Order Linear Boundary Value Problems Research Articles

B-spline method for second order RLC closed series circuit with small inductance value

In this article, we investigates the application of RLC closed series electric circuits. We form the second order linear differential equations for an electric circuit depends upon the Kirchhoffs Voltage laws. A circuit containing an inductance L or capacitor C and resistor R with current and voltage variable given by boundary value problem. Then we use the cubic B-spline functions on graded mesh to solve second order linear boundary value problems. This method gives the second ordered convergent. Numerical results (charge q) for different cases of L are presented and compared with exact solutions in the table form.

Positive Jacobian constraints for elliptic boundary value problems with piecewise-regular coefficients arising from multi-wave inverse problems

Multi-wave inverse problems are indirect imaging methods using the interaction of two different imaging modalities. One brings spatial accuracy, and the other contrast sensitivity. The inversion method typically involve two steps. The first step is devoted to accessing internal datum of quantities related to the unknown parameters being observed. The second step involves recovering the parameters themselves from the internal data. To perform that inversion, a typical requirement is that the Jacobian of fields involved does not vanish. A number of authors have considered this problem in the past two decades, and a variety of methods have been developed. Existing techniques require Hölder continuity of the parameters to be reconstructed. In practical applications, the medium may present embedded elements, with distinct physical properties, leading to discontinuous coefficients. In this article we explain how a Jacobian constraint can be imposed in the piecewise regular case, when the physical model is a divergence form second order linear elliptic 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.

Solution for a System of First-Order Linear Fuzzy Boundary Value Problems

In this paper, we consider homogeneous and non-homogeneous system of first order linear fuzzy boundary value problems (SFOLBVPs) under granular differentiability. Using the concept of horizontal membership function, we introduced the notion of first order granular differentiability for n-dimensional fuzzy functions. We present granular integral and its properties. Theorems on the existence and uniqueness of solutions for homogeneous and non-homogeneous SFOLFBVPs are proved. We develop an algorithm for solution of non-homogeneous SFOLBVPs under granular differentiability. We provide some examples to illustrate the validity of the proposed algorithm.

New results on the sign of the Green function of a two-point n-th order linear boundary value problem

This paper provides conditions for determining the sign of all the partial derivatives of the Green functions of n-th order boundary value problems subject to a wide set of homogeneous two-point boundary conditions, removing restrictions of previous results about the distance between the two extremes that define the problem. To do so, it analyzes the sign of the derivatives of the solutions of related two-point n-th order boundary value problems subject to n-1 boundary conditions by introducing a new property denoted by ‘hyperdisfocality’.

Reproducing kernel representation of the solution of second order linear three‐point boundary value problem

This paper investigates the analytical solutions and numerical solutions of second order linear boundary value problems with three‐point boundary conditions by means of the weak pre‐orthogonal adaptive Fourier decomposition (W‐POAFD) method introduced by Qian et al. Using the reproducing kernel functions of the Sobolev spaces on the unit interval , we construct a dictionary to implement the W‐POAFD. Based on orthonormalizataion, we can select the crucial parameters successively by adopting the weak maximal selection strategy of the W‐POAFD method. Then the numerical solutions have “sparse” representations, which converge to the analytic solution with fast convergence rate. Some numerical experiments demonstrate the high accuracy and efficiency of the W‐POAFD method in this three‐point boundary value problem.

Cubic spline solutions of the ninth order linear and non-linear boundary value problems

A lot of numerical formulations of physical phenomena contain 9th-order BVPs. The presented probe intends to consider the spline solutions of the 9th-order boundary value problems using Cubic B Spline(CBS). Ninth order boundary value problems arise in the study of laminar viscous flow in a semi-porous channel, astrophysics, hydrodynamic & hydro-magnetic stability. The derived technique is exceptionally useful and is appropriate for such kinds of linear and non-linear boundary value problems. Cubic B Spline(CBS) is successfully applied to mathematical models. The outcomes are contrasted to those presented in the literature, revealing that the introduced technique leads to an advisable estimation of the exact solution.

