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Register a new userThe decomposition method and Maple procedure for finding first integrals of nonlinear PDEs of any order with any number of independent variables
الطريقة التفكيكية والإجراء المابل للعثور على الأحداث الأولى للمعادلات التفاضلية غير الخطية من أي درجة مع أي عدد من المتغيرات المستقلة
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Authors
Yu.N. Kosovtsov
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في هذا البحث، نقترح طريقة جديدة بالنسبة لإيجاد حلول لبعض أنواع المعادلات التفاضلية الغير خطية في شكل مغلق. يستند الأسلوب إلى تقسيم المشغلات غير الخطية في عدد من المشغلات ذات أولويات منخفضة. ويظهر أن عملية التقسيم يمكن أن تكون بواسطة عمليات تكرارية، كل خطوة منها مختصرة إلى حل نظام معادلات التفاضلية الغير خطية لمتغير واحد. بالإضافة إلى ذلك، نجد على هذا الطريق التعبير المباشر للمعادلة الأولى للتفاضلية الغير خطية الأولية للتقسيم الأولي. ملاحظة قيمة أن هذه المعادلة الأولى للتفاضلية الخطية إذا كانت المعادلة الأولية للتفاضلية خطية في أعلى المشتقات. يتم تطبيق الأسلوب المطور في إجراء Maple، والذي يمكن حقًا حل العديد من معادلات التفاضلية مختلف الأولويات مع عدد مختلف من المتغيرات المستقلة. مثال على المعادلات التفاضلية الغير خطية مع حلولها العامة المحسوبة توضح الإمكانية لحل المعادلات التفاضلية الغير خطية بشكل تلقائي.
In present paper we propose seemingly new method for finding solutions of some types of nonlinear PDEs in closed form. The method is based on decomposition of nonlinear operators on sequence of operators of lower orders. It is shown that decomposition process can be done by iterative procedure(s), each step of which is reduced to solution of some auxiliary PDEs system(s) for one dependent variable. Moreover, we find on this way the explicit expression of the first-order PDE(s) for first integral of decomposable initial PDE. Remarkably that this first-order PDE is linear if initial PDE is linear in its highest derivatives. The developed method is implemented in Maple procedure, which can really solve many of different order PDEs with different number of independent variables. Examples of PDEs with calculated their general solutions demonstrate a potential of the method for automatic solving of nonlinear PDEs.
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