Higher Order Boundary Value Problems Research Articles

Macroscopic modeling of flexoelectricity-driven remanent polarization in piezoceramics

This paper presents a variational modeling framework for investigating the flexoelectricity-driven evolution of remanent polarization in piezoceramics. In small-scale electromechanical systems, strain gradients can exhibit polarization in dielectric materials via the direct flexoelectric effect. In ferroelectrics, it is reasonable to expect that a sufficiently large magnitude of the flexoelectricity leads to a switching of the domain structure and thus the material becomes remanently polarized. It is interesting to note that this means that poling would be able to occur in the absence of any external electrical source. This provides the motivation to gain a better understanding of this effect for a possible technical use in e.g. sensor applications. For this purpose, a macroscopic model is presented that couples flexoelectricity and ferroelectric domain switching processes. By embedding the model in the variational framework of the generalized standard materials (GSM), a minimum-type potential structure and thus a stable and efficient numerical treatment is obtained. A mixed finite element formulation based on the Helmholtz free energy is introduced to solve the higher-order flexoelectric boundary value problem. In order to realistically predict the flexoelectric material behavior, the model response is adapted to experimental results in literature obtained for a piezoceramic in a bending test. By simulations based on the adapted model the evolution of the flexoelectricity-driven remanent polarization in the vicinity of a notch is shown.

Series solutions to higher-order parametric boundary value problems

Series solutions to higher-order parametric boundary value problems

The improved residual power series method for the solutions of higher-order linear and nonlinear boundary value problems

AbstractIRPSM is the extended form of RPSM that gives an approximate solution to boundary value problems without requiring an exact solution. This method creates a truncated series for the determination of the missing initial conditions. The present study is conducted to evaluate semi-numerical solutions to differential equations using the Improved Residual Power Series Method (IRPSM). Our main objective is to check whether the proposed scheme is efficient for solving such equations or not. The solution of differential equations has been approximated using truncated residual power series. The method is used to solve differential equations for higher-order linear and non-linear boundary value problems. Absolute errors for the solved problems are calculated to ensure accuracy. It is also compared with some known methods, like the Differential Transform Method (DTM) and the Homotopy Perturbation Method (HPM). For the calculations, Mathematica 12.0 software is used. The computed results are compared to the exact solutions as well as the available results in the literature. The results showed that the results obtained by IRPSM are closely related to the exact solution when compared to the DTM and HPM results.

On the Solution of Caputo Fractional High-Order Three-Point Boundary Value Problem with Applications to Optimal Control

This research paper establishes the existence and uniqueness of solutions for a non-integer high-order boundary value problem, incorporating the Caputo fractional derivative with a non-local type boundary condition. The analytical approach involves the introduction of the fractional Green’s function. To analyze our findings effectively, we apply the Banach contraction fixed point theorem as the primary principle. Furthermore, we illustrate our results through the presentation of various examples.

Lyapunov and Hartman-type inequalities for higher-order discrete fractional boundary value problems

Lyapunov and Hartman-type inequalities for higher-order discrete fractional boundary value problems

A Lyapunov-type inequality for a class of higher-order fractional boundary value problems

A Lyapunov-type inequality for a class of higher-order fractional boundary value problems

UTILIZING PSEUDOINVERSE MATRIX TO SOLVE AN NONLINEAR BOUNDARY VALUE PROBLEM

In this paper, by using the concepts of pseudoinverse matrix combining with coincidence degree theory due to Mawhin and constructing suitable operators, we study the existence of solutions for a nonlinear higher-order boundary value problem at resonance case in R^n . An illustrative example is presented at the end of the paper to illustrate the validity of our results

A reliable algorithm for higher order boundary value problems

In this manuscript, extension of residual power series method (RPSM) is proposed for ordinary differential equations with boundary conditions. The extended algorithm is tested against different boundary value problems (BVPs), and results are compared with other schemes to show the reliability of proposed methodology. Analysis reveals that extended approach can easily handle BVPs and provide convergent series solution without discretization, perturbation or linearization. Hence, practically RPSM is better choice for scientists and researchers working in different field of engineering and sciences.

