Linear Boundary Value Problems Research Articles

New collocation approaches based on reproducing kernel functions for second order nonlinear boundary value problems

In the previous work, the reproducing kernel Hilbert space (RKHS) theory has been employed mainly to solve linear boundary value problems (BVPs). Reproducing kernel functions (RKFs) based techniques for nonlinear BVPs mainly applied iterative techniques. However, there is no error analysis for these methods. In this letter, we shall propose new collocation algorithms based on RKFs for dealing with second order nonlinear BVPs. And the detailed error analysis is provided.

Black-Box Solver for Numerical Simulations and Mathematical Modelling in Engineering Physics

This article presents a two-grid approach for developing a black-box iterative solver for a large class of real-life problems in continuum mechanics (heat and mass transfer, fluid dynamics, elasticity, electromagnetism, and others). The main requirements on this (non-)linear black-box solver are: (1) robustness (the lowest number of problem-dependent components), (2) efficiency (close-to-optimal algorithmic complexity), and (3) parallelism (a parallel robust algorithm should be faster than the fastest sequential one). The basic idea is to use the auxiliary structured grid for more computational work, where (non-)linear problems are simpler to solve and to parallelize, i.e., to combine the advantages of unstructured and structured grids: simplicity of generation in complex domain geometry and opportunity to solve (non-)linear (initial-)boundary value problems by using the Robust Multigrid Technique. Topics covered include the description of the two-grid algorithm and estimation of their robustness, convergence, algorithmic complexity, and parallelism. Further development of modern software for solving real-life problems justifies relevance of the research. The proposed two-grid algorithm can be used in black-box parallel software for the reduction in the execution time in solving (initial-)boundary value problems.

Linear Boundary-Value Problems for Systems of Integrodifferential Equations with Degenerate Kernel. Resonance Case for a Weakly Perturbed Boundary-Value problem

Linear Boundary-Value Problems for Systems of Integrodifferential Equations with Degenerate Kernel. Resonance Case for a Weakly Perturbed Boundary-Value problem

A 〖(2n+9)〗^thDegree Polynomial Approximation Method for Solving Eight Order Linear Boundary Value Problems

Galerking method, Collocation method and least squired method have been used for solving boundary value problems (BVPs) and from practical point of view almost any success of above mention method depends on the selection of basis functions. Also, Adomain decomposition and variational iteration methods have been applied for the solution of BVPs and required respectively calculating the Adomian’s polynomials and Lagrange multipliers. In this paper a simple method of Degree polynomial approximation solution were n is number of successive derivatives of the governing equation is suggested as an alternative of the above mentioned methods. The method consists of first obtaining from the governing equation its n successive derivatives and boundary conditions a linear differential system of equations evaluated at the boundary points. Next, an approximate solution in the form of polynomial of degree with unknown coefficients is assumed. To determine the unknown coefficients, the assumed solution is incorporate into the linear differential system which turns to be a linear system of equation with unknowns which is salvable uniquely. The method is tested for n=5 to four examples. It clear that from tables 3.1 to 3.4 and figures 3.1 to 3.4 the numerical outcomes agree with the exact solution and also better than some existing results in the literatures. it should be also noted that the accuracy of the method can be increased by increased n. All Numerical computation were performed using maple 2016 software.

A generalized CAS wavelet method for solving ψ-Caputo fractional differential equations

PurposeThe purpose of the present work is to introduce a wavelet method for the solution of linear and nonlinear psi-Caputo fractional initial and boundary value problem.Design/methodology/approachThe authors have introduced the new generalized operational matrices for the psi-CAS (Cosine and Sine) wavelets, and these matrices are successfully utilized for the solution of linear and nonlinear psi-Caputo fractional initial and boundary value problem. For the nonlinear problems, the authors merge the present method with the quasilinearization technique.FindingsThe authors have drived the orthogonality condition for the psi-CAS wavelets. The authors have derived and constructed the psi-CAS wavelets matrix, psi-CAS wavelets operational matrix of psi-fractional order integral and psi-CAS wavelets operational matrix of psi-fractional order integration for psi-fractional boundary value problem. These matrices are successfully utilized for the solutions of psi-Caputo fractional differential equations. The purpose of these operational matrices is to make the calculations faster. Furthermore, the authors have derived the convergence analysis of the method. The procedure of implementation for the proposed method is also given. For the accuracy and applicability of the method, the authors implemented the method on some linear and nonlinear psi-Caputo fractional initial and boundary value problems and compare the obtained results with exact solutions.Originality/valueSince psi-Caputo fractional differential equation is a new and emerging field, many engineers can utilize the present technique for the numerical simulations of their linear/non-linear psi-Caputo fractional differential models. To the best of the authors’ knowledge, the present work has never been introduced and implemented for psi-Caputo fractional differential equations.

