Linear Boundary Value Problems Research Articles

Approximate Solutions for Higher Order Linear and Nonlinear Boundary Value Problems

Approximate Solutions for Higher Order Linear and Nonlinear Boundary Value Problems

Modeling of a temperature field for extruder body

The paper considers the process of induction heating of the extruder body, the temperature of which determines the degree of heating of the polymer mixture in the zone of loading the dry mixture. A mathematical model of this process is formulated taking into account radiant heat transfer in the gap between the inductor and the case. An iterative numerical-analytical method is proposed for solving the corresponding nonlinear boundary value problem of housing heating, at the first iteration of which a linear boundary value problem is solved (without taking into account radiant heat transfer). At the subsequent stages, a nonlinear boundary value problem is solved. The iterative method is based on the application of integral transformations of the linear part of the problem, followed by an iterative scheme for finding a nonlinear problem. This scheme is based on the algorithms for the equivalent simplification of the expressions obtained by solving the problem. The results of mathematical modeling of the corresponding algorithms are presented.

Differential-algebraic boundary-value problems with the variable rank of leading-coefficient matrix

Conditions for the solvability of the linear boundary-value problem for systems of differential-algebraic equations with the variable rank of the leading-coefficient matrix and the corresponding solution construction procedure have been found.

EvoStencils: a grammar-based genetic programming approach for constructing efficient geometric multigrid methods

For many systems of linear equations that arise from the discretization of partial differential equations, the construction of an efficient multigrid solver is challenging. Here we present EvoStencils, a novel approach for optimizing geometric multigrid methods with grammar-guided genetic programming, a stochastic program optimization technique inspired by the principle of natural evolution. A multigrid solver is represented as a tree of mathematical expressions that we generate based on a formal grammar. The quality of each solver is evaluated in terms of convergence and compute performance by automatically generating an optimized implementation using code generation that is then executed on the target platform to measure all relevant performance metrics. Based on this, a multi-objective optimization is performed using a non-dominated sorting-based selection. To evaluate a large number of solvers in parallel, they are distributed to multiple compute nodes. We demonstrate the effectiveness of our implementation by constructing geometric multigrid solvers that are able to outperform hand-crafted methods for Poisson’s equation and a linear elastic boundary value problem with up to 16 million unknowns on multi-core processors with Ivy Bridge and Broadwell microarchitecture.

New results and applications on the existence results for nonlinear coupled systems

In this manuscript, we study a certain classical second-order fully nonlinear coupled system with generalized nonlinear coupled boundary conditions satisfying the monotone assumptions. Our new results unify the existence criteria of certain linear and nonlinear boundary value problems (BVPs) that have been previously studied on a case-by-case basis; for example, Dirichlet and Neumann are special cases. The common feature is that the solution of each BVPs lies in a sector defined by well-ordered coupled lower and upper solutions. The tools we use are the coupled lower and upper solutions approach along with some results of fixed point theory. By means of the coupled lower and upper solutions approach, the considered BVPs are logically modified to new problems, known as modified BVPs. The solution of the modified BVPs leads to the solution of the original BVPs. In our case, we only require the Nagumo condition to get a priori bound on the derivatives of the solution function. Further, we extend the results presented in (Franco et al. in Extr. Math. 18(2):153–160, 2003; Franco et al. in Appl. Math. Comput. 153:793–802, 2004; Franco and O’Regan in Arch. Inequal. Appl. 1:423–430, 2003; Asif et al. in Bound. Value Probl. 2015:134, 2015). Finally, as an application, we consider the fully nonlinear coupled mass-spring model.

Generalized Iteration Method via Green's Function for the Solution of the Fourth-order Boundary Value Problems

The aim of this paper is to extend and generalize Picard-Green’s fixed point iteration method for the solution of fourth-order Boundary Value Problems. Several numerical applications to linear and nonlinear fourth-order Boundary Value Problems are discussed to illustrate the main results.

