A High Order Solution of Three Dimensional TIME Dependent Nonlinear Convective-diffusive Problem Using Modified Variational Iteration Method

In this paper, we have achieved high order solution of a three dimensional nonlinear diffusive-convective problem using modified variational iteration method. The efficiency of this approach has been shown by solving two examples. All computational work has been performed in MATHEMATICA

New Approach for Solving Three Dimensional Space Partial Differential Equation

هذا البحث يعرض طريقة تحويل جديدة لحل معادلات تفاضلية جزئية لإيجاد الحلول الدقيقة المناسبة  في مجال اوسع و يمكن استخدامه لحل مسائل بدون اللجوء الى تقطيع و تردد المجال. يقرن التحويل الجديد مع طريقة الهوموتوبي المضطربة لحل معادلات تفاضلية جزئية من الرتبة الاولى ثلاثية الابعاد ذات شروط ابتدائية و من ثم اثبات تقارب تلك الحلول. تنفيذ الطريقة المقترحة اثبتت فائدتها في ايجاد الحلول المضبوطة.  التطبيق العملي اثبت تأثير الاسلوب و سهولة التنفيذ في ايجاد الحل المضبوط. اخيرا جميع البرامج نفذت باستخدام الماتلاب اصدار 7.12 This paper presents a new transform method to solve partial differential equations, for finding suitable accurate solutions in a wider domain. It can be used to solve the problems without resorting to the frequency domain. The new transform is combined with the homotopy perturbation method in order to solve three dimensional second order partial differential equations with initial condition, and the convergence of the solution to the exact form is proved. The implementation of the suggested method demonstrates the usefulness in finding exact solutions. The practical implications show the effectiveness of approach and it is easily implemented in finding exact solutions.        Finally, all algorithms in this paper are implemented in MATLAB version 7.12

Application of the bifurcation method to the modified Boussinesq equation

In this paper, we investigate the modified Boussinesq equation $$u_{tt}- u_{xx}-\varepsilon u_{xxxx}-3(u^2)_{xx}+3(u^2u_x)_{x}=0.$$ Firstly, we give a property of the solutions of the equation, that is, if $1+u(x, t)$ is a solution, so is $1-u(x, t)$. Secondly, by using the bifurcation method of dynamical systems we obtain some explicit expressions of solutions for the equation, which include kink-shaped solutions, blow-up solutions, periodic blow-up solutions and solitary wave solutions. Some previous results are extended