ordinary differential equation

In this paper, we introduce second derivative multistep collocation meth-ods for the numerical integration of ordinary differential equations (ODEs). These methods combine the concepts of both multistep methods and col-location methods, using second derivative of the solution in the collocation points, to achieve an accurate and efficient solution with strong stability properties, that is, A-stability for ODEs. Using the second-order deriva-tives leads to high order of convergency in the proposed methods. These methods approximate the ODE solution by using the numerical solution in some points in the r previous steps and by matching the function values and its derivatives at a set of collocation methods. Also, these methods utilize information from the second derivative of the solution in the colloca-tion methods. We present the construction of the technique and discuss the analysis of the order of accuracy and linear stability properties. Finally, some numerical results are provided to confirm the theoretical expecta-tions. A stiff system of ODEs, the Robertson chemical kinetics problem, and the two-body Pleiades problem are the case studies for comparing the efficiency of the proposed methods with existing methods.

In this paper, we will use the SIRD model to discuss the development of the Corona pandemic in Iraq; through a system of non-linear differential equations, we will use the (Runge-Kutta) method as a solution to the system of various non-linear equations such as the SIRD model, and the parameters used are based on This paper deals with confirmed cases of injury, recovery and deaths from the accurate data available for the period from (February 24, 2020) to (February 22, 2022), and we also present an estimate of the Basic Reproduction number (R0) for the (SIRD) model.

In this paper, the methods called Newton’s interpolation and Aitken’s methods were developed and examined. We use Newton’s interpolation and Aitken’s methods to find the exact and analytic results for three different types of nonlinear ordinary differential equations (NLODEs) of first and second order through illustrative examples. By using the new method, we successfully handle some class of nonlinear ordinary differential equations of first and second order in a simple and elegant way compared to Newton’s and Lagrange methods in previous studies. One can conclude that Newton’s interpolation and Aitken’s methods are easy to yield and implement actual precise results.

In the paper, the authors view some ordinary differential equations and their solutions from the angle of (the generalized) derivative polynomials and simplify some known identities for the Bernoulli numbers and polynomials, the Frobenius-Euler polynomials, the Euler numbers and polynomials, in terms of the Stirling numbers of the first and second kinds.

In this paper, we present an approximate method to solve the solution of the second kind Volterra integral equations. This method is based on a previous scheme, applied by Maleknejad ‎et al., ‎‎[K. Maleknejad ‎and Aghazadeh, Numerical solution of Volterra integral equations of the second kind with convolution kernel by using Taylor-series expansion method, ‎Appl. Math. Comput.‎ (2005)]‎ to gain the approximate solution of the second kind Volterra integral equations with convolution kernel and Maleknejad ‎et al. ‎[K. Maleknejad ‎and‎ T. Damercheli, Improving the accuracy of solutions of the linear second kind volterra integral equations system by using the Taylor expansion method, ‎Indian J. Pure Appl. Math.‎ (2014)] ‎to gain the approximate solutions of systems of second kind Volterra integral equations with the help of Taylor expansion method. The Taylor expansion method transforms the integral equation into a linear ordinary differential equation (ODE) which, in this case, requires specified boundary conditions. Boundary conditions can be determined using the integration technique instead of differentiation technique. This method is more  stable than derivative method and can be implemented to obtain an approximate solution of the Volterra integral equation with smooth and weakly singular kernels. An error analysis for the method is provided. A comparison between our obtained results and the previous results is made which shows that the suggested method is accurate enough and more ‎stable.‎

In this paper Legendre wavelet bases have been used for finding approximate solutions to singular boundary value problems arising in physiology. When the number of basis functions are increased the algebraic system of equations would be ill-conditioned (because of the singularity), to overcome this for large $M$, we use some kind of Tikhonov regularization. Examples from applied sciences are presented to demonstrate the efficiency and accuracy of the method.

ýIn this paperý, ýfirst we discuss a local stability analysis of model was introduced by Pý. ýJý. ýMumby etý. ýalý. ý(2007)ý, ýwith $\frac{gM^{2}}{M}$ as the functional response termý. ýWe conclude that the grazing intensity is the important parameter to control the existence or extinction of the coral reefý. ýNextý, ýwe consider this model under the influence of the time delay as the bifurcation parameterý. ýWe show that for small time delayý, ýthe stability type of the equilibria will not changeý, ýhowever for large enough time delayý, ýthe interior equilibrium point become unstable in contrast to the ODE caseý. ýAlso for some critical grazing intensity and the time delayý, ýa Hopf bifurcation occur and a nontrivial periodic orbit will appearý. ýFurther we discuss its corresponding stability switching directionsý.

We describe the construction of second derivative general linear methods (SGLMs) of orders five and six. We will aim for methods which are A--stable and have Runge--Kutta stability property. Some numerical results are given to show the efficiency of the constructed methods in solving stiff initial value problems.

In this article, we develop the Sinc-Galerkin method based on double exponential transformation for solving a class of weakly singular nonlinear two-point boundary value problems with nonhomogeneous boundary conditions. Also several examples are solved to show the accuracy efficiency of the presented method. We compare the obtained numerical results with results of the other existing methods in the literature. The results of this paper confirm that our method is very fast, simple and considerably accurate.

Long time integration of large stiff systems of initial value problems, arising from chemical reactions, demands efficient methods with good accuracy and extensive absolute stability region. In this paper, we apply second derivative general linear methods to solve some stiff chemical problems such as chemical Akzo Nobel problem, HIRES problem and OREGO problem.

Second derivative general linear methods (SGLMs) as an extension of general linear methods (GLMs) have been introduced to improve the stability and accuracy properties of GLMs. The coefficients of SGLMs are given by six matrices, instead of four matrices for GLMs, which are obtained by solving nonlinear systems of order and usually Runge--Kutta stability conditions. In this paper, we introduce a technique for construction of an special case of SGLMs which decreases the complexity of finding coefficients matrices.