Approximation Properties of Solutions to Multipoint Boundary-Value Problems

We consider a wide class of linear boundary-value problems for systems of $m$ ordinary differential equations of order $r$, known as general boundary-value problems. Their solutions $y:[a,b]\to \mathbb{C}^{m}$ belong to the Sobolev space $(W_1^{r})^m$, and the boundary conditions are given in the form $By=q$ where $B:(C^{(r-1)})^{m}\to\mathbb{C}^{rm}$ is an arbitrary continuous linear operator. We prove that a solution to such a problem can be approximated with an arbitrary precision in $(W_1^{r})^m$ by solutions to multipoint boundary-value problems with the same right-hand sides. These multipoint problems are built explicitly and do not depend on the right-hand sides of the general boundary-value problem. For these problems, we obtain estimates of errors of solutions in the normed spaces $(W_1^{r})^m$ and $(C^{(r-1)})^{m}$.

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.

Splines solutions of boundary value problems that arises in sculpturing electrical process of motors with two rotating mechanism circuit

This study manages the numeric roots of the 7th-order linear & nonlinear boundary value problems(BVPs) utilizing another CB(Cubic-B) spline strategy. Cubic Spline interpolation is a different type of spline interpolation which is utilized very frequently to escape the problem of Runge’s phenomenon. That technique provides an interpolating polynomial which is evener and has lesser error than former interpolating polynomials such as Lagrange polynomial and Newton polynomial. The primary thought is that we have altered the BVPs to deliver another framework arrangement of linear equations. We develop the class of numerical techniques for a particular selection of the factors that are associated with CB Spline. The end conditions associated with the BVPs are determined. For each problem, the results obtained by CB Spline is compared with the exact solution. The absolute error(AE) for every iteration is calculated. To show the higher level of preciseness of CB Spline, the absolute errors of the CB Spline has been compared with different techniques such as Modified Decomposition Method(MDM), Differential Transform Method(DTM), Homotopy Perturbation Method(HPM), Variational Iteration technique(VIT) and observed to be more accurate. Graphs that describe the graphical comparison of CB Spline at n = 5 and n = 10 are also included in this paper. The calculation created here isn’t just for the numeric roots of the 7th-order BVPs. It also evaluates the derivative to the 7th-order derivative of the specific solution. A few models are represented, which depicting the practicality and capability of the suggested conspire.

Experimental measurement and numerical validation of sloshing effects on resistance increase in head waves

Most sea trials of large merchant ships are carried out with the filling ballast water in cargo, and the trial environment may involve mild waves as well. It causes the ship to undergo a coupling effect between external wave excitation and internal sloshing motion, and an understanding of such coupling is important to analyze the components of the total resistance measured in sea trials. In this study, a blunt modified Wigley equipped with two inner tanks is selected to observe its hydrodynamic responses under head waves, using both experimental measurement and numerical computations. Ship is considered to have a forward motion at the Froude number of 0.2, and three filling conditions varying from low filling to high filling have been arranged for two inner tanks. For validation, numerical computation is performed based on the three-dimensional Rankine panel method with the linearized boundary value problems. Both seakeeping and internal sloshing flow are analyzed using the linear potential theory. The validation presents a sufficient agreement between experimental measurement and numerical results with or without the sloshing effect, and both two methods have shown that the sloshing coupling effect is generally insignificant for present ship model. However, when the wave encounter frequency becomes very close to the tank natural frequency, the medium filling tanks generate a strong coupling effect which can restrain ship motion responses as well as added resistance.

On the method of reflections

This paper aims at reviewing and analysing the method of reflections, which is an iterative procedure designed for solving linear boundary value problems set in multiply connected domains. Being based on a decomposition of the domain boundary, this method is particularly well-suited to numerical solvers relying on boundary integral representation. For both the sequential and parallel forms of the method appearing in the literature, we interpret the procedure in terms of projection operators. Using a Hilbert space setting and orthogonality, we prove the unconditional convergence of the sequential form and propose a modification of the parallel one that makes it unconditionally converging. Several examples of boundary value problems that enter such a framework are given, an alternative proof of convergence is provided in a case which does not. A few numerical tests conclude the study.

