Al-Tememe Convolution

Transcription

1 Al-Tememe Convolution Ali Hassan Mohammed University of Kufa, Faculty of Education for Girls, Department of Mat hematics, Iraq AlaaSallhHadi University of Kufa, Faculty of Education for Girls, Department of Mathematics, Iraq Hassan NademRasoul University of Kufa. Faculty of Computer Science and Math. Department of Mathematics, Iraq Abstract Our aim in this paper is to find the convolution of Al-Tememe transformation. And how we can use it us to solve some linear ordinary differential equations (LODE). 1. Introduction We will use the new idea in to find the convolution of Al-Tememe transformation which we will define it in a new method and so it will give us the ability to find for some functions more easily without using the partition method. As well as the Al-Tememe convolution is interest in solving some linear ordinary differential equations (LODE) by simpler way without using extended partition method. 2. Preliminaries Definition 1: Let is defind function at period then the integral transformation for whose it s symbol is defined as: Where a fixed function of two variables is called the kernel of the transformation, and are real numbers or, such that the integral above converges. Definition 2: The Al-Tememe transformation for the function is defined by the following integral: Such that this integral is convergent, is positive constant. Property 1: This transformation is characterized by the linear property, that is, Where are constants, the functions are defined when The Al-Tememe transform of some fundamental functions are given in table (1) 310 P a g e e d i t o g p c p u b l i s h i n g. c o m

2 From the Al-Tememe definition and the above table, we get: Theorem1: Table (1). If and is constant, then.see Definition 3: Let be a function where and is said to be an inverse for the Al-Tememe transformation and written as, where returns the transformation to the original function. Property 2: If,, and are constants then, Definition 4: The equation, Theorem 2: Where are constants and is a function of, is called Euler s equation. If the function is defined for and its derivatives are exist then: Definition 5: A function is piecewise continuous on an interval if the interval can be partitioned by a finite number of points such that: 311 P a g e e d i t o g p c p u b l i s h i n g. c o m

3 1. is continuous on each subinterval (, ), for 2. The function has jump discontinuity at, thus Note: A function is piecewise continuous on if it is piecewise continuous in for all Convolution of the Laplace Transform [4]: The convolution of two functions, and, defined for, plays an important role in a number of different physical applications. The convolution is given by the integral: (Convolution Theorem)[5]: If and are piecewise continuous functions on and of exponential order, then Definition 6 : Al-Tememe Convolution : Al-Tememe convolution of two functions, and, is defined for by: ; and are piecewise continuous on. Theorem 3: Let and be two functions of. The Al-Tememe convolution of and is also a function of denoted by and is geven by the relation: Proof: Let and noting that is fixed in the interior integral. If for for 312 P a g e e d i t o g p c p u b l i s h i n g. c o m

4 Note: If and then: Other basic properties of the convolution are as follows: 1), the convolution is commutative 2), c constant; 3) (associative property); 4) (distributive property) Prove (1): ; and are piecewise continuous on. Now, let If So, this convolution is commutative. Prove (2): By the same method we can prove Prove (3): So, this convolution is associative. Prove (4) 313 P a g e e d i t o g p c p u b l i s h i n g. c o m

5 Example 1: To find Firstly, we use the usual method: Now, we will use the convolution method and we get : Example 2 : To find We note that Now, we will use the convolution method and we get : 314 P a g e e d i t o g p c p u b l i s h i n g. c o m

6 Example 3: To find the solution of the differential equation : Take to both sides Take to both sides Now we will get the solution by the convolution method: Now to solve the equation (1) old-fashioned method: 315 P a g e e d i t o g p c p u b l i s h i n g. c o m

7 Example 4: To find the solution of the differential equation: Take to both sides Take to both sides Example 5: To find the solution of the differential equation: Take to both sides 316 P a g e e d i t o g p c p u b l i s h i n g. c o m

8 Take to both sides References: [1] Gabriel Nagy, Ordinary Differential Equations Mathematics Department, Michigan State University, East Lansing, MI, October 14, [2] Mohammed, A.H., Athera Nema Kathem, Solving Euler s Equation by Using New Transformation, Karbala university magazine for completely sciences,volume (6), number (4. (2008). [3] PhD, Andr as Domokos, Differential Equations Theory and Applications, California State University, Sacramento, Spring, [4] Urs Graf, Applied Laplace Transforms and z-transforms for Scientists and Engineers, [5] William F. Trench, Elementary Differential Equations Trinity University, P a g e e d i t o g p c p u b l i s h i n g. c o m