations which uses an elliptic equation for the pressure in lieu of the divergence equation for the velocity is shown to be equivalent to the usual formulation if the boundary conditions are treated correctly. The stability of numerical methods which use this formulation is demonstrated. Part II: We consider the numerical solution of the stream function vorticity formulation of the two dimensional incompressible Navier-Stokes equations for unsteady flows on a domain with rigid walls. The no-slip boundary conditions on the velocity components at the rigid walls are prescribed. In the stream function vorticity formulation these become two boundary conditions on the stream function and there is no explicit boundary condition on the vorticity. The accuracy of the numerical approximations to the stream function and the vorticity is investigated.The common approach in calculations is to employ second order accurate finite difference approximations for all the space derivatives and the boundary conditions together with a time marching procedure involving iteration at each time step to satisfy the boundary conditions. With such schemes the vorticity may be only first order accurate. Higher order approximations to the no-slip boundary conditions have frequently been used to overcome this problem. A one dimensional initial boundary value problem containing the salient features is proposed and analysed. With the use of this model, the behaviour observed in calculations is explained.

Three dimensional initial boundary value problem of the Navier-Stokes equation is considered. The equation is split in an Euler equation and a non-stationary Stokes equation within each time step. Unlike the conventional approach, we apply a non-homogeneous Stokes equation instead of homogeneous one. Under the hypothesis that the original problem possesses a smooth solution, the estimate of theHs+1 norm, 0≦s<3/2, of the approximate solutions and the order of theL2 norm of the errors is obtained.

A new definition of conditional invariance for boundary value problems involving a wide range of boundary conditions (including initial value problems as a special case) is proposed. It is shown that other definitions worked out in order to find Lie symmetries of boundary value problems with standard boundary conditions, followed as particular cases from our definition. Simple examples of direct applicability to the nonlinear problems arising in applications are demonstrated. Moreover, the successful application of the definition for the Lie and conditional symmetry classification of a class of (1 + 2)-dimensional nonlinear boundary value problems governed by the nonlinear diffusion equation in a semi-infinite domain is realised. In particular, it is proven that there is a special exponent, k ≠ -2, for the power diffusivity uk when the problem in question with non-vanishing flux on the boundary admits additional Lie symmetry operators compared to the case k ≠ -2. In order to demonstrate the applicability of the symmetries derived, they are used for reducing the nonlinear problems with power diffusivity uk and a constant non-zero flux on the boundary (such problems are common in applications and describing a wide range of phenomena) to (1 + 1)-dimensional problems. The structure and properties of the problems obtained are briefly analysed. Finally, some results demonstrating how Lie invariance of the boundary value problem in question depends on the geometry of the domain are presented.

Transform solution for the two dimensional problem of waves in liquid filled viscoelastic tubes

Stretching of polycarbonate films leads to the formation of shear bands in the necking zone [1]. Standard viscoplastic material models render mesh size dependent results, which requires a mathematical regularization. To this end, we present a finite strain gradient theory for a viscoplastic, isotropic material model where we extend the model presented in [2] to a micromorphic model by introducing a new micromorphic variable as an additional degree of freedom with its first gradient [3, 4]. The variable here has the meaning of a micro plastic strain, and is coupled with the macro plastic by a micro penalty term, forcing the macro‐plastic strain to be close to the micro‐plastic strain for the targeted shear band regularization effect. We have implemented the model equations as a three dimensional initial boundary value problem in an in house FE‐tool, to simulate different geometries with different thickness and to compare it the experimental tests. The analysis is performed for a uniaxial tensile geometry as well as for a biaxial tensile geometry. The numerical examples show the ability of the model to regularize the shear bands and solve the problem of localization.

The coupling of the homotopy perturbation method (HPM) and the variational iteration method (VIM) is a strong technique for solving higher dimensional initial boundary value problems. In this article, after a brief explanation of the mentioned method, the coupled techniques are applied to one-dimensional heat transfer in a rectangular radial fin with a temperature-dependent thermal conductivity to show the effectiveness and accuracy of the method in comparison with other methods. The graphical results show the best agreement of the three methods; however, the amount of calculations of each iteration for the combination of HPM and VIM was reduced markedly for multiple iterations. It was found that the variation of the dimensionless temperature strongly depends on the dimensionless small parameter \({\varepsilon_1}\). Moreover, as the dimensionless length increases, the thermal conductivity of the fin decreases along the fin.

<abstract> This paper studies the existence and uniqueness of solutions of a non local initial boundary value problem of a singular two dimensional nonlinear fractional order partial differential equation involving the Caputo fractional derivative by employing the functional analysis. We first establish for the associated linear problem a priori estimate and prove that the range of the operator generated by the considered problem is dense. The technique of obtaining the a priori bound relies on the construction of a suitable multiplicator. From the resulted a priori estimate, we can establish the solvability of the associated linear problem. Then, by applying an iterative process based on the obtained results for the associated linear problem, we establish the existence, uniqueness and continuous dependence of the weak solution of the considered nonlinear problem. </abstract>

Multiple nontrivial solutions for a class of higher dimensional discrete boundary value problems

Difference equations have many applications in discrete dynamical system [1,2]. In the previous article [3], the author has successfully solved the (one dimensional) boundary value problems involving heat distributions in lumped rods by difference equation method. In this paper, the author extends this method to solve the two dimensional problems, and find that the results are very near to that obtained by differential equation method in forms, and is relatively simpler to that by the later.

In the present paper, identification elliptic problem in a Hilbert space with Dirichlet and integral boundary conditions is discussed. Identification problem is reduced to auxiliary nonlocal boundary value problem with nonlocal integral condition. Operator approach is used to prove stability and coercive stability inequalities for solution of source identification elliptic problem in the case of a self-adjoint positive definite operator in a differential equation. As applications, four mixed boundary value problems for strongly elliptic multidimensional partial differential equation are investigated. Theorems on stability of solutions of these boundary value problems are established.

In this paper, we consider direct and inverse problems for the two-dimensional wave equation. The direct problem is an initial boundary value problem for this equation with nonlocal boundary conditions. In the inverse problem, it is required to find the time-variable coefficient at the lower term of the equation. The classical solution of the direct problem is presented in the form of a biorthogonal series in eigenvalues and associated functions, and the uniqueness and stability of this solution are proven. For solution to the inverse problem, theorems of existence in local, uniqueness in global, and an estimate of conditional stability are obtained. The problems of determining the right-hand sides and variable coefficients at the lower terms from initial boundary value problems for second-order linear partial differential equations with local boundary conditions have been studied by many authors. Since the nonlinearity is convolutional, the unique solvability theorems in them are proven in a global sense. In the works, the method of separation of variables is used to find the classical solution of the direct problem in the form of a biorthogonal series in terms of eigenfunctions and associated functions. The nonlocal integral condition is used as the overdetermination condition with respect to the solution of the direct problem. The direct problem reduces to equivalent integral equations of the Fourier method. To establish integral inequalities, the generalized Gronwall-Bellman inequality is used. We obtain an a priori estimate of the solution in terms of an unknown coefficient, that are useful for studying the inverse problem.