Numerical solution of first order linear fuzzy differential equations using Leapfrog method

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1 IOSR Journal of Mathematics (IOSR-JM) e-issn: , p-issn: X. Volume, Issue 5 Ver. I (Sep-Oct. 4), PP 7- Numerical solution of first order linear fuzz differential equations using Leapfrog method S. Sekar, K. Prabhavathi (Department of Mathematics, Government Arts College (Autonomous), Salem 636 7, India) (Department of Mathematics, Bannari Amman Institute of Technolog, Sathamangalam 638 4, India) Abstract : This paper presents numerical solutions of first order linear fuzz differential equations using Leapfrog method. The obtained discrete solutions are compared with single-term Haar wavelet series (STHWS) method []. Error graphs and error calculation tables are presented to highlight the efficienc of the Leapfrog method. This method can be implemented in the digital computers and take an time of solutions. Kewords: Fuzz differential equations, Fuzz initial value problems, Haar wavelet series, Leapfrog method, Single-term Haar wavelet series method. I. Introduction S. Abbasband and T. Allahviranloo [] addressed knowledge about dnamical sstems modelled b differential equations is often incomplete or vague. It concerns, for example, parameter values, functional relationships, or initial conditions. The well-known methods for solving analticall or numericall initial value problems can onl be used for finding a selected sstem behavior, e.g., b fixing the unknown parameters to some plausible values. The topics of fuzz differential equations, which attracted a growing interest for some time, in particular, in relation to the fuzz control, have been rapidl developed recent ears. The concept of a fuzz derivative was first introduced b S. L. Chang, L. A. Zadeh in [3]. It was followed up b D. Dubois, H. Prade in [4], who defined and used the extension principle. Other methods have been discussed b M. L. Puri, D. A. Ralescu in [5] and R. Goetschel, W. Voxman in [6]. Fuzz differential equations and initial value problems were regularl treated b O. Kaleva in [7] and [8], S. Seikkala in [9]. A numerical method for solving fuzz differential equations has been introduced b M. Ma, M. Friedman, A. Kandel in [] via the standard Euler method. The structure of this paper is organized as follows. In section, the proposed leapfrog method is explained in detailed. In section 3, some basic results on fuzz numbers and definition of a fuzz derivative, which have been discussed b S. Seikkala in [9], are given. In section 4, we define the problem that is a fuzz initial value problem. Its numerical solution is of the main interest of this work. Solving numericall the fuzz differential equation b the Leapfrog method is discussed in section 5. The proposed algorithm is illustrated b some examples in section 5 and the conclusion is in section 6. II. Leapfrog Method The most familiar and elementar method for approximating solutions of an initial value problem is Euler s Method. Euler s Method approximates the derivative in the form of d f t,, t, R b a finite difference quotient t t h t / h. We shall usuall discretize the independent variable in equal increments: tn tn h, n,,..., t. Henceforth we focus on the scalar case, N =. Rearranging the difference quotient gives us the corresponding approximate values of the dependent variable: n n hf t n, n, n,,..., t To obtain the leapfrog method, we discretize t n as in tn tn h, n,,..., t, but we double the time interval, h, and write the midpoint approximation h t h t h t in the form t h t h t / h and then discretize it as follows: n n hf t n, n, n,,..., t 7 Page

2 Numerical solution of first order linear fuzz differential equations using Leapfrog method The leapfrog method is a linear m = -step method, with a, a, b, b and b. It uses slopes evaluated at odd values of n to advance the values at points at even values of n, and vice versa, reminiscent of the children s game of the same name. For the same reason, there are multiple solutions of the leapfrog method with the same initial value. This situation suggests a potential instabilit present in multistep methods, which must be addressed when we analze them two values, and, are required to initialize solutions of n n hf t n, n, n,,..., t uniquel, but the analtical problem d f t,, t, R onl provides one. Also for this reason, one-step methods are used to initialize multistep methods. III. Preliminaries A parallelogram fuzz number u is defined b four real numbers k < l < m < n, where the base of the parallelogram is the interval [k, n] and its vertices at x = l, x = m. Parallelogram fuzz number will be written as u = (k, l, m, n). The membership function for the parallelogram fuzz number u = (k,, m, n) is defined as the following : x k, k x l l k ux, l x m u () x n, m x n m n we will have : () u > if k > ; () u > if > ; (3) u > if m > & (4) u > if n >. Let us denote R F b the class of all fuzz subsets of R (i.e. u :R [, ]) satisfing the following properties: (i) u RF, u is normal, i.e. x R with u x. (ii) u RF, u is convex fuzz set, i.e. utx t min u x, u, t,, x, R. (iii) u RF, u is upper semi continuous on R. (iv) x R; u x is compact, where A denotes the closure of A. Then R F is called the space of fuzz numbers.obviousl R R F. Here R R F is understood as R = {{x}; x is usual real number}. We define the r-level set, xr; [u]r = {x \ u(x) r}, r ; () Clearl, [u] = { x \ u(x) > } is compact, which is a closed bounded interval and we denote b [u]r= [u(r),u(r)]. It is clear that the following statements are true. u r is a bounded left continuous non decreasing function over [,],.. u r is a bounded right continuous non increasing function over [,], 3. u r u r for all r (,], for more details see [],[3]. Let D: R F R F R+ U, D (u, v) =Sup r[,] max u r v r, u r r numbers, where [u] r = [ u r, u r ],[v] r = [ v r, v r v, be Hausdorff distance between fuzz ]. The following properties are well-known: D(u + w, v + w) = D(u, v), u, v, w RF,, D(k.u, k.v) = k D(u, v), k R, u, v RF, D(u + v, w + e) D(u,w) + D(v,e), u, v, w, e R and (R F, D) is a complete metric space. F 8 Page

