Initial-boundary Value Research Articles

Initial Boundary Value and Inverse Coefficient Problems for One-Dimensional Fractional Diffusion Equation in a Half-Line

Initial Boundary Value and Inverse Coefficient Problems for One-Dimensional Fractional Diffusion Equation in a Half-Line

A generalized Sasa–Satsuma equation on the half line: From Dirichlet to Neumann map

In this paper, we study the initial-boundary value (IBV) problem for a generalized Sasa–Satsuma equation with [Formula: see text] Lax pair by Fokas unified method on the half line. Based on the analyticity and asymptotics of the eigenfunctions, the IBV problem is formulated as a Riemann–Hilbert (RH) problem. Further, the global relation among IBVs is established and the map from the Dirichlet boundary value to Neumann boundary value is obtained.

Partial Fourier Transform Methods to Solve the Solution Formula of Stokes Equation in Half-Space

Fluids are a shape of a matter which have substance liquids, gases and plasmas. In our daily life, fluids become important part, such as part of our blood and also help our body getting nutrients. It is well known that fluid motion can be described in mathematical model in especially in form of partial differential equations (PDE) and called as Navier Stokes Equations (NSE). The Navier Stokes equation is derived from balance of conservation of mass and conservation of momentum. In this paper, we consider the solution formula of the linearized of the Navier Stokes Equation (NSE) with the initial boundary value (IBV) problem in half space without surface tension. The model problem under consideration covers of non-linear fluid type. We solve the solution formula of velocity and density of the model problem by using Fourier transform and partial Fourier transform method. The strategy geting the solution of the model problem is based on the analysis of some resolvent of the model problem which obtained by using Laplace transform of the Stokes equations. Therefore, In particular, the formula of velocity and density of the Stokes equation are obtained.

Partial Fourier Transform Methods to Solve the Solution Formula of Stokes Equation in Half-Space

Fluids are a shape of a matter which have substance liquids, gases and plasmas. In our daily life, fluids become important part, such as part of our blood and also help our body getting nutrients. It is well known that fluid motion can be described in mathematical model in especially in form of partial differential equations (PDE) and called as Navier Stokes Equations (NSE). The Navier Stokes equation is derived from balance of conservation of mass and conservation of momentum. In this paper, we consider the solution formula of the linearized of the Navier Stokes Equation (NSE) with the initial boundary value (IBV) problem in half space without surface tension. The model problem under consideration covers of non-linear fluid type. We solve the solution formula of velocity and density of the model problem by using Fourier transform and partial Fourier transform method. The strategy geting the solution of the model problem is based on the analysis of some resolvent of the model problem which obtained by using Laplace transform of the Stokes equations. Therefore, In particular, the formula of velocity and density of the Stokes equation are obtained.

NODAL REACTIONS AND COEFFICIENTS OF THE STIFFNESS MATRIX OF A FINITE ELEMENT BASED ON THE REPRESENTATION OF DISPLACEMENTS BY POLYNOMIALS

The study of prismatic bodies with constants along one of the coordinates of mechanical and geometric parameters is most appropriate to conduct on the basis of the semianalytical finite element method (NMSE). Its essence is a combination of finite element sampling and decomposition of displacements in the characteristic direction by a system of trigonometric coordinate functions.In [8, 15], a variant of the semivanalytic finite element method for the calculation of prismatic bodies when used as a system of coordinate functions of Fourier series was developed. The use of trigonometric series provides maximum efficiency of the semi-analytical finite element method, however, only the boundary conditions corresponding to the object's support on an absolutely rigid in its plane and flexible diaphragm can be satisfied at the ends of the body.As a result of the performed researches, the basis of the representation of displacements by polynomials is obtained, which allows to significantly expand the range of boundary conditions at the ends of the body. In this case, it isnot possible to reduce the solution of the initial spatial boundary value problem to a sequence of two-dimensional problems, so a reasonable choice of appropriate polynomials becomes especially important.Both the conditionality of the matrix of the system of separate equations and, consequently, the convergence of integration algorithms for its solution, and the universality of the approach to the possibility of satisfying different variants of boundary conditions at the ends of the body depend on their correct choice.In addition, the question of methods of integration in the calculation of the coefficients of the stiffness matrix of a finite element (CE), which is quite common, due to the significant complexity of this procedure.