New Cubic B-Spline Approximation for Solving Linear Two-Point Boundary-Value Problems

In this study, we introduce a new cubic B-spline (CBS) approximation method to solve linear two-point boundary value problems (BVPs). This method is based on cubic B-spline basis functions with a new approximation for the second-order derivative. The theoretical new approximation for a second-order derivative and the error analysis have been successfully derived. We found that the second-order new approximation was O(h3) accurate. By using this new second-order approximation, the proposed method was O(h5) accurate. Four numerical problems consisting of linear ordinary differential equations and trigonometric equations with different step sizes were performed to validate the accuracy of the proposed methods. The numerical results were compared with the least squares method, finite difference method, finite element method, finite volume method, B-spline interpolation method, extended cubic B-spline interpolation method and the exact solutions. By finding the maximum errors, the results consistently showed that the proposed method gave the best approximations among the existing methods. We also found that our proposed method involved simple implementation and straightforward computations. Hence, based on the results and the efficiency of our method, we can say that our method is reliable and a promising method for solving linear two-point BVPs.

Second Chebyshev wavelets (SCWs) method for solving finite-time fractional linear quadratic optimal control problems

In this paper, we present an indirect computational procedure based on the truncated second kind Chebyshev wavelets for finding the solutions of Caputo fractional time-invariant linear optimal control systems which the functional cost consists of the finite-time quadratic cost function. Utilizing the operational matrices of second Chebyshev wavelets (SCWs) of the Riemann–Liouville fractional integral, the corresponding linear two-point boundary value problem (TPBVP), obtained from the fractional Euler–Lagrange equations, is reduced to a coupled Sylvester-type matrix equation. An equivalent linear matrix form using the Kronecker product is constructed. The upper bound of the error of the SCWs approximation and the convergence of the proposed method are investigated. Low computational complexity and flexible accuracy are two important superiorities of this approach. Numerical experiments provide satisfactory results compared to the exiting techniques.

Boundary influence on the fields and arrangement of the needle domains in barium titanate single crystals

The superposition theory is adopted to study the boundary value problem of needle domains in barium titanate single crystals, which decomposes into an infinite-medium solution and a complementary solution. The model of an equivalent edge dislocation and line charge is adopted to calculate the fields due to the discrete needle domains in an infinite medium. The complementary solutions are obtained from a linear elastic boundary value problem using the finite element method. The influence of boundaries on the fields and arrangement of the needles in thin barium titanate single crystals is analysed, by considering traction-free boundaries and using the plane stress approximation. According to the principle of minimum potential energy, it is found that pairs of parallel needle domains can be stabilized if the effective charge associated with the needle tips is reduced to 48% of the material’s full polarization charge. The evolution of comb-like arrays of needle domains under compressive mechanical load is simulated, and the results are found to be qualitatively similar to experimental observations which have not previously been explained theoretically.

Thermal wave propagation in a functionally graded annular fin with fixed base

Thermal wave propagation and efficiency analysis of thin axially symmetric functionally graded annular fins with a fixed base boundary condition are examined. All material properties are graded along the radial direction using the Mori–Tanaka homogenization scheme to design a realistic and physically meaningful graded material model and create a smooth distribution of elastic and thermal material properties. The novelty of this research is handling a more realistic annular fin, which has sufficient strength in practice and significantly reduces weight, using two different metal materials with the elasticity-based Mori–Tanaka model. These conditions result in two-point linear boundary value problems that are numerically solved by the pseudospectral Chebyshev method. The temperature distributions, thermal stresses and the efficiency in the functionally graded annular fin are compared with fins made of steel only and aluminium only.