Adaptive multigrid method for quantum eigenvalue problems

In this paper, a new type of adaptive finite element method is proposed for nonlinear eigenvalue problems in electronic structure calculations based on the multilevel correction method and adaptive multigrid method. Different from the classical adaptive finite element method for electronic structure calculations which needs to solve a nonlinear eigenvalue model directly in each adaptive finite element space, our approach only needs to solve a linear elliptic boundary value problem by adaptive multigrid method in each adaptive space, and then correct the eigenpair approximation by solving a small-scale nonlinear eigenvalue model in a low-dimensional correction space. Since solving large-scale nonlinear eigenvalue model is avoided which has exponentially increased computational time, the efficiency of electronic structure calculations can be improved evidently. In addition, the corresponding convergence analysis of the proposed adaptive multigrid algorithm are also derived theoretically and numerically.

Node's residual descent method for linear elastic boundary value problems

This study proposes a node's residual descent method (NRDM) for the linear elasticity boundary value problem. In this method, the boundary information and deformation states are attached to the nodes, and the partial differential equations (PDEs) are then transformed into a generalized difference form with the first-order generalized finite difference (GFD) scheme. Furthermore, this study thoroughly describes the NRDM, confirms the accuracy and convergence by numerical examples, performs an error analysis, and discusses the influence of computational parameters. Results demonstrate that NRDM has high computational accuracy and superliner convergence rate. The double derivation is implemented to solve the second-order PDE boundary value problem with the first-order GFD scheme, thus simplifying the requirement for star connectivity and reducing the computational complexity. NRDM is a matrix-free iterative method that eliminates the task of assembling large global sparse matrices, thus avoiding the numerous matrix–matrix product operations. Rather than solving large systems of linear equations, many extremely small matrix multiplications are present during iterations, which are extremely friendly to parallel computing and can be adapted to the computation of ultra-multi-degree-of-freedom problems. The computational accuracy and convergence rate are considerably improved in NRDM compared to the classical GFD method.

Kernel free boundary integral method for 3D incompressible flow and linear elasticity equations on irregular domains

A second-order accurate kernel-free boundary integral method is presented for Stokes flow and linear elasticity boundary value problems on three-dimensional irregular domains. It solves equations in the framework of boundary integral equations, whose corresponding discrete forms are well-conditioned and solved by the GMRES method. A notable feature of this approach is that the boundary or volume integrals encountered in BIEs are indirectly evaluated by a Cartesian grid-based method, which includes discretizing corresponding simple interface problems with a MAC scheme, correcting discrete linear systems to reduce large local truncation errors near the interface, solving the modified system by a CG method together with an FFT-based Poisson solver. No extra work or special quadratures are required to deal with singular or hyper-singular boundary integrals and the dependence on the analytical expressions of Green’s functions for the integral kernels is completely eliminated. Numerical results are given to demonstrate the efficiency and accuracy of the Cartesian grid-based method.

An extension of Krasnoselskii's cone fixed point theorem for a sum of two operators and applications to nonlinear boundary value problems

"The purpose of this work is to establish a new generalized form of the Krasnoselskii type compression-expansion fixed point theorem for a sum of an expansive operator and a completely continuous one. Applications to three non- linear boundary value problems associated to second order differential equations of coincidence type are included to illustrate the main results."

A new approach for solving boundary and initial value problems by coupling the he method and Sawi transform

This paper discusses and implements a newly developed technique using the He method with Sawi Transform. The main aim is to solve some initial and boundary problems. This combination exhibits an accurate strategy to obtain a precise solution for linear and nonlinear problems. To validate the proposed Hybrid method, a 4- examples are discussed, these including: Burger’s equation, telegraph equation, Kelin-Gordan equation, Duffing oscillator with cubic nonlinear term. The obtained results improve the exactness and the accuracy of the proposed combinations, and the proposed method is capable to solve a large number of linear and nonlinear initial and boundary value problems.

Boundary Value Problem for Impulsive Delay Fractional Differential Equations with Several Generalized Proportional Caputo Fractional Derivatives

A scalar nonlinear impulsive differential equation with a delay and generalized proportional Caputo fractional derivatives (IDGFDE) is investigated. The linear boundary value problem (BVP) for the given fractional differential equation is set up. The explicit form of the unique solution of BVP in the special linear case is obtained. This formula is a generalization of the explicit solution of the case without any delay as well as the case of Caputo fractional derivatives. Furthermore, this integral form of the solution is used to define a special proportional fractional integral operator applied to the determination of a mild solution of the studied BVP for IDGFDE. The relation between the defined mild solution and the solution of the BVP for the IDGFDE is discussed. The existence and uniqueness results for BVP for IDGFDE are proven. The obtained results in this paper are a generalization of several known results.

An Implicit Finite Difference Scheme and Neural Network Approach for Non-Newtonian Nanofluid Flow Using Induced Magnetic Field

The aim of this contribution is to propose a numerical scheme for solving linear and nonlinear boundary value problems. The scheme is implicit and it is constructed on three grid points. The stability of the proposed implicit scheme is provided. In addition to this, a mathematical model for heat and mass transfer using induced magnetic field (IMF) is modified. Furthermore, this model is transformed into boundary value problems by employing similarity transformations. The dimensionless model of boundary value problems is solved using the proposed numerical scheme. The scheme is applied with a combination of a shooting approach and an iterative method. From the obtained results, it can be seen that velocity profile declines with enhancing Weissenberg number. The results are also compared with those given in past research. In addition to this, a neural network approach is applied that is based on the input and outputs of the considered model with specified values of parameters.