Singular Perturbation Method Applied to Power Factor Correction Converter Application

A linear discrete stable control system is considered. The Power Factor Correction (PFC) converter to allow independent control of current and voltage. It converter are fast and slow states to inheres sty present small parameters inductor and capacitor its computes stiffness and to include switching ripple effects. As an alternative a Singular Perturbation Method (SPM) is presented Initial Value Problem (IVP) and Boundary Value Problem (BVP). It is applied to two state switching power converters to provide rigorous justification of the time scale separation. It is modeled as a one parameter singularly perturbed system. SPM consists of an outer series solution and one boundary layer correction (BLC) solution. A boundary layer correction is required to recover the initial conditions lost in the process of degeneration and to improve the solution. SPM is carried out up to second-order approximate solution for the PFC converter model for IVP and BVP. The results are compared with the exact solution (between with and without parameters). The results substantiate the application.

K-Version of Finite Element Method for BVPs and IVPs

The paper presents k-version of the finite element method for boundary value problems (BVPs) and initial value problems (IVPs) in which global differentiability of approximations is always the result of the union of local approximations. The higher order global differentiability approximations (HGDA/DG) are always p-version hierarchical that permit use of any desired p-level without effecting global differentiability. HGDA/DG are true Ci, Cij, Cijk, hence the dofs at the nonhierarchical nodes of the elements are transformable between natural and physical coordinate spaces using calculus. This is not the case with tensor product higher order continuity elements discussed in this paper, thus confirming that the tensor product approximations are not true Ci, Cijk, Cijk approximations. It is shown that isogeometric analysis for a domain with more than one patch can only yield solutions of class C0. This method has no concept of finite elements and local approximations, just patches. It is shown that compariso of this method with k-version of the finite element method is meaningless. Model problem studies in R2 establish accuracy and superior convergence characteristics of true Cijp-version hierarchical local approximations presented in this paper over tensor product approximations. Convergence characteristics of p-convergence, k-convergence and pk-convergence are illustrated for self adjoint, non-self adjoint and non-linear differential operators in BVPs. It is demonstrated that h, p and k are three independent parameters in all finite element computations. Tensor product local approximations and other published works on k-version and their limitations are discussed in the paper and are compared with present work.

An initial-boundary value problem for the general three-component nonlinear Schrödinger equations on a finite interval

Abstract The general three-component nonlinear Schrödinger (gtc-NLS) equations are completely integrable and contain the self-focusing, defocusing and mixed cases, which are applied in many physical fields. In this paper, we would like to use the Fokas method to explore the initial-boundary value (IBV) problem for the gtc-NLS equations with a $4\times 4$ matrix Lax pair on a finite interval based on the inverse scattering transform. The solutions of the gtc-NLS equations can be expressed using the solution of a $4\times 4$ matrix Riemann–Hilbert (RH) problem constructed in the complex $k$-plane. The jump matrices of the RH problem can be explicitly found in terms of three spectral functions related to the initial data, and the Dirichlet–Neumann boundary data, respectively. The global relation between the distinct spectral functions is also proposed to derive two distinct but equivalent types of representations of the Dirichlet–Neumann boundary value problems. Particularly, the relevant formulae for the boundary value problems on the finite interval can generate ones on the half-line as the length of the interval closes to infinity. Finally, we also analyse the linearizable boundary conditions for the Gel’fand–Levitan–Marchenko representation. These results will be useful to further study the solution properties of the IBV problem of the gtc-NLS system by using the Deift–Zhou’s nonlinear steepest descent method and some numerical methods.

A Comparative Study on Classical Fourth Order and Butcher Sixth Order Runge-Kutta Methods with Initial and Boundary Value Problems

In this paper, it is discussed about Runge-Kutta fourth-order method and Butcher Sixth order Runge-Kutta method for approximating a numerical solution of higher-order initial value and boundary value ordinary differential equations. The proposed methods are most efficient and extolled practically for solving these problems arising indifferent sector of science and engineering. Also, the shooting method is applied to convert the boundary value problems to initial value problems. Illustrative examples are provided to verify the accuracy of the numerical outcome and compared the approximated result with the exact result. The approximated results are found in good agreement with the result of the exact solution and firstly converge to more accuracy in the solution when step size is very small. Finally, the error with different step sizes is analyzed and compared to these two methods.

A gradient maximum principle of solutions for a quasilinear parabolic equation

This short note is concerned with a quasilinear diffusion equation under initial and Neumann boundary value condition. To be more precise, the authors establish a gradient maximum principle of classical solutions via the maximum principle and Hopf’s lemma. The result generalizes a recent work obtained by Kim (Proc Amer Math Soc 145:1203–1208, 2017).

