Linear Two-point Boundary Research Articles

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.

Thermal stress analysis of functionally graded solid and hollow thick-walled structures with heat generation

PurposeThis paper aims to present thermal and mechanical stresses in solid and hollow thick-walled cylinders and spheres made of functionally graded materials (FGMs) under the effect of heat generation.Design/methodology/approachConstant internal temperature and convective external conditions in hollow bodies along with internal heat generation with a combination of outer convective conditions in solid bodies are investigated individually. The effect of the heat convection coefficient on solid bodies is additionally discussed. The variation of the FGM properties in the radial direction is adapted to the Mori–Tanaka homogenization schemes, which produces irregular and two-point linear boundary value problems that are numerically solved by the pseudospectral Chebyshev method.FindingsIt has been shown that the selection of the mixtures of FGMs has to be made correctly to keep the thermal and mechanical loads acting on objects at low levels.Originality/valueIn this study, both solid and hollow functionally graded cylinders and spheres for different boundary conditions that are as their engineering applications are examined with the proposed method. The results have demonstrated that the pseudospectral Chebyshev method has high accuracy, low calculation costs and ease of application and can be easily adapted to such engineering problems.

Privacy preserving outsourcing algorithm for two-point linear boundary value problems

<p>One of a powerful application in the age of cloud computing is the outsourcing of scientific computations to cloud computing which makes cloud computing a very powerful computing paradigm, where the customers with limited computing resource and storage devices can outsource the sophisticated computation workloads into powerful service providers. One of scientific computations problem is Two-Point Boundary Value Problems(BVP) is a basic engineering and scientific problem, which has application in various domains. In this paper, we propose a privacy-preserving, verifiable and efficient algorithm for Two-Point Boundary Value Problems in outsourcing paradigm. We implement the proposed schema on the customer side laptop and using AWS compute domain elastic compute cloud (EC2) for the cloud side.</p>

A Unified Approach for Solving Linear and Nonlinear Odd-Order Two-Point Boundary Value Problems

In this article, we propose new spectral solutions for odd-order two-point boundary value problems. A numerical algorithm based on the use of collocation methods is implemented and analyzed. A new matrix of derivatives of certain Legendre basis polynomials is derived to serve in deriving the proposed algorithm. Our algorithm has the advantage that it is applicable to both linear and nonlinear odd-order two-point boundary value problems. The convergence and error analysis of the suggested method are investigated. Five illustrative examples supported with comparisons are displayed to ensure the efficiency and applicability of the proposed algorithm.

Parametric septic splines approach for the solution of linear sixth-order two-point boundary value problems

Parametric splines, which reduces to seven-degree polynomial splines, have been used to develop a class of numerical methods for the solution of linear sixth-order two-point boundary value problems. Convergence analysis of the methods is discussed through standard procedure. It is shown that the present methods give approximations, which are better than those produced by other splines and domain decomposition methods. Three numerical examples are given to illustrate the practical usefulness of the new approach.

Approximate Solutions of Singular Two-Point BVPs Using Legendre Operational Matrix of Differentiation

Exact and approximate analytical solutions of linear and nonlinear singular two-point boundary value problems (BVPs) are obtained for the first time by the Legendre operational matrix of differentiation. Different from other numerical techniques, shifted Legendre polynomials and their properties are employed for deriving a general procedure for forming this matrix. The accuracy of the technique is demonstrated through several linear and nonlinear test examples.

New Spectral Second Kind Chebyshev Wavelets Algorithm for Solving Linear and Nonlinear Second-Order Differential Equations Involving Singular and Bratu Type Equations

A new spectral algorithm based on shifted second kind Chebyshev wavelets operational matrices of derivatives is introduced and used for solving linear and nonlinear second-order two-point boundary value problems. The main idea for obtaining spectral numerical solutions for these equations is essentially developed by reducing the linear or nonlinear equations with their initial and/or boundary conditions to a system of linear or nonlinear algebraic equations in the unknown expansion coefficients. Convergence analysis and some efficient specific illustrative examples including singular and Bratu type equations are considered to demonstrate the validity and the applicability of the method. Numerical results obtained are compared favorably with the analytical known solutions.

Computation of Green’s functions for boundary value problems with Mathematica

This paper is devoted to construct an algorithm that allows us to calculate the explicit expression of the Green’s function related to a nth-order linear ordinary differential equation, with constant coefficients, coupled with two-point linear boundary conditions. We develop this algorithm by making a Mathematica package.

Quarter-sweep Modified SOR Iterative Algorithm and Cubic Spline Basis for the Solution of Second Order Two-point Boundary Value Problems

The aim of this study is to describe the formulation of Quarter-Sweep Modified Successive Over-Relaxation (QSMSOR) iterative method using cubic polynomial spline scheme for solving second order two-point linear boundary value problems. To solve the problems, a linear system will be constructed via discretization process by using cubic spline approximation equation. Then the generated linear system has been solved using the proposed QSMSOR iterative method to show the superiority over Full-Sweep Modified Successive Over-Relaxation (FSMSOR) and Half-Sweep Modified Successive Over-Relaxation (HSMSOR) methods. Computational results are provided to illustrate the effectiveness of the proposed method.

