GPs and end of life decisions : views and experiences

The views and experiences of GPs with respect to end of life (EoL) care are seldom addressed. The aim of this article is to better understand this aspect of care. A cross-sectional survey of all doctors in the country was designed and set up. The overall response was 396 (39.7%), 160 of which were GPs. 28.7% of GPs received no formal training in palliative medicine. 89.8% of respondents declared that their religion was important in EoL care. 45.3% agreed with the right of a patient to decide whether or not to hasten the EoL. 70.5% agreed that physicians should aim to preserve life. 15% of GPs withdrew or withheld treatment in the care of these patients. 41.1% had intensified analgesia at EoL. 7.5% had sedated patients at EoL. Lastly, 89.1% GPs would never consider euthanasia. Significant correlation (p< 0.05) was observed between considering euthanasia, using sedation, importance of religion and patients’ rights in EoL. A thematic analysis of comments highlighted the importance of the topic and feeling uncomfortable in EoL care. In conclusion there needs to be more training in palliative care. GPs believe in preserving life, would not consider euthanasia but do not shun intensification of analgesia at the end of life. There might be some misunderstanding with respect to the role of sedation at the EoL. GPs need legal and moral guidance in EoL care, in the absence of which, their religion is used as a guide.peer-reviewe

A Parameterized multi-step Newton method for solving systems of nonlinear equations

We construct a novel multi-step iterative method for solving systems of nonlinear equations by introducing a parameter. to generalize the multi-step Newton method while keeping its order of convergence and computational cost. By an appropriate selection of theta, the new method can both have faster convergence and have larger radius of convergence. The new iterative method only requires one Jacobian inversion per iteration, and therefore, can be efficiently implemented using Krylov subspace methods. The new method can be used to solve nonlinear systems of partial differential equations, such as complex generalized Zakharov systems of partial differential equations, by transforming them into systems of nonlinear equations by discretizing approaches in both spatial and temporal independent variables such as, for instance, the Chebyshev pseudo-spectral discretizing method. Quite extensive tests show that the new method can have significantly faster convergence and significantly larger radius of convergence than the multi-step Newton method.Peer ReviewedPostprint (author's final draft

A collocation method based on the bernoulli operational matrix for solving nonlinear BVPs which arise from the problems in calculus of variation

A new collocation method is developed for solving BVPs which arise from the problems in calculus of variation. These BVPs result from the Euler-Lagrange equations, which are the necessary conditions of the extremums of problems in calculus of variation. The proposed method is based upon the Bernoulli polynomials approximation together with their operational matrix of differentiation. After imposing the collocation nodes to the main BVPs, we reduce the variational problems to the solution of algebraic equations. It should be noted that the robustness of operational matrices of differentiation with respect to the integration ones is shown through illustrative examples. Complete comparisons with other methods and superior results confirm the validity and applicability of the presented method

An efficient spectral approximation for solving several types of parabolic PDEs with nonlocal boundary conditions

The problem of solving several types of one-dimensional parabolic partial differential equations (PDEs) subject to the given initial and nonlocal boundary conditions is considered. The main idea is based on direct collocation and transforming the considered PDEs into their associated algebraic equations. After approximating the solution in the Legendre matrix form, we use Legendre operational matrix of differentiation for representing the mentioned algebraic equations clearly. Three numerical illustrations are provided to show the accuracy of the presented scheme. High accurate results with respect to the Bernstein Tau technique and Sinc collocation method confirm this accuracy

Numerical Solution of Nonlinear Fractional Volterra Integro-Differential Equations via Bernoulli Polynomials

This paper presents a computational approach for solving a class of nonlinear Volterra integro-differential equations of fractional order which is based on the Bernoulli polynomials approximation. Our method consists of reducing the main problems to the solution of algebraic equations systems by expanding the required approximate solutions as the linear combination of the Bernoulli polynomials. Several examples are given and the numerical results are shown to demonstrate the efficiency of the proposed method

Convergence Analysis of Legendre Pseudospectral Scheme for Solving Nonlinear Systems of Volterra Integral Equations

We are concerned with the extension of a Legendre spectral method to the numerical solution of nonlinear systems of Volterra integral equations of the second kind. It is proved theoretically that the proposed method converges exponentially provided that the solution is sufficiently smooth. Also, three biological systems which are known as the systems of Lotka-Volterra equations are approximately solved by the presented method. Numerical results confirm the theoretical prediction of the exponential rate of convergence

A collocation method based on the Bernoulli operational matrix for solving highorder linear complex differential equations in a rectangular domain,”

This paper contributes a new matrix method for the solution of high-order linear complex differential equations with variable coefficients in rectangular domains under the considered initial conditions. On the basis of the presented approach, the matrix forms of the Bernoulli polynomials and their derivatives are constructed, and then by substituting the collocation points into the matrix forms, the fundamental matrix equation is formed. This matrix equation corresponds to a system of linear algebraic equations. By solving this system, the unknown Bernoulli coefficients are determined and thus the approximate solutions are obtained. Also, an error analysis based on the use of the Bernoulli polynomials is provided under several mild conditions. To illustrate the efficiency of our method, some numerical examples are given

Fourier operational matrices of differentiation . . .

This paper introduces Fourier operational matrices of differentiation and transmission for solving high-order linear differential and difference equations with constant coefficients. Moreover, we extend our methods for generalized Pantograph equations with variable coefficients by using Legendre Gauss collocation nodes. In the case of numerical solution of Pantograph equation, an error problem is constructed by means of the residual function and this error problem is solved by using the mentioned collocation scheme. When the exact solution of the problem is not known, the absolute errors can be computed approximately by the numerical solution of the error problem. The reliability and efficiency of the presented approaches are demonstrated by several numerical examples, and also the results are compared with different methods

Numerical Solution of Nonlinear Fractional Volterra Integro-Differential Equations via Bernoulli Polynomials

This paper presents a computational approach for solving a class of nonlinear Volterra integro-differential equations of fractional order which is based on the Bernoulli polynomials approximation. Our method consists of reducing the main problems to the solution of algebraic equations systems by expanding the required approximate solutions as the linear combination of the Bernoulli polynomials. Several examples are given and the numerical results are shown to demonstrate the efficiency of the proposed method