Mathematical Researches
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بررسی وجود و یگانگی جواب نوعی معادلۀ دیفرانسیل با مشتقات جزئی غیرخطی مختلط در فضاهای توابع پیوسته هلدر و سوبولف
The Existence and Uniqueness of the Solution of a Nonlinear Partial Differential Equation in the Eontinuous Functions of Holder and Sobolev Spaces
جبر
alg
مقاله مستقل
Original Manuscript
<span style="font-family:B Nazanin;"><span style="font-size:10.0pt;">در این مقاله به بررسی وجود و یگانگی جواب نوعی معادله دیفرانسیل با مشتقات جزئی غیرخطی در فضای مختلط بهصورت</span></span><span dir="LTR"><span style="font-size:10.0pt;"> </span></span><span style="font-family:B Nazanin;"><span style="font-size:10.0pt;"></span></span><br>
<span style="font-family:Calibri,sans-serif;"><span style="font-size:11.0pt;"><img alt="" chromakey="white" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image001.png" > </span></span><span style="font-size:9.0pt;"></span><br>
<span style="font-family:B Nazanin;"><span style="font-size:10.0pt;">که در آن</span></span><span style="font-family:B Nazanin;"><span style="font-size:10.0pt;"> ­­</span></span><br>
<span style="font-family:Calibri,sans-serif;"><span style="font-size:11.0pt;"><img alt="" chromakey="white" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image002.png" > </span></span><span dir="RTL"><span style="font-family:B Nazanin;"><span style="font-size:10.0pt;"></span></span></span><br>
<span style="font-family:B Nazanin;"><span style="font-size:10.0pt;">می­پردازیم. در ابتدا با تبدیل معادله </span></span><span style="font-family:B Nazanin;"><span style="font-size:10.0pt;">دیفرانسیل با مشتقات جزئی غیرخطی</span></span><span style="font-family:B Nazanin;"><span style="font-size:10.0pt;"> در فضای مختلط به یک معادله دیفرانسیل خطی از مرتبۀ اول، وجود جواب را نشان می­دهیم. سپس با به­کارگیری عملگرهای انتگرالی تکین ضعیف و قوی و ویژگی تابع هولومورفی، همارزی جواب معادله دیفرانسیل با مشتقات جزئی غیرخطی را با یک دستگاه معادلات انتگرالی تکین نشان می­دهیم. هم­چنین، جواب معادله دیفرانسیل با مشتقات جزئی غیرخطی مختلط را در فضای توابع پیوسته هلدر یک مرتبه مشتق­پذیر و </span></span><span style="font-family:B Nazanin;"><span style="font-size:10.0pt;">فضای سوبولف بهترتیب روی </span></span><span style="font-family:B Nazanin;"><span style="font-size:10.0pt;">ناحیهای</span></span><span style="font-family:B Nazanin;"><span style="font-size:10.0pt;"> کراندار و با مساحت متناهی </span></span><span style="font-family:B Nazanin;"><span style="font-size:10.0pt;">بررسی می­کنیم. با بیان شرط لیپ شیتس بهصورت جداگانه در هر دو فضای توابع پیوسته هلدر یک مرتبۀ مشتق­پذیر و فضای سوبولف و </span></span><span style="font-family:B Nazanin;"><span style="font-size:10.0pt;">با تکیه بر ویژگی تابع انقباض و قضیه نقطه ثابت باناخ، یگانگی جواب معادله دیفرانسیل با مشتقات جزئی غیرخطی مختلط مورد نظر در این مقاله اثبات می­شود.<a href="./files/site1/files/62/4JoveiniAkbari.pdf">./files/site1/files/62/4JoveiniAkbari.pdf</a></span></span>
<strong>Introduction</strong><br>
Complex analysis is a comparatively active branch in mathematics which has grown significantly. A deep look at the implications of continuity, derivative and integral in complex analysis and their relation with partial differential equations determines the importance of establishing the relation between complex analysis and the theory of partial differential equations.<br>
There are three aspects of the aims of complex analysis:<br>
First it is possible to interpret the peculiarities of holomorphic functions as properties of solutions of specials systems of partial differential equations. Secondly, complex analysis becomes applicable to general classes of differential equations, not only to special ones. And thirdly, the new general complex analysis is able to construct solutions and to describe the properties of given solutions with the help of solutions of corresponding problems for holomorphic functions. The third aspect is significant, since in the case of nonlinear equations it essentially means their reduction to linear problems. In addition this third aspect shows that the general complex analysis is able to make use of results of classical function theory and of such results which originally have not been connected with partial differential equations.<br>
One of the main goals of complex analysis is its systematic application in the branch of the theory of differential equations. By using the contraction function, the Gauss-Oatrogradski integral formula and the Banach fixed point theorem of the complex space, the classical and interesting method of existence and uniqueness of the solution of differential equations with nonlinear partial derivatives are smooth, which shows the importance of complex analysis in the branch of the theory of differential equations. In recent years, complex analysis and their relationship with partial differential equations have attracted a number of researchers.<br>
In this paper, we determine the existence and uniqueness of a solution of a nonlinear partial differential equation in the form<br>
<img alt="" height="19" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image002.gif" width="116" ><br>
where<br>
<img alt="" height="38" src="file:///C:Users1AppDataLocalTempmsohtmlclip11clip_image004.gif" width="539" ><br>
<strong>Material and methods</strong><br>
Firstly, we transform the nonlinear partial differential equation in a complex space to a linear differential equation of the first order and show the existence of the solution. Then, we consider the equivalence of the solution of the nonlinear partial differential equation with a singular integral equation system by using weak and strong singular integral operators and the property of the holomorphic function. Also, we consider the solution of the nonlinear partial differential equation on a bounded domain with finite area in the Holder space and Sobolev space.<br>
<strong>Results and discussion</strong><br>
We obtain the uniqueness of the solution of the nonlinear partial differential equation by applying the Lipshitz condition in the Holder space and Sobolev space based on the contraction function and the Banach fixed point theorem.<br>
<strong>Conclusion</strong><br>
In this paper, we discuss on the existence and uniqueness of the solution of the nonlinear partial differential equation in the continuous functions of Holder and Sobolev spaces in general case in complex space. We deduce the proposed method can be extended in other spaces.<a href="./files/site1/files/62/4Abstract(1).pdf">./files/site1/files/62/4Abstract(1).pdf</a><br>
معادلۀ دیفرانسیل جزئی غیرخطی, توابع پیوسته هلدر, فضای سوبولف, قضیه نقطه ثابت باناخ.
Nonlinear partial differential equation, Continuous functions of Holder, Sobolev space, Banach fixed point theorem.
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http://mmr.khu.ac.ir/browse.php?a_code=A-10-505-1&slc_lang=fa&sid=1
Mozhgan
Akbari
مژگان
اکبری
m_akbari@guilan.ac.it
10031947532846003739
10031947532846003739
Yes
University of Guilan
دانشگاه گیلان، دانشکدۀ علوم ریاضی، گروه ریاضی محض
Fateme
Joveini
فاطمه
جوینی
f_joveini@phd.guilan.ac.ir
10031947532846003740
10031947532846003740
No
University of Guilan
دانشگاه گیلان، دانشکدۀ علوم ریاضی، گروه ریاضی محض