Galerkin Method and Its Residual Correction with Modified Legendre Polynomials

Accuracy and error analysis is one of the significant factors in computational science. This study employs the Galerkin method to solve second order linear or nonlinear Boundary Value Problems (BVPs) of Ordinary Differential Equations (ODEs) with modified Legendre polynomials to seek numerical solutions. The residual function of a differential operator is used as non-homogeneous term information of an error differential equation. The Galerkin approximation is then improved or corrected by solving the error differential equation by the Galerkin method using the same polynomials. Thus we apply the double layer Galerkin method to a variety of instances. We compare approximate solutions with exact ones and results available in the literature, and in every case, we find better accuracy.

The solution of fifth and sixth order linear and non linear boundary value problems by the Improved Residual Power Series Method

In this paper we solve some fifth and sixth order boundary value problems (BVPs) by the improved residual power series method (IRPSM). IRPSM is a method that extends the residual power series method (RPSM) to (BVPs) without requiring exact solution. The presented method is capable to handle both linear and nonlinear boundary value problems (BVPs) effectively. The solutions provided by IRPSM are compared with the actual solution and with the existing solutions. The results demonstrate that the approach is extremely accurate and dependable.

New Spline Method for Solving Linear Two-Point Boundary Value Problems

In this research, second order linear two-point boundary value problems are treated using new method based on hybrid cubic B-spline. The values of the free parameter,Gamma , are chosen via optimization. The value of the free parameter plays an important role in giving accurate results. Optimization of this parameter is carried out. This method is tested on four examples and a comparison with cubic B-spline, trigonometric cubic B-spline and extended cubic B-spline methods has been carried out. The examples suggest that this method produces more accurate results than the other three methods. The numerical results are presented to illustrate the efficiency of our method.

The principal eigenvalue of some nth order linear boundary value problems

The purpose of this paper is to present a procedure for the estimation of the smallest eigenvalues and their associated eigenfunctions of nth order linear boundary value problems with homogeneous boundary conditions defined in terms of quasi-derivatives. The procedure is based on the iterative application of the equivalent integral operator to functions of a cone and the calculation of the Collatz–Wielandt numbers of such functions. Some results on the sign of the Green functions of the boundary value problems are also provided.

A stable and efficient technique for linear boundary value problems by applying kernel functions

In this paper, the basis functions are firstly constructed by employing the reproducing kernel functions (RKFs). Based on these basis functions and an appropriate choice of the collocation nodes, a novel collocation approach is proposed for second order linear boundary value problems (BVPs). Compared with the existing RKFs based methods for BVPs, the outstanding features of the present approach are its good stability and high accuracy. The effectiveness of the approach is verified through some numerical experiments.

Approximate Solutions for Higher Order Linear and Nonlinear Boundary Value Problems

Approximate Solutions for Higher Order Linear and Nonlinear Boundary Value Problems

Numerical solutions of higher order boundary value problems via wavelet approach

This paper presents a numerical scheme based on Haar wavelet for the solutions of higher order linear and nonlinear boundary value problems. In nonlinear cases, quasilinearization has been applied to deal with nonlinearity. Then, through collocation approach computing solutions of boundary value problems reduces to solve a system of linear equations which are computationally easy. The performance of the proposed technique is portrayed on some linear and nonlinear test problems including tenth, twelfth, and thirteen orders. Further convergence of the proposed method is investigated via asymptotic expansion. Moreover, computed results have been matched with the existing results, which shows that our results are comparably better. It is observed from convergence theoretically and verified computationally that by increasing the resolution level the accuracy also increases.

