The solution of a nonclassic problem for one-dimensional hyperbolic equation using the decomposition procedure

In this article, we propose a new approach for solving an initial–boundary value problem with a non-classic condition for the one-dimensional wave equation. Our approach depends mainly on Adomian's technique. We will deal here with new type of nonlocal boundary value problems that are the solution of hyperbolic partial differential equations with a non-standard boundary specification. The decomposition method of G. Adomian can be an effective scheme to obtain the analytical and approximate solutions. This new approach provides immediate and visible symbolic terms of analytic solution as well as numerical approximate solution to both linear and nonlinear problems without linearization. The Adomian's method establishes symbolic and approximate solutions by using the decomposition procedure. This technique is useful for obtaining both analytical and numerical approximations of linear and nonlinear differential equations and it is also quite straightforward to write computer code. In comparison to traditional procedures, the series-based technique of the Adomian decomposition technique is shown to evaluate solutions accurately and efficiently. The method is very reliable and effective that provides the solution in terms of rapid convergent series. Several examples are tested to support our study.     

Using the iterative reproducing kernel method for solving a class of nonlinear fractional differential equations

In previous works, we have devoted to using the reproducing kernel methods solving integer order differential equations, based on the review of previous works, in this paper, we mainly present a method for solving a class of higher order fractional differential equations with general boundary value problems by using Taylor formula into reproducing kernel space. Its analytical solution is represented in the form of series. The analytical solution and approximate solution obtained by this method is given and it is uniformly converge to the exact solution and its corresponding derivatives. The numerical examples are studied to demonstrate the accuracy of the present method.     

ATOMFT: solving ODEs and DAEs using Taylor series

Taylor series methods compute a solution to an initial value problem in ordinary differential equations by expanding each component of the solution in a long Taylor series. The series terms are generated recursively using the techniques of automatic differentiation. The ATOMFT system includes a translator to transform statements of the system of ODEs into a FORTRAN 77 object program that is compiled, linked with the ATOMFT runtime library, and run to solve the problem. We review the use of the ATOMFT system for nonstiff and stiff ODEs, the propagation of global errors, and applications to differential algebraic equations arising from certain control problems, to boundary value problems, to numerical quadrature, and to delay problems.     

The Decomposition Method For Solving Of A Class Of Singular Two-Point Boundary Value Problems

In this paper, a class of nonlinear singular two-point boundary value problems are solved by using Adomian's decomposition methods. The approximate solution of this problem is calculated in the form of series with easily computable components. Finally, we give a nonlinear numerical example.     

A new algorithm based on differential transform method for solving multi-point boundary value problems

In this work, an efficient algorithm based on the differential transform method is applied to solve the multi-point boundary value problems. The solution obtained by using the proposed method takes the form of a convergent series with easily computable components. Several numerical examples, both linear and nonlinear, are given to testify the validity and applicability of the proposed method. Comparisons are made between the present method and the other existing methods.     

Thermal/structural dynamic analysis via approximate analytical approach

The present paper uses a classical Galerkin weighted residual formulation to obtain the approximate analytical solution of a thermally loaded beam executing free flexural vibrations. The approach used is one where the time variable is considered in the same manner as the spatial variable, and is included in the basis functions. The basis functions used in the approach are polynomials obtained from the terms of a power series, with the condition of nullity on the boundary. This choice simplifies the algebraic manipulations considerably and yields close form expressions for components of the system matrix. The latter also simplifies the numerical computation of coefficients of the approximating polynomial. The approach provides benefits in terms of increased accuracy and lower computational costs.     

A numerical study of annular dump diffuser flows

A series of flow calculations were conducted for a model annular dump diffuser. Because the presence of the multichanneled flow configuration in this model diffuser required careful numerical treatment, various numerics issues were studied. Among those issues, the influences of the numerical schemes and their interaction with the grid distribution are investigated. Proper grid distribution was crucial for obtaining an appropriate numerical solution. Grid distributions for this flow configuration were found to significantly affect the computing time as well as the characteristics of the calculated solutions. In addition, the downstream boundary conditions needed to yield the unique solution are discussed. Different flow characteristics caused by the change of downstream boundary conditions were observed. The importance of the flow profile at the dump diffuser inlet is also identified via the theory data comparison.     

