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پژوهش های ریاضی، جلد ۴، شماره ۱، صفحات ۹۱-۱۰۴
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| عنوان فارسی |
حل عددی معادلات انتگرال ولترا تصادفی منفرد ضعیف با استفاده از روش ماتریس عملیاتی چندجمله ای های اویلر |
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| چکیده فارسی مقاله |
در این مقاله به حل عددی معادلات انتگرال ولترا تصادفی منفرد ضعیف با استفاده از ماتریسهای عملیاتی چندجملهایهای اویلر میپردازیم. برای این منظور ابتدا ماتریس عملیاتی و ماتریس عملیاتی تصادفی چندجملهایهای اویلر را به‌دست میآوریم. تمامی توابع موجود در معادلۀ انتگرال ولترا تصادفی را با استفاده از سری چندجملهایهای اویلر تقریب زده و سپس از ماتریسهای عملیاتی چندجملهایهای اویلر استفاده میکنیم. با این کار حل معادلۀ انتگرال ولترا تصادفی منفرد ضعیف به حل یک دستگاه معادلات جبری تبدیل میشود، که با روشی مناسب میتوان آن را حل کرد. سپس کران بالای خطای روش مطرح شده برای حل معادلات انتگرال ولترا تصادفی منفرد ضعیف را اثبات میکنیم. هم‌چنین برای بررسی دقت و کارایی روش چند مثال عددی ارائه میشود. نتایج ارائه شده با استفاده از اجرا کردن برنامه‌ای نوشته شده در نرمافزار متلب به‌دست آمده است.
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| کلیدواژه‌های فارسی مقاله |
معادلات انتگرال ولترا تصادفی، معادلات انتگرال ولترا منفرد ضعیف، ماتریسهای عملیاتی، چندجملهایهای اویلر |
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| عنوان انگلیسی |
Numerical Solution of Weakly Singular Ito-Volterra Integral Equations via Operational Matrix Method based on Euler Polynomials |
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| چکیده انگلیسی مقاله |
Introduction
Many problems which appear in different sciences such as physics, engineering, biology, applied mathematics and different branches can be modeled by using deterministic integral equations. Weakly singular integral equation is one of the principle type of integral equations which was introduced by Abel for the first time. These problems are often dependent on a noise source which are neglected due to poor computational tools. So, it is satisfactory to use stochastic models to describe the behaviour of them. Thus, researchers added uncertainty term in the deterministic models and this leads to the stochastic models such as stochastic partial differential equations or stochastic integral equations. Since 1960, by increasing computational power, some random factors are inserted to deterministic integral equations and are created various kinds of stochastic integral equations such as Ito-Volterra integral equations, Ito-Fredholm integral equations, or weakly singular Ito-Volterra integral equations. In more cases, the analytical solution of these equations do not exist or finding their analytic solution is very difficult. Thus, presenting an accurate numerical method is an essential requirement in numerical analysis. Numerical solution of stochastic integral equations because the randomness has its own difficulties. In recent years, some different basis functions have been used to estimate the solution of stochastic integral equations. In this paper, we develop operational matrix method based on Euler polynomials to solve weakly singular Ito-Volterra integral equations. Euler polynomials have received considerable attention in dealing with various problems and equations.
Material and methods
In this scheme, we first calculate operational matrix of integration and stochastic operational matrix of integration based on Euler polynomials and then by using these matrices, weakly singular Ito-Volterra integral equation is transformed to a system of algebraic equations which can be solved via a suitable numerical method.
Results and discussion
We solve some test examples by using present technique to demonstrate the efficiency, high accuracy and the simplicity of the present method, then compare the proposed method with block-pulse method. The reported results demonstrate that there is a good agreement between approximate solution and exact solution. Also, the numerical results reported in the tables indicate that the accuracy improve by increasing the N. Therefore, to get more accurate results, using the larger N is recommended. Note that, obtained results confirm that proposed method enables us to find some more reasonable approximate solutions than block-pulse method.
Conclusion
The following conclusions were drawn from this research. - Coefficients of the approximate function via Euler polynomials are found very easily and therefore many calculations are reduced.
- Euler polynomials are simple basis functions, so proposed method is easy to implement and it is a powerful mathematical tool to obtain the numerical solution of various kind of problems with little additional works.
- The main characteristic of this method is that it reduces tht considered problem to a system of algebraic equations which can be easily solved by using direct method or iterative method../files/site1/files/41/7Extended_Abstract.pdf
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| کلیدواژه‌های انگلیسی مقاله |
Stochastic integral equations, Volterra integral equations, Singular integral equations, Operational matrix method, Euler polynomials |
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| نویسندگان مقاله |
فرشید میرزایی | Farshid Mirzaee
Faculty of Mathematical Sciences and Statistics, Malayer University, Malayer, Iran
دانشگاه ملایر، دانشکدۀ علوم ریاضی و آمار، گروه ریاضی
نسرین صمدیار | Nasrin Samadyar;
Faculty of Mathematical Sciences and Statistics, Malayer University, Malayer, Iran
دانشگاه ملایر، دانشکدۀ علوم ریاضی و آمار، گروه ریاضی
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| نشانی اینترنتی |
http://mmr.khu.ac.ir/browse.php?a_code=A-10-238-25&slc_lang=fa&sid=1 |
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| زبان مقاله منتشر شده |
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| نوع مقاله منتشر شده |
مقاله استخراج شده از طرح پژوهشی |
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