Graded mesh modified backward finite difference method for two parameters singularly perturbed second-order boundary value problems

In the literature so far, for Caputo fractional boundary value problems of order 2q when 1<2q<2, the problems use the same boundary conditions of the integer-order differential equation of order ‘2’. In addition, they only use the left Caputo derivative in computing the solution of the Caputo boundary value problem of order 2q. Further, even the initial conditions for a Caputo fractional differential equation of order nq use the corresponding integer-order initial conditions of order ‘n’. In this work, we establish that it is more appropriate to use the Caputo fractional initial conditions and Caputo fractional boundary conditions for sequential initial value problems and sequential boundary value problems, respectively. It is to be noted that the solution of a Caputo fractional initial value problem or Caputo fractional boundary value problem of order ‘nq’ will only be a Cnq solution and not a Cn solution on its interval. In this work, we present a methodology to compute the solutions of linear sequential Caputo fractional differential equations using initial and boundary conditions of fractional order kq, k=0,1,…(n−1) when the order of the fractional derivative involved in the differential equation is nq. The Caputo left derivative can be computed only when the function can be expressed as f(x−a). Then the Caputo right derivative of the same function will be computed for the function f(b−x). Further, we establish that the relation between the Caputo left derivative and the Caputo right derivative is very essential for the study of Caputo fractional boundary value problems. We present a few numerical examples to justify that the Caputo left derivative and the Caputo right derivative are equal at any point on the Caputo function’s interval. The solution of the linear sequential Caputo fractional initial value problems and linear sequential Caputo fractional boundary value problems with fractional initial conditions and fractional boundary conditions reduces to the corresponding integer initial and boundary value problems, respectively, when q=1. Thus, we can use the value of q as a parameter to enhance the mathematical model with realistic data.

There are few numerical techniques available to solve third order boundary value problems. In this paper, we present Galerkin weighted residual method for constructing the numerical solution of third order linear and nonlinear boundary value problems with two point boundary conditions. The method is formulated as a rigorous matrix form. Several numerical examples of both linear and nonlinear boundary value problems in the literature are presented to illustrate the reliability and efficiency of the proposed method. The present method is quite efficient and yields better results when compared with the existing methods in the literature. Keywords: Galerkin method, Third order linear and nonlinear BVPs, Bernstein Polynomials.

Based on the meshless local natural neighbor interpolation method (MLNNI), a novel solution procedure is developed for the analysis of nonlinear steady and transient heat transfer of two-dimensional structures in this paper. Nonlinearities arising from temperature dependence of material properties and nonlinear boundary conditions have been taken into account. The present method is developed based on the natural neighbor interpolation (NNI) for constructing shape functions at scattered nodes. The three-node triangular FEM shape function is employed as the test function, which reduces the orders of integrands involved in domain integrals. Due to the delta function property of the natural neighbor shape functions, there is no need to employ special techniques to enforce the essential boundary conditions. The backward difference method is employed for the time integration scheme in transient analysis and the Newton–Raphson iterative procedure is required at each time step. Three numerical examples with different geometries and boundary conditions are presented at the end to demonstrate the validity and accuracy of the proposed method for the solution of a wide class of nonlinear steady and transient heat transfer problems.

Background/Objectives: Numerically solving second order Boundary Value Problems (BVPs) directly is an area that has drawn much attention in recent literature and hence the general aim of this article. Methods/Statistical Analysis: Block methods are developed using Taylor series approach and are applied to directly approximate the considered second order Boundary Value Problems. Certain specifications of the block method developed involves it being of step number k = 4 and order p = 5. This block method consists of mathematical expressions that will simultaneously provide results at the grid points and derivatives of the grid points for the k-step method. Findings: After the development of the new block method in this article, certain Boundary Value Problems are considered and the new block method is adopted to solve it numerically and comparison was made with their exact solution. The results show the new block method show superiority in comparison to other numerical methods of same order p = 5 adopted by other authors in previous literature. Hence, this new block method can be considered a more suitable numerical approach to approximate the numerical solutions of second order BVPs. Applications/Improvements: This block method is suitable for application in solving numerically second order BVPs with closer accuracy to the exact or analytic solutions of the differential equations under consideration.

Efficient algorithms for analyzing the singularly perturbed boundary value problems of fractional order

We consider the numerical solution of the well known Falkner-Skan problem, which is third order nonlinear boundary value problem. The approach we are going to follow is to first transform the third order boundary value problem on the semi-finite domain into a second order nonlinear boundary value problem on a finite domain through introducing a special transformation. The resulting two-point boundary value problem is then treated numerically using the sinc-collocation method which is known to converge exponentially. Numerical results will be presented for various values of the parameters representing various types of flows. Comparison with the work of others will also be done to show the accuracy of the Sinc method.