Нелинейные колебания поверхности раздела двух жидкостей при угловых колебаниях бака

The development of rocket and space technology in recent years has led to the widespread use of various cryogenic liquids. To increase the shelf life of cryo-products on board spacecraft or in tankers of future space filling stations, it is proposed to create a certain stock of cryo-product, which is simultaneously in a two-phase or three-phase state and forms layers of liquid. The paper considers the problem in a nonlinear formulation about the oscillations of the interface of a two-layer liquid in an arbitrary axisymmetric cavity of a solid body performing angular oscillations around a horizontal axis. For the considered class of cavities with an arbitrary bottom and a lid, the nonlinear problem is reduced to the sequential solution of linear boundary value problems. Nonlinear differential equations describing the oscillations of the interface between two liquids in the vicinity of the main resonance are obtained. In the case of a circular cylindrical cavity with flat bottoms, solutions of boundary value problems in the form of cylindrical functions were used to calculate linear and nonlinear hydrodynamic coefficients depending on the depth and density of the upper liquid.

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.

АЛГОРИТМ РЕШЕНИЯ КРАЕВОЙ ЗАДАЧИ ДЛЯ СУЩЕСТВЕННО НАГРУЖЕННЫХ ДИФФЕРЕНЦИАЛЬНЫХ УРАВНЕНИЙ

A linear boundary value problem for essentially loaded differential equations is considered. Using the properties of essentially loaded differential we reduce the considering problem to a two-point boundary value problem for loaded differential equations. This problem is investigated by parameterization method. We offer algorithm for solving to boundary value problem for the system of loaded differential equations. This algorithm includes of the numerical solving of the Cauchy problems for system of the ordinary differential equations and solving of the linear system of algebraic equations. For numerical solving of the Cauchy problem we apply the Runge–Kutta method of 4th order. The proposed numerical implementation is illustrated by example.

An efficient numerical approach for solving two-point fractional order nonlinear boundary value problems with Robin boundary conditions

This article proposes new strategies for solving two-point Fractional order Nonlinear Boundary Value Problems (FNBVPs) with Robin Boundary Conditions (RBCs). In the new numerical schemes, a two-point FNBVP is transformed into a system of Fractional order Initial Value Problems (FIVPs) with unknown Initial Conditions (ICs). To approximate ICs in the system of FIVPs, we develop nonlinear shooting methods based on Newton’s method and Halley’s method using the RBC at the right end point. To deal with FIVPs in a system, we mainly employ High-order Predictor–Corrector Methods (HPCMs) with linear interpolation and quadratic interpolation (Nguyen and Jang in Fract. Calc. Appl. Anal. 20(2):447–476, 2017) into Volterra integral equations which are equivalent to FIVPs. The advantage of the proposed schemes with HPCMs is that even though they are designed for solving two-point FNBVPs, they can handle both linear and nonlinear two-point Fractional order Boundary Value Problems (FBVPs) with RBCs and have uniform convergence rates of HPCMs, mathcal{O}(h^{2}) and mathcal{O}(h^{3}) for shooting techniques with Newton’s method and Halley’s method, respectively. A variety of numerical examples are demonstrated to confirm the effectiveness and performance of the proposed schemes. Also we compare the accuracy and performance of our schemes with another method.

Multilevel correction adaptive finite element method for solving nonsymmetric eigenvalue problems

Large-scale nonsymmetric eigenvalue problems are common in various fields of science and engineering computing. However, their efficient handling is challenging, and research on their solution algorithms is limited. In this study, a new multilevel correction adaptive finite element method is designed for solving nonsymmetric eigenvalue problems based on the adaptive refinement technique and multilevel correction scheme. Different from the classical adaptive finite element method, which requires solving a nonsymmetric eigenvalue problem in each adaptive refinement space, our approach requires solving a symmetric linear boundary value problem in the current refined space and a small-scale nonsymmetric eigenvalue problem in an enriched correction space. Since it is time-consuming to solve a large-scale nonsymmetric eigenvalue problem directly in adaptive spaces, the proposed method can achieve nearly the same efficiency as the classical adaptive algorithm when solving the symmetric linear boundary value problem. In addition, the corresponding convergence and optimal complexity are verified theoretically and demonstrated numerically.