Analytical solution of gas lubricated slider microbearing

The analytical solution of a two-dimensional, isothermal, compressible gas flow in a slider microbearing is presented. A higher order accuracy of the solution is achieved by applying the boundary condition of Kn ² order for the velocity slip on the wall, together with the momentum equation of the same order (known as the Burnett equation). The analytical solution is obtained by the perturbation analysis. The order of all terms in continuum and momentum equations and in boundary conditions is evaluated by incorporating the exact relation between the Mach, Reynolds and Knudsen numbers in the modelling procedure. Low Mach number flows in microbearing with slowly varying cross-sections are considered, and it is shown that under these conditions the Burnett equation has the same form as the Navier–Stokes equation. Obtained analytical results for pressure distribution, load capacity and velocity field are compared with numerical solutions of the Boltzmann equation and some semi-analytical results, and excellent agreement is achieved. The model presented in this paper is a useful tool for the prediction of flow conditions in the microbearings. Also, its results are the benchmark test for the verifications of various numerical procedures.     

Continuum solvation model: Computation of electrostatic forces from numerical solutions to the Poisson-Boltzmann equation

A rigorous formulation of the solvation forces (first derivatives) associated with the electrostatic free energy calculated from numerical solutions of the linearized Poisson-Boltzmann equation on a discrete grid is described. The solvation forces are obtained from the formal solution of the linearized Poisson-Boltzmann equation written in terms of the Green function. An intermediate region for the solute-solvent dielectric boundary is introduced to yield a continuous solvation free energy and accurate solvation forces. A series of numerical tests show that the calculated forces agree extremely well with finite-difference derivatives of the solvation free energy. To gain a maximum efficiency, the nonpolar contribution to the free energy is expressed in terms of the discretized grid used for the electrostatic problem. The current treatment of solvation forces can be used to introduce the influence of a continuum solvation model in molecular mechanics calculations of large biological systems.     

A new numerical method on American option pricing

Mathematically, the Black-Scholes model of American option pricing is a free boundary problem of partial differential equation. It is well known that this model is a nonlinear problem, and it has no closed form solution. We can only obtain an approximate solution by numerical method, but the precision and stability are hard to control, because the singularity at the exercise boundary near expiration date has a great effect on precision and stability for numerical method. We propose a new numerical method, FDA method, to solve the American option pricing problem, which combines advantages the Semi-Analytical Method and the Front-Fixed Difference Method. Using the FDA method overcomes the difficulty resulting from the singularity at the terminal of optimal exercise boundary. A large amount of calculation shows that the FDA method is more accurate and stable than other numerical methods.     

Constraint variational schemes for analysis of rectangular plates on an elastic half space

The flexural interaction of a rectangular thin elastic plate resting in smooth contact with an isotropic homogeneous elastic half space is analysed by using constraint variational schemes. The deflected shape of the plate is represented by a double power series of spatial variables with a set of generalized coordinates. The contact stresses are expressed in terms of the generalized coordinates by discretizing the contact area into several rectangular regions and solving an appropriate flexibility equation based on generalized Boussinesq's solution. Using the representations adopted for displacement and contact stresses, a constraint energy functional is constructed to determine the generalized coordinates. The constraint term in the variational functional corresponds to plate edge boundary conditions and formulations corresponding to both Lagrange multiplier and penalty types are presented. It is noted that for the present class of problems, penalty type formulations are numerically efficient. The convergence and numerical stability of the solution scheme is confirmed. Selected numerical results are presented to illustrate the dependence of flexural response of plate on the governing parameters of the plate-half space system.     

Comparison of numerical methods for solving Liapunov matrix equations†

A numerical comparison is made of most published methods for solving the linear matrix equations which arise when a quadratic form Liapunov function is applied to a constant linear system (continuous or discrete, real or complex). Generally, for the real equations direct methods are satisfactory for systems of order ten or less, whereas for larger order systems iterative methods (based upon expressing the solution in terms of an infinite series) are to be preferred. For the complex equations the most convenient numerical method uses an explicit representation for the solution in terms of the eigenvalues and vectors of the system matrix. If the system matrix is in companion form then algorithms taking account of this structure offer minor improvements.     

Dual series representation and its applications to a string subjected to support motions

Dual series representation (DSR) for the dynamic response of a finite elastic body subjected to boundary traction and boundary support excitations is proposed in this paper. To confirm the validity of the present model, a string subjected to support motions is solved. Four analytical methods including (1) a diamond rule, (2) a series solution with the quasi-static decomposition method, (3) DSR by the Cesáro sum technique, and (4) DSR by the Stokes' transformation method are presented. It is found that the numerical results obtained by using these four methods are in good agreement, and that both the Cesáro sum and Stokes' transformation regularization techniques can extract the finite part of the divergent series. The advantages and disadvantages of these four methods are discussed. In comparison with the quasi-static decomposition method and the Cesáro sum technique, the Stokes' transformation is the best way not only because it is free from calculation of the quasi-static solution, but also because its convergence rate is as fast as that of the mode acceleration method.     