Initial-boundary value problem for the Hirota equation posed on a finite interval

In this paper, we turn our attention to a nonhomogeneous initial boundary value problem for Hirota equation posed on a bounded interval (0,L). In particular, the explicit solution formula of linear nonhomogeneous boundary value problem is established by Laplace transform. Using space L2(0,T;H0s−1(0,1)), which is preparing for trilinear estimates and Lions-Magenes interpolation theorem, we prove the local existence, uniqueness, and Lipschitz continuous in C(0,T;Hs(0,1))∩L2(0,T;Hs+1(0,1)) corresponding to the initial and boundary data. Moreover, the local solution extends to a global one by a priori bound.

Boundary value problems with rough boundary data

We consider linear boundary value problems for higher-order parameter-elliptic equations, where the boundary data do not belong to the classical trace spaces. We employ a class of Sobolev spaces of mixed smoothness that admits a generalized boundary trace with values in Besov spaces of negative order. We prove unique solvability for rough boundary data in the half-space and in sufficiently smooth domains. As an application, we show that the operator related to the linearized Cahn–Hilliard equation with dynamic boundary conditions generates a holomorphic semigroup in Lp(R+n)×Lp(Rn−1).

VARIATIONAL ITERATION METHOD FOR SOLVING FUZZY BOUNDARY VALUE PROBLEMS

Abstract: 
 This paper will find the approximate solution to the linear and nonlinear fuzzy boundary value problems using the variational iteration method. The numerical scheme is based on analyzing the fuzzy problem into two crisp sub-problems. The first is for the upper solution and the second is for the lower solution of the fuzzy solution. Also, the convergence of the obtained variational iteration formula has been proved to converge to the exact solution of the problem under consideration in each illustrative example, since there is no general formula that may be obtained for the correction functional related to the problem under consideration which is due to the variation of the general Lagrange multiplier from one problem to another.
 Keywords:Fuzzy boundary value problems, variational iteration method, solution of fuzzy differential equations using variational iteration method.

An asymptotic homogenization formula for complex permittivity and its application

The $\mathbb R$-linear boundary value problem in a multiply connected domain on a flat torus is considered. This problem is closely related to the Riemann-Hilbert problem on analytic functions. The considered problem arises in the homogenization procedure of random media with complex constants which express the permittivity of components. A new asymptotic formula for the effective permittivity tensor is derived. The formula contains location of inclusions in symbolic form. The application of the derived formula to investigation of the morphology of the tumor cells in disordered biological media is discussed.

Solvability for Optimal Classical Continuous Control Problem Controlling by Quaternary Hyperbolic Boundary Value Problem

This work is concerned with studying the solvability for optimal classical continuous control quaternary vector problem that controls by quaternary linear hyperbolic boundary value problem. The existence of the unique quaternary state vector solution for the quaternary linear hyperbolic boundary value problem is studied and demonstrated by employing the method of Galerkin, where the classical continuous control quaternary vector is Known. Also, the existence theorem of an optimal classical continuous control quaternary vector related to the quaternary linear hyperbolic boundary value problem is demonstrated. The existence of a unique solution to the adjoint quaternary linear hyperbolic boundary value problem associated with the quaternary linear hyperbolic boundary value problem is formulated and studied. The directional derivative for the cost functional is derived. Finally, the necessary optimality theorem for the optimal classical continuous control quaternary vector is proved.

The Classical Continuous Optimal Control for Quaternary parabolic boundary value problem

The aim of this paper is to study the quaternary classical continuous optimal control for a quaternary linear parabolic boundary value problems(QLPBVPs). The existence and uniqueness theorem of the continuous quaternary state vector solution for the weak form of the QLPBVPs with given quaternary classical continuous control vector (QCCCV) is stated and proved via the Galerkin Method. In addition, the existence theorem of a quaternary classical continuous optimal control vector governinig by the QLPBVPs is stated and demonstrated. The Fréchet derivative for the cost function is derived. Finally, the necessary conditions for the optimality theorem of the proposed problem is stated and demonstrated.

On the Solvability of a Linear Boundary Value Problem with Impulse Effects for Differential System

On the Solvability of a Linear Boundary Value Problem with Impulse Effects for Differential System

Uniformly convergent scheme for fourth-order singularly perturbed convection-diffusion ODE

This paper contemplates a numerical investigation of the convection-diffusion type's fourth-order singularly perturbed linear and nonlinear boundary value problems. First, the considered linear fourth-order differential equation is converted into a strongly/weakly coupled singularly perturbed system (depending on the coefficient of the first-order derivative) of two ordinary differential equations with Dirichlet boundary conditions to solve the problem numerically. One of the equations is free from the perturbation parameter in the system. To obtain the solution for this system, we propose a numerical method of quadratic B-splines on an exponentially graded mesh. Convergence analysis shows that the proposed numerical scheme is second-order uniformly convergent in the discrete maximum norm. The nonlinear differential equation is linearized using the quasilinearization technique, and then the proposed approach is applied to the linearized problem. The theoretical outcomes are validated by executing the proposed method on three test problems.