An Integral Transform Associated with Boundary Conditions Containing an Eigenvalue Parameter

In this research, the power series method for numerical solutions of boundary value problems of Class M is being presented. Taylor series method was used to transformed a boundary value problem of class M to its equivalent initial value problem. A new standard and exact finite difference scheme were developed and applied to boundary value problem. We also developed a series which eventually approximates the solution to the boundary value problem of Class M. We model a reallife problem involving radioactive substances into a differential equation and solved it using different step length. Comparison of the numerical solutions of the scheme was made with their theoretical (exact) solutions on some sampled initial value problems. The scheme developed have been shown to be efficient, ingenious and computationally reliable.

This paper studies a boundary value problem of nonlinear second-order q-difference equations with non-separated boundary conditions. As a first step, the given boundary value problem is converted to an equivalent integral operator equation by using the q-difference calculus. Then the existence and uniqueness of solutions of the problem is proved via the resulting integral operator equation by means of Leray-Schauder nonlinear alternative and some standard fixed point theorems. Our approach is simpler than the one involving the typical series solution form of q-difference equations. The results corresponding to a second-order q-difference equation with anti-periodic boundary conditions appear as a special case. Furthermore, our results reduce to the corresponding results for classical second-order boundary value problems with non-separated boundary conditions in the limit q → 1, which provides a useful check.2010 Mathematics Subject Classification. 39A05, 39A13.

In this paper, we study solvability of new classes of nonlocal boundary value problems for the Laplace equation in a ball. The considered problems are multidimensional analogues (in the case of a ball) of classical periodic boundary value problems in rectangular regions. To study the main problem, first, for the Laplace equation, we consider an auxiliary boundary value problem with an oblique derivative. This problem generalizes the well-known Neumann problem and is conditionally solvable. The main problems are solved by reducing them to sequential solution of the Dirichlet problem and the problem with an oblique derivative. It is proved that in the case of periodic conditions, the problem is conditionally solvable; and in this case the exact condition for solvability of the considered problem is found. When boundary conditions are specified in the anti-periodic conditions form, the problem is certainly solvable. The obtained general results are illustrated with specific examples.

Boundary value problems (BVP) arise in many mathematical model of frontier science. In short, a BVP is a second order ordinary differential equation with boundary conditions prescribed at two distinct points. Usually, we prescribe three types of boundary conditions: Dirichlet, Neumann and Mixed. Among them mixed boundary conditions is little difficult to approach. One of the method to solve the BVP with mixed boundary conditions is the shooting method. In this method, we first convert the BVP into an initial value problems (IVP) and then solve it. We then check whether the solution of IVP is a solution of the BVP. There are standard ways of solving IVPs. Recently, some alternate methods are also used to solve them. For example, several authors use Haar wavelet. Frames are considered as an extension to wavelet. In this paper, we construct operational matrices from a frame and then apply them to solve the IVP which is obtained from the given BVP by shooting method. This method is simple and easy to use because we construct the operational matrix once, store it and then apply it any times we want. Moreover, we also show that the given operational matrix method converges quadratically.

This article aims to develop a uniform method for solving plane elastic boundary value problems, PEBVPs. Stress or displacement vectors are assumed to be given on the boundary in the classical formulation of PEBVP, while non– classical formulations include boundary conditions in terms of orientations of stresses, forces or displacements. It is shown that the approximation of the complex potentials by the linear combinations of holomorphic functions can be used to obtain solutions in these cases and that the other well known numerical methods can be represented as particular cases of this approach. Two examples are presented.

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.

Eigenvalues of a general class of boundary value problem with derivative-dependent nonlinearity

Boundary value problems in transport with mixed and oblique derivative boundary conditions, steady state and transient solutions

Existence of solutions of two-point boundary value problems for nonlinear systems

An effective numerical technique for solving a special class of ordinary difference equations

A frozen Jacobian iterative method is proposed for solving systems of nonlinear equations. In particular, we are interested in solving the systems of nonlinear equations associated with initial value problems (IVPs) and boundary value problems (BVPs). In a single instance of the proposed iterative method DEDF, we evaluate two Jacobians, one inversion of the Jacobian and four function evaluations. The direct inversion of the Jacobian is computationally expensive, so, for a moderate size, LU factorization is a good direct method to solve the linear system. We employed the LU factorization of the Jacobian to avoid the direct inversion. The convergence order of the proposed iterative method is at least eight, and it is nine for some particular classes of problems. The discretization of IVPs and BVPs is employed by using Jacobi-Gauss-Lobatto collocation (J-GL-C) method. A comparison of J-GL-C methods is presented in order to choose best collocation method. The validity, accuracy and the efficiency of our DEDF are shown by solving eleven IVPs and BVPs problems.

ABSTRACTThe inspiration for this study originates from a recognized research gap within the broader collection of studies on nanofluids, with a specific focus on their interactions with different surfaces and boundary conditions (BCs). The primary purpose of this research is to use an artificial neural network to examine the combination of Alumina‐Engine oil‐based nanofluid flow subject to electro‐magnetohydrodynamic effects, within a porous medium, and over a stretching surface with an impermeable structure under convective BCs. The flow model incorporates Thermophoresis and Brownian motion directly from Buongiorno's model. Accounting for the porous medium's effect, the model integrates the Forchheimer number (depicting local inertia) and the porosity factor developed in response to the presence of the porous medium. The conversion of governing equations into non‐linear ordinary differential systems is achieved by implementing transformations. A highly non‐linear ordinary differential system's final system is solved using a numerical scheme (Runge–Kutta fourth‐order). Findings indicate that the porosity factor positively impacts the skin friction and the momentum boundary layer. The influence suggests an increment in the frictional force and a decline in the velocity profile. The volume fraction, Prandtl number, and magnetic number significantly impact the flow profiles. The skin friction data is tabulated with some physical justifications.

Numerical solution of two-point boundary value problems

Analysis of some geotechnical problems using finite-element methods requires the implementation of a nonlinear model for soil materials, to better represent their actual behaviour. Constitutive modelling of soil mass behaviour and material interfaces is an essential component of the solution of boundary and initial value problems. The hyperbolic model is one of the most frequently used non-linear models for predicting the behaviour of soils in boundary value problems. The parameters of this model for specific soils need to be determined experimentally. This paper presents the results of extensive laboratory testing carried out on three soils from the Eastern Province of Saudi Arabia: sand, marl and sabkha. The tests used to obtain these results were the triaxial compression test, the hydrostatic (isotropic compression) test, and the direct shear test. Additionally, other tests were used to obtain various physical properties needed for the complete characterisation of these soils. Parameters of the hyperbolic models for non-linear tangent Young's and bulk moduli are presented. These parameters compare well with those reported in the literature. They are incorporated in the hyperbolic model and used to back-predict the stress–strain behaviour of the investigated soils. The calibrated models are found to predict soil behaviour very well.

Dual extremum principles and error bounds for a class of boundary value problems