تعميم طرق جديدة لتحويلات لابلاس لحل المعادلات التفاضلية الجزئية الخطية من ا لرتبة الثانية بمعاملات ثابتة مع او بدون شروط == Generalized New Methods Of Laplace Transformations To Solve Linear Partial Differential Equations Of Second Order With Constant Coefficients With Or Without Conditions
Author name:
نور علي حسين
Supervisor name:
علي حسن محمد
General topic:
Mathematics
Specific topic:
Mathematics
Degree:
Master
University:
University of Kufa - College Of Education For Girls - Department Of Mathematics
Language:
English
University location:
Najaf
First pages:
27T390 - p.pdf
Abstract:
في عملنا نقترح طريقتين لحل المعادلات التفاضلية الجزئية الخطية باستخدام تحويلات لابلاس : - 1 - الطريقة الاولى بشروط ابتدائية ، هذه الطريقة تتلخص كما ياتي : - نفرض اننا نمتلك معادلة تفاضلية جزئية من الرتبة صيغتها العامة تكتب كما ياتي : - حيث ثوابت ، دالة مستمرة تحويل لابلاس لها يمكن ايجاده ، وكذلك بعض الشروط الابتدائية معروفة .لايجاد المعادلة التفاضلية (1) ناخذ تحويل لابلاس لطرفيها وبعد تعويض الشروط ، نحصل على معادلة تفاضلية اعتيادية من الرتبة . صيغتها العامة بحيث ، ثوابت ودالة الى ، .بعد حل المعادلة (2) نحصل على حيث دالة الى ودالة الى و.الان ،بما ان ، عليهمعادلة(4)، تمثل حل المعادلة التفاضلية الجزئية (1) والذي ممكن ان يكتب بالصيغة | In our work we suggesting two methods for solving L.P.D.Es by using Laplace transformations : - 1 - The first method with initial conditions , this method is summarized as follow : Suppose we have a P.D.E of order , which general form can be written as follows : - ...( 1) Where are constants, is a continuous function whose Laplace transformation can be determined, and also some initial conditions are defined.To find a solution of the D.E (1) we can take Laplace transformation to both sides of it and after substituting the conditions, we obtain an O.D.E of order .The general form of it is …(2) s.t. , are constants and are function of and .After solving eq.(2), we get …(3)Where is function of and is function of and .Now, since , thus …(4)Eq. (4), represents the solution of the P.D.E (1), which can be written by the form : …(5)Such that are functions of and that are functions of , whose numbers equal to the degree of .2 - The second method without any conditions, to explain this method, Suppose we have a L.P.D.E of second order with constant coefficients which has the general form …(6)which doesn’t obey any initial and boundary conditions and the Laplace transformation of is known.To solve eq.(6) we take L.T to both sides we get : …(7) Eq.(7) is O.D.E of the second order with constant coefficientsWhere is a polynomial of represents the denominator of L.T of the function , and is the function of and .To solve the O.D.E(7), we substitute any constant assuming instead of ,thus we have where is the function of and and resultant from solution of D.E (7), is polynomial of .since , thus we have …(8), number of depends on the degree of .Where are functions of , and are functions of , may contain one or more of the functions and .To find the forms of the functions , we take derivatives of eq.(8) w.r.t twice and, also w.r.t twice and w.r.t and to get .We substitute and in P.D.E (6) and by equality coefficients of both sides we get the values of , and by this way we obtain the solution of the P.D.E (6).