Method of functional parametrization for solving a semi-periodic initial problem for fourth-order partial differential equations

A semi-periodic initial boundary-value problem for a fourth-order system of partial differential equations is considered. Using the method of functional parametrization, an additional parameter is carried out and the studied problem is reduced to the equivalent semi-periodic problem for a system of integro-differential equations of hyperbolic type second order with functional parameters and integral relations. An interrelation between the semi-periodic problem for the system of integro-differential equations of hyperbolic type and a family of Cauchy problems for a system of ordinary differential equations is established. Algorithms for finding of solutions to an equivalent problem are constructed and their convergence is proved. Sufficient conditions of a unique solvability to the semi-periodic initial boundary value problem for the fourth-order system of partial differential equations are obtained.

Parabolic Boundary Value Problems in Unlimited Piecewise Homogeneous Wedge-Shaped Hollow Cylinder

The unique exact analytical solutions of parabolic boundary value problems of mathematical physics in unbounded by variable z piecewise-homogeneous by radially variable r wedge-shaped by an angularly variable φ hollow cylinder were constructed at first time by the method of classical integral and hybrid integral transforms in combination with the method of main solutions (matrices of influence and Green matrices) in the proposed article. The cases of assigning on the verge of the wedge the boundary conditions of Dirichlet and Neumann and their possible combinations (Dirichlet — Neumann, Neumann — Dirichlet) are considered. Finite integral Fourier transform by an angular variable, a Fourier integral transform on a Cartesian axis by an applicative variable and a hybrid integral transform of the Hankel type of the second kind on a segment of the polar axis with n points of conjugation were used to construct classic solutions of investigated initial-boundary value problems. The consistent application of integral transforms by geometric variables allows us to reduce the three-dimensional initial boundary-value problems of conjugation to the Cauchy problem for a regular linear inhomogeneous 1st order differential equation whose unique solution is written in a closed form. The application of inverse integral transforms restores explicitly the solution of the considered problems through their integral image.

Analytical-Numerical Study of Finite-Time Blow-up of the Solution to the Initial-Boundary Value Problem for the Nonlinear Klein–Gordon Equation

An analytical-numerical approach is used to study the finite-time blow-up of the solution to the initial boundary-value problem for the nonlinear Klein–Gordon equation. An analytical analysis yields an upper estimate for the blow-up time of the solution with an arbitrary positive initial energy. With the use of this a priori information, the blow-up process is numerically analyzed in more detail. It is shown that the numerical analysis of the blow-up of the solution makes it possible to improve the analytical estimate and to detect local blow-up with respect to the spatial variable.

The Generalized Carrier–Greenspan Transform for the Shallow Water System with Arbitrary Initial and Boundary Conditions

We put forward a solution to the initial boundary value (IBV) problem for the nonlinear shallow water system in inclined channels of arbitrary cross-section by means of the generalized Carrier-Greenspan hodograph transform (Rybkin et al., 2014). Since the Carrier-Greenspan transform, while linearizing the shallow water system, seriously entangles the IBV in the hodograph plane, all previous solutions required some restrictive assumptions on the IBV conditions, e.g., zero initial velocity, smallness of boundary conditions. For arbitrary non-breaking initial conditions in the physical space, we present an explicit formula for equivalent IBV conditions in the hodograph plane, which can readily be treated by conventional methods. Our procedure, which we call the method of data projection, is based on the Taylor formula and allows us to reduce the transformed IBV data given on curves in the hodograph plane to the equivalent data on lines. Our method works equally well for any inclined bathymetry (not only plane beaches) and, moreover, is fully analytical for U-shaped bays. Numerical simulations show that our method is very robust and can be used to give express forecasting of tsunami wave inundation in narrow bays and fjords

Waves Generated on Running Stream due to a Disturbance on Sea Bed

In this paper we have studied the effect of running stream on the waves generated by disturbance on the sea bed. The problem is formulated as an initial cum boundary value problem with Laplace equation as the governing differential equation. Employing Laplace transform in time and Fourier transform in space, the free surface depression is obtained which is then evaluated asymptotically at a large distance and after a long time after the occurrence of the disturbance.The asymptotic form of the free surface depression is studied graphically for different values of the parameter. It is observed that in absence of running stream, the free surface depression exhibits uniform oscillatory behaviour at large distance from the occurrence of the disturbance and also after a long lapse of time. However in the presence of the uniform current it is found that if the velocity of the running stream is less than a critical value then there occurs an interaction between the wave and the stream due to which there is a significant change in the asymptotic form of free surface depression.