High-Order Numerical Method for Two-Point Boundary Value Problems

In this paper the two-point boundary value problem is transformed into general first-order ordinary differential equation system through introduction of conditions of an integral character to supplement the simultaneous set of first-order equations. A new discrete approximation of a high-order compact difference scheme is presented for the first-order system. It is a block-bidiagonal profile and removes the limits of other high-order discrete schemes at the interval ends. The numerical tests of a seventh-order compact difference scheme show that the proposed scheme is very convenient and efficient for linear and nonlinear two-point boundary value problems.

On the method of complementary functions for linear and nonlinear two-point boundary value problems

On the method of complementary functions for linear and nonlinear two-point boundary value problems

Numerical analytic method for the case of singular matrices in boundary conditions

A modification of the numerical analytic method of successive approximations is proposed for investigating the existence and constructing solutions of systems of nonlinear ordinary differential equations with two-point linear boundary conditions. The method is tailored to the case where the matrices contained in boundary conditions are singular, but their linear combination is nonsingular.

Strong Stability of Compact Discrete Boundary Value Problems via Exact Discretizations

This paper is concerned with strong stability inequalities (with respect to discrete and continuous Sobolev norms) for general linear mth order two-point boundary value problems and their discretizations by compact finite-difference schemes. The well-known stability of compact, consistent approximations on irregular grids is demonstrated here by a direct connection with the strong stability inequality of the continuous problem. The connection is made by means of exact discretizations, which are compact schemes that are exactly satisfied by the true solution of the original problem. The analysis involves certain generalized splines related to the Green’s function for the problem.

The Operator Compact Implicit Method for Fourth Order Ordinary Differential Equations

Two fourth order accurate approximations to general linear fourth order two-point boundary value problems with Dirichlet boundary conditions are evaluated. The first is a new implementation of the operator compact implicit (OCI) method. Its derivation will be given along with proofs of theoretical convergence and stability properties. In particular, it will be shown that it has no formal cell Reynolds number limitations. The second is centered second order differencing with Richardson extrapolation. Overall computational efficiency and the results of several numerical tests are discussed. Some comments are made on extending the OCI method to other boundary conditions and to nonlinear problems.

Analytic pseudoorthogonalization methods for linear two-point boundary value problems illustrated by the Orr-Sommerfeld equation

Two pseudoorthogonalization schemes for the solution of two-point linear boundary eigenvalue problems are discussed, which serve, during numerical integration, to keep the solutions linearly independent, without disturbing their inherent analytic dependence upon parameters and initial conditions. Solutions of the Orr-Sommerfeld equation are given as examples.

Numerical solution of two-point boundary value problems

A new algorithm is presented for the numerical solution of linear and non-linear two-point boundary value problems with explicit type of boundary conditions. It is a systematic iterative method which reduces the problem to a corresponding initial value problem. The method has been tested to work for a set of 36 simultaneous differential equations 4 of which were non-linear. Extension of this method for the more complicated case of implicit boundary conditions will be discussed in a separate paper.

A sixth order method for the integration of two point boundary value problems

A finite difference method for obtaining sixth order accurate approximation to the solution of the two-point linear boundary value problem is given. The convergence of the method is proved. Numerical results for a typical problem are tabulated and in each case the observed error is compared with its theoretical estimate. The numerical results are compared with those obtained from an earlier method of the author and the method of Noumerov.

A numerical method of solving second-order linear differential equations with two-point boundary conditions

ABSTRACTA direct method of integrating the equation y″ + g(x) y = h(x), with the two-point linear boundary conditions y′(a) + αy(a) = A, y′(b) + βy(b) = B, is based on the factorization of the equation into two first-order linear equations v′ − sv = h and y′ + sy = v, where s is a solution of the Riccati equation s′ − s2 = g. The first-order equations for v and y are integrated in succession, one in the direction of x increasing, and one in the direction of x decreasing, one boundary condition being used in each of these integrations. The appropriate solution of the Riccati equation is determined by the boundary condition at the end of the range from which the integration of the equation for v is started. The process is compared with the matrix factorization method of Thomas and Fox, and its stability discussed.

Procedure for the Machine or Numerical Solution of Ordinary Linear Differential Equations for Two-Point Linear Boundary Values

Procedure for the Machine or Numerical Solution of Ordinary Linear Differential Equations for Two-Point Linear Boundary Values

Procedure for the machine or numerical solution of ordinary linear differential equations for two-point linear boundary values

Procedure for the machine or numerical solution of ordinary linear differential equations for two-point linear boundary values