Solving Whitham-Broer-Kaup-Like Equations Numerically by using Hybrid Differential Transform Method and Finite Differences Method

           الهدف من هذا البحث هو استخدام طريقة هجينة لطريقتين قويتين, هما التحويل التفاضلي وطريقة الفروقات المنتهية, وللحصول على حل لمعادلات Whitham-Broer-Kaup-Likeالثنائيةوالتي يناقش سلوكيات انتشار موجات المياه الضحلة. حيث يتم التحقق من كفاءة الطريقة على مثل هذه المسائل من خلال أخذ معلمات وشروط ابتدائية. النتائج العددية موضحة ببعدين وثلاثة أبعاد والتي تم الحصول عليها باستخدام هذه الطريقة ومقارنتها مع الحلول التحليلية تبين أن هناك دقة كبيرة لاقتراب هذه النتائج مع النتائج التحليلية.This paper aims to propose a hybrid approach of two powerful methods, namely the differential transform and finite difference methods, to obtain the solution of the coupled Whitham-Broer-Kaup-Like equations which arises in shallow-water wave theory. The capability of the method to such problems is verified by taking different parameters and initial conditions. The numerical simulations are depicted in 2D and 3D graphs. It is shown that the used approach returns accurate solutions for this type of problems in comparison with the analytic ones

(R1966) Semi Analytical Approach to Study Mathematical Model of Atmospheric Internal Waves Phenomenon

This research aims to study atmospheric internal waves which occur within the fluid rather than on the surface. The mathematical model of the shallow fluid hypothesis leads to a coupled nonlinear system of partial differential equations. In the shallow flow model, the primary assumption is that vertical size is smaller than horizontal size. This model can precisely replicate atmospheric internal waves because waves are dispersed over a vast horizontal area. A semi-analytical approach, namely modified differential transform, is applied successfully in this research. The proposed method obtains an approximate analytical solution in the form of convergent series without any linearization, perturbation, or calculation of unneeded terms, which is a significant advantage over other existing methods. To test the effectiveness and accuracy of the proposed method, obtained results are compared with Elzaki Adomain Decomposition Method, Modified Differential Transform Method, and Homotopy Analysis Method

New developments in Functional and Fractional Differential Equations and in Lie Symmetry

Delay, difference, functional, fractional, and partial differential equations have many applications in science and engineering. In this Special Issue, 29 experts co-authored 10 papers dealing with these subjects. A summary of the main points of these papers follows:Several oscillation conditions for a first-order linear differential equation with non-monotone delay are established in Oscillation Criteria for First Order Differential Equations with Non-Monotone Delays, whereas a sharp oscillation criterion using the notion of slowly varying functions is established in A Sharp Oscillation Criterion for a Linear Differential Equation with Variable Delay. The approximation of a linear autonomous differential equation with a small delay is considered in Approximation of a Linear Autonomous Differential Equation with Small Delay; the model of infection diseases by Marchuk is studied in Around the Model of Infection Disease: The Cauchy Matrix and Its Properties. Exact solutions to fractional-order Fokker–Planck equations are presented in New Exact Solutions and Conservation Laws to the Fractional-Order Fokker–Planck Equations, and a spectral collocation approach to solving a class of time-fractional stochastic heat equations driven by Brownian motion is constructed in A Collocation Approach for Solving Time-Fractional Stochastic Heat Equation Driven by an Additive Noise. A finite difference approximation method for a space fractional convection-diffusion model with variable coefficients is proposed in Finite Difference Approximation Method for a Space Fractional Convection–Diffusion Equation with Variable Coefficients; existence results for a nonlinear fractional difference equation with delay and impulses are established in On Nonlinear Fractional Difference Equation with Delay and Impulses. A complete Noether symmetry analysis of a generalized coupled Lane–Emden–Klein–Gordon–Fock system with central symmetry is provided in Oscillation Criteria for First Order Differential Equations with Non-Monotone Delays, and new soliton solutions of a fractional Jaulent soliton Miodek system via symmetry analysis are presented in New Soliton Solutions of Fractional Jaulent-Miodek System with Symmetry Analysis