A New Algorithm Based on Differential Transform Method for Solving Partial Differential Equation System with Initial and Boundary Conditions

In this paper, a new Fourier-differential transform method (FDTM) based on differential transformation method (DTM) is proposed. The method can effectively and quickly solve linear and nonlinear partial differential equations with initial boundary value (IBVP). According to boundary condition, the initial condition is expanded into a Fourier series. After that, the IBVP is transformed to an iterative relation in K-domain. The series solution or exact solution can be obtained. The rationality and practicability of the algorithm FDTM are verified by comparisons of the results obtained by FDTM and the existing analytical solutions.

Initial-boundary value and inverse problems for subdiffusion equations in ℝ^N

Initial-boundary value and inverse problems for subdiffusion equations in ℝ^N

PARABOLIC BOUNDARY VALUE PROBLEMS IN A PIECEWISE HOMOGENEOUS WEDGE-SHAPED SOLID CYLINDER

The unique exact analytical solutions of parabolic boundary value problems of mathematical physics in piecewise homogeneous wedge-shaped solid cylinder were constructed at first time by the method of integral and hybrid integral transforms in combination with the method of main solutions (matrices of influence and Green matrices). The cases of assigning on the verge of the wedge the boundary conditions of Dirichlet and Neumann and their possible combinations (Dirichlet – Neumann, Neumann – Dirichlet) are considered. Finite integral Fourier transform by an angular variable $\varphi \in (0; \varphi_0)$, a Fourier integral transform on the Cartesian segment $(-l_1;l_2)$ by an applicative variable $z$ and a hybrid integral transform of the Hankel type of the first kind on a segment $(0;R)$ of the polar axis with $n$ points of conjugation by an radial variable $r$ were used to construct solutions of investigated initial-boundary value problems. The consistent application of integral transforms by geometric variables allows us to reduce the three-dimensional initial boundary-value problems of conjugation to the Cauchy problem for a regular linear inhomogeneous 1st order differential equation whose unique solution is written in a closed form. The application of inverse integral transforms restores explicitly the solution of the considered problems through their integral image. The structure of the solution of the problem in the case of setting the Neumann boundary conditions on the wedge edges is analyzed. Exact analytical formulas for the components of the main solutions are written and the theorem on the existence of a single bounded classical solution of the problem is formulated. The obtained solutions are algorithmic in nature and can be used (using numerical methods) in solving applied problems.

The Ablowitz-Ladik system on a graph

This paper presents an approach to study initial-boundary value (IBV) problems for integrable nonlinear differential-difference equations (DDEs) posed on a graph. As an illustrative example, we consider the Ablowitz–Ladik system posed on a graph that is constituted by N semi-infinite lattices (edges) connected through some boundary conditions. We first show that analyzing this problem is equivalent to analyzing a certain matrix IBV problem; then we employ the unified transform method (UTM) to analyze this matrix IBV problem. We also compare our results with some previously known studies. In particular, we show that the inverse scattering method (ISM) for the integrable DDEs on the set of integers can be recovered from the UTM applied to our N = 2 graph problem as a particular case, and the non-local reductions of integrable DDEs can be obtained as local reductions from our results.

Initial-boundary value problem for the spin-1 Gross-Pitaevskii system with a 4 × 4 Lax pair on a finite interval

In this paper, we explore the initial-boundary value (IBV) problem for an integrable spin-1 Gross-Pitaevskii system with a 4 × 4 Lax pair on the finite interval x ∈ [0, L] by extending the Fokas unified approach. The solution of this three-component system can be expressed by means of the solution of a 4 × 4 matrix Riemann-Hilbert (RH) problem formulated in the complex spectral k-plane. Furthermore, the relevant jump matrices with explicit (x, t)-dependence of the matrix RH problem can be explicitly obtained using three spectral functions {s(k), S(k), SL(k)} arising from the initial data and Dirichlet-Neumann boundary conditions at x = 0, L, respectively. The global relation is also presented and used to deduce two distinct but equivalent types of representations [i.e., one via the large k of asymptotics of the eigenfunctions and another one in terms of the Gel’fand-Levitan-Marchenko (GLM) approach] for the Dirichlet and Neumann boundary value problems. In particular, the formulae for IBV problem on the finite interval can be extended to the ones on a half-line as the length L of the interval approaches to infinity. Moreover, we also give the linearizable boundary conditions for the GLM representations.