A New Transformation Method for Solving High-Order Boundary Value Problems

The main purpose of this work is to provide a new approximation method, the so-called parameterised differential transform method (PDTM), for solving high-order boundary value problems (HOBVPs). Our method is based on the classical differential transform method and differs from it by calculating the coefficients of the solution, which has the form of a series. We applied the proposed new method to fourth-order boundary value problems to substantiate it. The resulting solution is graphically compared with the exact solution and the solutions obtained by the classical DTM and ADM methods.

Extended Residual Power Series Algorithm for Boundary Value Problems

In this article, modification of the residual power series method (RPSM) is proposed for higher order boundary value problems (BVPs). The proposed algorithm is tested against various linear and nonlinear BVPs of orders nine up to thirteen. For the efficiency check of RPSM, obtained series solutions are compared with other available results in the literature. Analysis indicates that RPSM is better in terms of accuracy as compared to other mentioned schemes. As RPSM is applicable to BVPs without linearization, discretization, and perturbation, hence practically it is the best suitable solution tool for more complex BVPs in science and engineering.

Multiple positive solutions for higher-order fractional integral boundary value problems with singularity on space variable

In this article, based on further investigation of the properties of the Green function, together with a fixed point theorem due to Avery-Peterson, height functions on special bounded sets are constructed to obtain the existence of triple positive solutions for higher-order fractional integral boundary value problems. The nonlinearity permits singularities both on the time and the space variables.

On the dynamics of high-order beams with vibration absorbers

In this work, a general method is proposed to solve a wide class of high-order boundary value problems governing the dynamics of beams equipped with vibration absorbers. The method relies on one-dimensional beam theories and theory of generalized functions to model the reaction forces of the absorbers. The key step consists in deriving, via Laplace transform, the Green’s function of a linear differential operator of any order n, which involves even-order terms and constant coefficients. Next, the Green’s function, obtained in exact and elegant analytical form, serves as a basis to derive the exact solutions of boundary-value problems of order n governing the frequency response of beams with absorbers, under concentrated/distributed loads. Remarkably, the solutions are derived for any order n of the boundary-value problem, providing the exact frequency response of beams of relevant engineering interest as composite beams, twisted beams, coupled bending-torsion beams and multiple-beam systems including any number of beams; moreover, they are readily implementable for any number/positions of absorbers and loads. To illustrate the method, a case study is selected: a boundary value problem of order n=10 governing the frequency response of a coupled bending-torsion beam with asymmetric cross section and no warping effects. For this problem, an additional validation of the proposed analytical framework is obtained by the theory of distributions, while a comparison with an exact classical approach demonstrates correctness and advantages for applications with multiple absorbers.

Higher-order p-Laplacian boundary value problems with resonance of dimension two on the half-line

We apply the extension of coincidence degree to obtain sufficient conditions for the existence of at least one solution for a class of higher-order p-Laplacian boundary value problems with two-dimensional kernel on the half-line. The result obtained improves and generalizes some of the known results on p-Laplacian boundary value problems in the literature. We also validate our result with an example.

Cubic splines solutions of the higher order boundary value problems arise in sandwich panel theory

An inventive strategy is bestowed here to acquire the numeral roots of nonlinear boundary value problems(BVPs) of 14th-order utilizing cubic splines. Two cubic splines; Polynomial and non-polynomial, are exploited to find out the solutions of nonlinear boundary value problems(BVPs) of 14th-order. The strategies embraced in this work depend on cubic polynomial spline(CPS) and cubic non-polynomial spline(CNPS) strategy in combination of the decomposition procedure. The prescribed method transforms the boundary value problem to a system of linear equations. The algorithms we are going to develop in this paper are not only simply the approximation solution of the 14th order boundary value problems using cubic polynomial spline(CPS) and cubic non-polynomial spline(CNPS) but also describe the estimated derivatives of 1st order to 14th order of the analytic solution at the same time. These strategies will be operated on three problems to evidence the handiness of the technique by means of step size h = 1/5. The exactness of this method for detailed investigation is equated with the precise solution and conveyed through tables. To reveal the efficiency of our outcomes, the AEs (absolute errors) of the CPS and CNPS have been contrasted with Adomian Decomposition Method and Differential Transform Method and our results discovered to be more precise.

