Accurate spectral solutions of first and second-order initial value problems by the ultraspherical wavelets-Gauss collocation method

In this paper, we present an ultraspherical wavelets-Gauss collocation method for obtaining direct solutions of first- and second-order nonlinear differential equations subject to homogenous and nonhomogeneous initial conditions. The properties of ultraspherical wavelets are used to reduce the differential equations with their initial conditions to systems of algebraic equations, which then must be solved by using suitable numerical solvers. The function approximations are spectral and have been chosen in such a way that make them easy to calculate the expansion coefficients of the thought-for solutions. Uniqueness and convergence of the proposed function approximation is discussed. Four illustrative numerical examples are considered and these results are comparing favorably with the analytic solutions and proving more accurate than those discussed by some other existing techniques in the literature

A Pseudospectral Algorithm for Solving Multipantograph Delay Systems on a Semi-Infinite Interval Using Legendre Rational Functions

A new Legendre rational pseudospectral scheme is proposed and developed for solving numerically systems of linear and nonlinear multipantograph equations on a semi-infinite interval. A Legendre rational collocation method based on Legendre rational-Gauss quadrature points is utilized to reduce the solution of such systems to systems of linear and nonlinear algebraic equations. In addition, accurate approximations are achieved by selecting few Legendre rational-Gauss collocation points. The numerical results obtained by this method have been compared with various exact solutions in order to demonstrate the accuracy and efficiency of the proposed method. Indeed, for relatively limited nodes used, the absolute error in our numerical solutions is sufficiently small

FLAVONOIDS FROM SUGAR BEET LEAVES AS HEPATOPROTECTIVE AGENT

Objective: This work was designed to investigate the activity of Beta vulgaris (B. vulgaris) extract against hepatotoxicity induced by (carbon tetrachloride) CC14 in male rats.Methods: Hepatoprotective study was performed on rats, divided into different groups; control healthy rats, the group received B. vulgaris extract, intoxicated rats by CC14, CCl4 group treated with alcoholic leaves extract, and CCl4 intoxicated rats treated with silymarin. The evaluation was done through measuring liver function indices and oxidative stress markers.Results: The activities of Alanine Transferase (ALT), Aspartate Transferase (AST), Alkaline Phosphatase (ALP), and gamma-glutamyl transferase (GGT) increased by 187.07, 52.37, 50.58, and 94.59% respectively in CCl4 group from control. Supplementation of beet extract decreased this elevation to 10.83, 26.43, 17.07 and 37.21% for the previous parameters respectively. The values obtained of the enzymes activity return nearly to that of control values, also a histopathological investigation of liver confirmed the results obtained.Conclusion: Beet showed a remarkable anti-hepatotoxic activity against CC14 induced hepatic damageKeywords: B. vulgaris, Hepatoprotective, Flavonoids, Liver function, Antioxidant enzyme, Histopatholog

Efficient Solutions of Multidimensional Sixth-Order Boundary Value Problems Using Symmetric Generalized Jacobi-Galerkin Method

This paper presents some efficient spectral algorithms for solving linear sixth-order two-point boundary value problems in one dimension based on the application of the Galerkin method. The proposed algorithms are extended to solve the two-dimensional sixth-order differential equations. A family of symmetric generalized Jacobi polynomials is introduced and used as basic functions. The algorithms lead to linear systems with specially structured matrices that can be efficiently inverted. The various matrix systems resulting from the proposed algorithms are carefully investigated, especially their condition numbers and their complexities. These algorithms are extensions to some of the algorithms proposed by Doha and Abd-Elhameed (2002) and Doha and Bhrawy (2008) for second- and fourth-order elliptic equations, respectively. Three numerical results are presented to demonstrate the efficiency and the applicability of the proposed algorithms

STUDY OF THE POSSIBLE ANTIHYPERTENSIVE AND HYPOLIPIDEMIC EFFECTS OF AN HERBAL MIXTURE ON L-NAME-INDUCED HYPERTENSIVE RATS

Objective: Hypertension is a chronic medical condition. Diet can improve blood pressure control and decrease the risk of health complication.Methods: In this study, four plants: Roselle, Marjoram, Chamomile, and Doum were extracted by water. Equal portions of them were mixed. Lethaldose 50% of the mixture was assayed; the dose which did not cause any mortality was 266.94 mg/100 g body weight. Animals were classified into fivegroups: Negative control group, positive control group where hypertension was induced by L-name, two groups treated with two doses of the mixture,and a group treated with prazosin as a standard treatment. Treatment of hypertensive rats continued for 4 successive weeks.Results: Treatment with the mixture showed a significant reduction in blood pressure of hypertensive rats, as well as serum cholesterol, low-densitylipoprotein-cholesterol, and urea levels when compared to positive control group.Conclusion: The results obtained suggest that the aqueous extract is efficient as an antihypertensive and hypolipidemic agent.Keywords: Rats, Aqueous extract, Hypertension, Hyperlipidemia, L-name

A pseudo-spectral scheme for systems of two-point boundary value problems with left and right sided fractional derivatives and related integral equations

We target here to solve numerically a class of nonlinear fractional two-point boundary value problems involving left- and right-sided fractional derivatives. The main ingredient of the proposed method is to recast the problem into an equivalent system of weakly singular integral equations. Then, a Legendre-based spectral collocation method is developed for solving the transformed system. Therefore, we can make good use of the advantages of the Gauss quadrature rule. We present the construction and analysis of the collocation method. These results can be indirectly applied to solve fractional optimal control problems by considering the corresponding Euler-Lagrange equations. Two numerical examples are given to confirm the convergence analysis and robustness of the scheme. © 2021 Tech Science Press. All rights reserved.Russian Foundation for Basic Research, РФФИ: 19-01-00019The Russian Foundation for Basic Research (RFBR) Grant No. 19-01-00019