This paper focuses on the comparison of various numerical solutions for Initial Value Problem(IVP) and Boundary Value Problem(BVP) in differential equations. The practical problem is solved by comparing each numerical method solution. In the part of IVP problem, this paper established an Susceptible Infectious Recovered Model (SIR) to measure the disease situation in India during the new coronavirus period of 2020-2021, and calculated the corresponding parameters through machine learning genetic algorithm(GA). The differential equations are solved by using first and second-order Taylor expansions(TM), Euler(EM), improved Euler(ME), Runge-Kutta 2(RK2), optional Runge-Kutta(RK2*), and Runge-Kutta 4(RK4) methods. Then, based on the model, the situation of the novel coronavirus epidemic in the next year is predicted; In the part of BVP problem, this paper establishes a linear spring model(LSM) to measure the displacement of elastomer under the force distribution, that is, the force density, by using a simulation algorithm(SA) to expand a small amount of measurement data, and by using machine learning algorithm linear regression(LR) to calculate the spring stiffness coefficient. Numerical solutions are calculated by using forward, center and backward finite difference method(FDM), Galerkin method in variational method(VGLM), collocation method(CM) and Galerkin method in finite element method(FEM), and mean square error(MSE) is used to evaluate the model. Finally, the analytical solutions especially Laplace Transformation(LT) of the two models are given to compare the numerical solutions.

A pragmatic combination of time-integration methods is presented. In particular, the backward difference method is changed into a modified Crank-Nicolson method with a doubling of the time step. This combination is specially effective in the case of linear analysis, since it simplifies the element assembly and the forward elimination process. As compared with the backward difference method, this time integration method has the potential to significantly reduce computation time with improved accuracy.

In this work, a technique based on combination between collocation and spline method along with the shooting method is proposed for the solution of boundary value problem of order 2. Numerical results show that the method is simple and effective.

This chapter presents existence theory for second order boundary value problems on infinite intervals. There are two major approaches in the literature to establish existence of solutions to boundary value problems on infinite intervals. The first approach is based on a diagonalization process whereas the second is based on the Furi-Pera fixed point theorem. Both approaches will be presented in this chapter. In Section 1.2 we list several examples from the real world phenomena which motivate the study of boundary value problems on infinite intervals. In Section 1.3 we discuss some infinite interval problems which date back to 1896 and state several fundamental results which are well known.

We study the regularity of the solutions of second order boundary value problems on manifolds with boundary and bounded geometry. We first show that the regularity property of a given boundary value problem $(P, C)$ is equivalent to the uniform regularity of the natural family $(P_x, C_x)$ of associated boundary value problems in local coordinates. We verify that this property is satisfied for the Dirichlet boundary conditions and strongly elliptic operators via a compactness argument. We then introduce a uniform Shapiro-Lopatinski regularity condition, which is a modification of the classical one, and we prove that it characterizes the boundary value problems that satisfy the usual regularity property. We also show that the natural Robin boundary conditions always satisfy the uniform Shapiro-Lopatinski regularity condition, provided that our operator satisfies the strong Legendre condition. This is achieved by proving that "well-posedness implies regularity" via a modification of the classical "Nirenberg trick". When combining our regularity results with the Poincare inequality of (Ammann-Grosse-Nistor, preprint 2015), one obtains the usual well-posedness results for the classical boundary value problems in the usual scale of Sobolev spaces, thus extending these important, well-known theorems from smooth, bounded domains, to manifolds with boundary and bounded geometry. As we show in several examples, these results do not hold true anymore if one drops the bounded geometry assumption. We also introduce a uniform Agmon condition and show that it is equivalent to the coerciveness. Consequently, we prove a well-posedness result for parabolic equations whose elliptic generator satisfies the uniform Agmon condition.

On the Accuracy of Gaussian Approximation to the Distribution Functions of Sums of Independent Variables

The scattering of surface water waves by bottom undulation in the presence of a permeable vertical barrier is investigated for its solution. A mixed boundary value problem (BVP) arises here in a natural way while examining this physical problem. Regular perturbation analysis is employed to determine the solution of the BVP. By utilizing this analysis the given BVP reduces to two different BVPs up to first order. The solution of the zeroth order BVP is obtained with the aid of eigenfunction expansion method in conjunction with least-squares approximation. The first order BVP is solved with the help of the Green's integral theorem and the physical quantities, namely the reflection and transmission coefficients, are obtained in the form of integrals which involve the bottom undulation and the solution of the zeroth order BVP. A particular form of the bottom undulation which closely resembles to some obstacles made by nature due to sedimentation and ripple growth of sand, is considered to evaluate these integrals. The variation of these coefficients is examined for different values of the porous effect parameter, barrier length, number of ripples and ripple amplitude.

We present two collocation algorithms based on interpolating and approximating subdivision schemes for the solution of fourth order boundary value problems arising in the mathematical modeling of viscoelastic, and inelastic flows, deformation of beams, arches, and load bearing members like street lights and robotic arms in multipurpose engineering systems. Numerical examples are given to illustrate the algorithms. We conclude that approximating schemes based collocation algorithms give better solution than interpolating schemes based collocation algorithms. Main purpose of this paper is to explore and seek the applications of interpolating and approximating subdivision schemes in the field of boundary value problems along with intrinsic comparison of the results obtained by algorithms based on these schemes. A comparison with other approaches of this type of boundary value problems in order to see the advantages of the proposed methods is also given.

A priori estimation of the systematic error of consistently derived theories for thin structures