Solution of third order linear and nonlinear boundary value problems of integro-differential equations using Haar Wavelet method

In this paper, numerical solution of third order integro-differential equation with boundary conditions is given utilizing Haar collocation technique. Both nonlinear and linear integro-differential equations are solved using this method. The third order derivative is approximated using Haar functions in both nonlinear and linear integro-differential equations. Integration is used to obtain the expression of lower order derivatives as well as the solution for the unknown function. The Gauss elimination approach is utilized for linear systems and Broyden approach is adopted for nonlinear systems. Validation and convergence of the proposed approach are illustrated using some examples. At various collocation and gauss points, the maximum absolute and root mean square errors are compared to the exact solution. The convergence rate is also measured using different numbers of nodal points, and it is nearly equal to 2.

Sumudu Transform and Variational Iteration Method to Solve Two Point Second Order Linear Boundary Value Problems

Sumudu Transform and Variational Iteration Method to Solve Two Point Second Order Linear Boundary Value Problems

Cubic spline fractal solutions of two-point boundary value problems with a non-homogeneous nowhere differentiable term

Fractal interpolation functions (FIFs) supplement and subsume all classical interpolants. The major advantage by the use of fractal functions is that they can capture either the irregularity or the smoothness associated with a function. This work proposes the use of cubic spline FIFs through moments for the solutions of a two-point boundary value problem (BVP) involving a complicated non-smooth function in the non-homogeneous second order differential equation. In particular, we have taken a second order linear BVP: y′′(x)+Q(x)y′(x)+P(x)y(x)=R(x) with the Dirichlet’s boundary conditions, where P(x) and Q(x) are smooth, but R(x) may be a continuous nowhere differentiable function. Using the discretized version of the differential equation, the moments are computed through a tridiagonal system obtained from the continuity conditions at the internal grids and endpoint conditions by the derivative function. These moments are then used to construct the cubic fractal spline solution of the BVP, where the non-smooth nature of y′′ can be captured by fractal methodology. When the scaling factors associated with the fractal spline are taken as zero, the fractal solution reduces to the classical cubic spline solution of the BVP. We prove that the proposed method is convergent based on its truncation error analysis at grid points. Numerical examples are given to support the advantage of the fractal methodology.

The Sign of the Green Function of an n-th Order Linear Boundary Value Problem

This paper provides results on the sign of the Green function (and its partial derivatives) of an n-th order boundary value problem subject to a wide set of homogeneous two-point boundary conditions. The dependence of the absolute value of the Green function and some of its partial derivatives with respect to the extremes where the boundary conditions are set is also assessed.

Coupled System of Boundary Value Problems by Galerkin Method with Cubic B-Splines

Coupled system of second order linear and nonlinear boundary value problems occur in various fields of Science and Engineering including heat and mass transfer. In the formulation of the problem, any one of 81 possible types of boundary conditions may occur. These 81 possible boundary conditions are written as a combination of four boundary conditions. To solve a coupled system of boundary value problem with these converted boundary conditions, a Galerkin method with cubic Bsplines as basis functions has been developed. The basis functions have been redefined into a new set of basis functions which vanish on the boundary. The nonlinear boundary value problems are solved with the help of quasilinearization technique. Several linear and nonlinear boundary value problems are presented to test the efficiency of the proposed method and found that numerical results obtained by the present method are in good agreement with the exact solutions available in the literature.

A novel wavelet based hybrid method for finding the solutions of higher order boundary value problems

Abstract In this paper, a new wavelet based hybrid method is developed for obtaining the solution of higher order linear and nonlinear boundary value problems. The proposed method is based on approximation of solution by non-dyadic wavelets family with dilation factor 3. Discretization of domain is done by collocation method. The nonlinearities in boundary value problems are tackled by Quasi-linearization technique. Eleven numerical experiments are performed on linear and nonlinear boundary value problems with order ranging from eighth to twelfth to prove the successful application of the proposed method. Also, the obtained solutions are compared with exact and numerical solutions available in the literature to prove the efficiency of the method over other methods.