Numerical Study of a Nonlinear High Order Boundary Value Problems Using Genocchi Collocation Technique

Numerical Study of a Nonlinear High Order Boundary Value Problems Using Genocchi Collocation Technique

Collision Probability of Debris Clouds Based on Higher-Order Boundary Value Problems

Debris clouds stemming from the breakup of objects pose a threat to spacecraft. Here, a method based on higher-order boundary value problems (BVPs) is presented to analyze the hazard of debris clouds. The higher-order transfer map from initial velocity space to final position space is obtained with the use of differential algebra. An adaptive homotopic scheme is proposed to solve the perturbed BVP, where the transfer map is used to correct the residuals. Then, higher-order expansions of the BVP solution and the Jacobian with respect to the final position are derived, and the debris density inside a spacecraft envelope is calculated with the expansions. The simulation results show that the proposed higher-order method contributes to accelerating the convergence of perturbed BVP solvers. The density transformation method based on the map is validated by Monte Carlo simulations. The effects of perturbations on the collision risk are shown.

A Metaheuristic Optimization Algorithm for Solving Higher-Order Boundary Value Problems

An effective metaheuristic algorithm to solve the higher-order boundary value problems, called a genetic programming technique is presented. In this paper, a genetic programming algorithm, which depends on the syntax tree representation, is employed to obtain the analytical solutions of higher- order differential equations with the boundary conditions. The proposed algorithm can be produce an exact or approximate solution when the classical methods lead to unsatisfactory results. To illustrate the efficiency and accuracy of the designed algorithm, several examples are tested. Finally, the obtained results are compared with the existing methods such as the homotopy analysis method, the B-Spline collocation method and the differential transform method.

AN ENHANCED WAVELET BASED METHOD FOR NUMERICAL SOLUTION OF HIGH ORDER BOUNDARY VALUE PROBLEMS

The Legendre wavelet collocation method (LWCM) is suggested in this study for solving high-order boundary value problems numerically. Eighth, tenth, and twelfth-order examples are used as test problems to ensure that the technique is efficient and accurate. In comparison to other approaches, the numerical results obtained using LWCM demonstrate that the method's accuracy is very good. The results indicate that the method requires less computational effort to achieve better results.

Galerkin Method for Nonlinear Higher-Order Boundary Value Problems Based on Chebyshev Polynomials

In the current paper, we develop an algorithm to approximate the analytic solution for the nonlinear boundary value problems in higher-order based on the Galerkin method. Chebyshev polynomials are introduced as bases of the solution. Meanwhile, some theorems are deducted to simplify the nonlinear algebraic set resulted from applying the Galerkin method, while Newton’s method is used to solve the resulting nonlinear system. Numerous examples are presented to prove the usefulness and effectiveness of this algorithm in comparison with some other methods.

Fictitious centre RBF method for high order BVPs in multiply connected domains

We propose a Kansa-radial basis function (RBF) collocation method for the solution of 2D and 3D high order (i.e. of order 2N where N∈N,N≥3) boundary value problems (BVPs) in multiply connected domains. These problems include N boundary conditions (BCs), and thus we need to select N distinct sets of boundary centres. One set is positioned on the physical boundary of the problem while the rest are placed outside the exterior boundary of the domain. The efficacy of the proposed approach is demonstrated by applying it to a 2D and a 3D high order BVP in multiply connected domains.