An Analytic Solution for Riccati Matrix Delay Differential Equation using Coupled Homotopy-Adomian Approach

في هذا البحث تناولنا طريقة فعالة وجديدة وهي الدمج بين طريقتي الهوموتوبي و الادوميان مع إستخدام مفهوم المعادلات التفاضلية الاعتيادية التباطئية لحل معادلة المصفوفات التباطئية لمعادلة ريكاتي والحصول على حل تقريبي دقيق جدا قليل الخطأ ويقترب من الحل المظبوط او الحل الحقيقي. في هذه الطريقة تم الكشف على نتائج ادق خلال فترة التأخير التي مرت بها المعادلة. أيضا في هذه الطريقة ان الاقتراب للحل الحقيقي يأخذ منطقة او فترة اوسع وكلما استمرينا ب التكرارا حيث نحصل على نتائج دقيقة جدا ويكون الخطأ جدا صغير.An efficient modification and a novel technique combining the homotopy concept with  Adomian decomposition method (ADM) to obtain an accurate analytical solution for Riccati matrix delay differential equation (RMDDE) is introduced  in this paper  . Both methods are very efficient and effective. The whole integral part of ADM is used instead of the integral part of homotopy technique. The major feature in current technique gives us a large convergence region of iterative approximate solutions .The results acquired by this technique give better approximations for a larger region as well as previously. Finally, the results conducted via suggesting an efficient and easy technique, and may be addressed to other non-linear problems

Sumudu decomposition method for Solving fractional-order Logistic differential equation

In This paper, we propose a numerical algorithm for solving nonlinear fractional-order Logistic differential equation (FLDE) by using Sumudu decomposition method (SDM). This method is a combination of the Sumudu transform method and decomposition method. We have apply the concepts of fractional calculus to the well known population growth modle inchaotic dynamic. The fractional derivative is described in the Caputosense. The numerical results shows that the approach is easy to implement and accurate when applied to various fractional differentional equations. &nbsp

(R1885) Analytical and Numerical Solutions of a Fractional-Order Mathematical Model of Tumor Growth for Variable Killing Rate

This work intends to analyze the dynamics of the most aggressive form of brain tumor, glioblastomas, by following a fractional calculus approach. In describing memory preserving models, the non-local fractional derivatives not only deliver enhanced results but also acknowledge new avenues to be further explored. We suggest a mathematical model of fractional-order Burgess equation for new research perspectives of gliomas, which shall be interesting for biomedical and mathematical researchers. We replace the classical derivative with a non-integer derivative and attempt to retrieve the classical solution as a particular case. The prime motive is to acquire both analytical and numerical solutions to the posed problem. At first, we employ the transform method, and then the Adomian decomposition method to obtain the solutions that shall be useful to provide information about the effect of medical care in the annihilation of gliomas. Finally, we discuss the applicability of this model with numerical simulations and graphical representations

A new analytic solution for fractional chaotic dynamical systems using the differential transform method

AbstractNonlinear differential equations with fractional derivatives give general representations of real life phenomena. In this paper, a modification of the differential transform method (DTM) for solving the nonlinear fractional differential equation is introduced for the first time. The new algorithm is simple and gives an accurate solution. Moreover the new solution is continuous and analytic on each subinterval. A fractional Chen system is considered, to demonstrate the efficiency of the algorithm. The results obtained show good agreement with the generalized Adams–Bashforth–Moulton method

Modified Step Variational Iteration Method for Solving Fractional Biochemical Reaction Model

A new method called the modification of step variational iteration method (MoSVIM) is introduced and used to solve the fractional biochemical reaction model. The MoSVIM uses general Lagrange multipliers for construction of the correction functional for the problems, and it runs by step approach, which is to divide the interval into subintervals with time step, and the solutions are obtained at each subinterval as well adopting a nonzero auxiliary parameter ℏ to control the convergence region of series' solutions. The MoSVIM yields an analytical solution of a rapidly convergent infinite power series with easily computable terms and produces a good approximate solution on enlarged intervals for solving the fractional biochemical reaction model. The accuracy of the results obtained is in a excellent agreement with the Adam Bashforth Moulton method (ABMM)