Postbuckling analysis of plates on an elastic foundation by the boundary element method

The postbuckling behavior of plates on an elastic foundation is studied by using the boundary element method (BEM). A new fundamental solution of lateral deflection is derived through the resolution theory of a differential operator, and a set of boundary element formulae in incremental form is presented. By using these formulae, the BEM solution procedure becomes relatively simple. The results of a number of numerical examples are compared with existing solutions and good agreement is observed. It shows that the proposed method is effective for solving the postbuckling problems of plates with arbitrary shape and various boundary conditions.     

Application of fractional order theory of magneto-thermoelasticity to an infinite perfect conducting body with a cylindrical cavity

A fractional model of the equations of generalized magneto-thermoelasticity for a perfect conducting isotropic thermoelastic media is given. This model is applied to solve a problem of an infinite body with a cylindrical cavity in the presence of an axial uniform magnetic field. The boundary of the cavity is subjected to a combination of thermal and mechanical shock acting for a finite period of time. The solution is obtained by a direct approach by using the thermoelastic potential function. Laplace transform techniques are used to derive the solution in the Laplace transform domain. The inversion process is carried out using a numerical method based on Fourier series expansions. Numerical computations for the temperature, the displacement and the stress distributions as well as for the induced magnetic and electric fields are carried out and represented graphically. Comparisons are made with the results predicted by the generalizations, Lord–Shulman theory, and Green–Lindsay theory as well as to the coupled theory.

     

Series solution to the Pochhammer-Chreeequation and comparison with exact solutions

In this study, a decomposition method for approximating the solution of the Pochhammer-Chreeequation is implemented. By using this scheme, explicit exact solution is calculated in the form of a convergent power series with easily computable components. To illustrate the application of this method numerical results are derived by using the calculated components of the decomposition series. The obtained results are found to be in good agreement with the exact solutions known for some special cases.     

基于有限元的一类stefan问题数值模型研究

该文给出基于有限元方法的一类一维stefan问题的数值求解过程及算法.模型的建立基于已知的相变界面和固定边界处测得的温度和热流.模型的精度通过与Neumann获得的解析解的比较而得到验证.文中所讨论的模型可以用于反Stefan问题中自由边界的实时跟踪或者控制.最后,比较了已有的有限元模型,给出了仿真结果.     

A method for solving nonlinear time-dependent drainage model

The purpose of this paper is to present a method for solving nonlinear time-dependent drainage model. This method is based on the perturbation theory and Laplace transformation. The proposed technique allows us to obtain an approximate solution in a series form. The computed results are in good agreement with the results of Adomian decomposition method. Results are presented graphically and in tabulated forms to study the efficiency and accuracy of method. The present approach provides a reliable technique, which avoids the tedious work needed by classical techniques and existing numerical methods. The nonlinear time-dependent drainage model is solved without linearizing or discretizing the nonlinear terms of the equation. The method does not require physically unrealistic assumptions, linearization or discretization in order to find the solutions of the given problems.     

An efficient numerical technique for the solution of nonlinear singular boundary value problems

In this work, a new technique based on Green’s function and the Adomian decomposition method (ADM) for solving nonlinear singular boundary value problems (SBVPs) is proposed. The technique relies on constructing Green’s function before establishing the recursive scheme for the solution components. In contrast to the existing recursive schemes based on the ADM, the proposed technique avoids solving a sequence of transcendental equations for the undetermined coefficients. It approximates the solution in the form of a series with easily computable components. Additionally, the convergence analysis and the error estimate of the proposed method are supplemented. The reliability and efficiency of the proposed method are demonstrated by several numerical examples. The numerical results reveal that the proposed method is very efficient and accurate.     

Solution of time-domain Maxwell equation with PML by using modified Laguerre polynomials

In this work, a stable numerical algorithm proposed by Chung et al. for the time-domain Maxwell equations is generalized. The time-domain Maxwell equations are solved by expressing the transient behaviors in terms of the modified Laguerre polynomials, and then the original equations of the initial value and boundary value can be transformed into a series of problems independent of the time variable. In this case the method of finite difference （FD）, the finite element method （FEM）, the method of moment （MoM）, etc. or the combination of these methods can be used to solve the problems. Finally, a numerical model is provided for the scattering problem with perfect matched layer （PML） by using FD. The comparison between the results of the proposed method and FDTD is presented to verify the proposed new method.