3 Numerical solution of first order linear fuzz differential equations using Leapfrog method IV. Fuzz Initial Value Problems Consider a first-order fuzz initial value differential equation is given b t f t, t, t t, T, t, (3) where is a fuzz function of t, f (t, ) is a fuzz function of the crisp variable t and the fuzz variable, is the fuzz derivative of and t is a parallelogram or a parallelogram shaped fuzz number. We denote the fuzz function b,. It means that the r-level set of (t) for t t, Tis t r t; r, t; r, t t ; r, t ; r, r, r we write f t; f t;, f t; and f t; Ft,,, f t; Gt,,. Because of t f t, we have f t; t; r Ft ; t; r, t; r (4) f t; t; r G t; t; r, t; r (5) B using the extension principle, we have the membership function f(t; (t))(s) = Sup{(t)( )\s = f( t, )}, s R (6) so fuzz number f(t; (t)). From this it follows that f t; t r f t, t; r, f t, t; r, r ; (7) where f t t; r min f t, u\ u t r f t t; r max f t, u\ u t, (8), r (9) Definition 4. A function f: R R F is said to be fuzz continuous function, if for an arbitrar fixed t R and, such that t t D f t, f t exists. Throughout this paper we also consider fuzz functions which are continuous in metric D. Then the continuit of f(t, (t); r) guarantees the existence of the definition of f(t, (t); r) for t t, T and r, []. Therefore, the functions G and F can be definite too. V. Numerical Examples Consider a first-order fuzz initial value differential equation is given b In this section, the exact solutions and approximated solutions obtained b Leapfrog method and STHWS method. To show the efficienc of the Leapfrog method, we have considered the following problem taken from [], along with the exact solutions. The discrete solutions obtained b the two methods, Leapfrog method and STHWS method; the absolute errors between them are tabulated and are presented in Table and Table. To distinguish the effect of the errors in accordance with the exact solutions, graphical representations are given for selected values of r and are presented in Fig. to Fig. 6 for the following problem, using three dimensional effects. Example 5. Consider the initial value problem [] t tf t, t,,..r e,.5.r e The exact solution at t =. is given b Y.; r..r e e.5.5,.5.r ee, r Example 5. Consider the fuzz initial value problem [] 9 Page

4 Numerical solution of first order linear fuzz differential equations using Leapfrog method t t, t I,,.75.5r,.5.5r, r. The exact solution is given b Y t; r ; re t, Y t; r ; re t which at t = Y ; r.75.5r e,.5.5r e, r. Example 5.3 Consider the fuzz initial value problem [] t c, t c where c i, for i =, are triangular fuzz numbers. The exact solution is given b Y t; r lr tanw rt, Y t; r l r tanw r t, with l w r c r/ c r, l r c r/ c r, r c r/ c r, w r c r/ c r,,, where c r c, r, c, r and c r c, r, c, r c, r.5.5r, c, r.5.5r, c,r.75.5r, c, r.5.5r, The r-level sets of t are Y t; r c,r sec w rt, Y t; r c rsec w rt,,,, which defines a fuzz number. We have ft, ; r min c. u c u t; r, t; r, c c, r, c, r, c c,r, c, r, f t, ; r maxc. u c u t; r, t; r, c c r, c r, c c r, c r.,, Table : Error Calculations STHWS Error Example 5. Example 5. Example 5.3 r..e-7.e-7 6.E-7 6.E-7.E-7.E-8..E-7.E-7 7.E-7 7.E-7.E-7.E E-7 3.E-7 8.E-7 8.E-7 3.E-7 3.E E-7 4.E-7 9.E-7 9.E-7 4.E-7 4.E E-7 5.E-7.E-6.E-6 5.E-7 5.E E-7 6.E-7.E-6.E-6 6.E-7 6.E E-7 7.E-7.E-6.E-6 7.E-7 7.E E-7 8.E-7.3E-6.3E-6 8.E-7 8.E E-7 9.E-7.4E-6.4E-6 9.E-7 9.E-8.E-6.E-6.5E-6.5E-6.E-6 9.9E-8 Table : Error Calculations Leapfrog Error Example 5. Example 5. Example 5.3 r..e-7.e-7 6.E-7 6.E-7.E-7.E-8..E-7.E-7 7.E-7 7.E-7.E-7.E E-7 3.E-7 8.E-7 8.E-7 3.E-7 3.E-8,,,, Page

5 Numerical solution of first order linear fuzz differential equations using Leapfrog method.4 4.E-7 4.E-7 9.E-7 9.E-7 4.E-7 4.E E-7 5.E-7.E-6.E-6 5.E-7 5.E E-7 6.E-7.E-6.E-6 6.E-7 6.E E-7 7.E-7.E-6.E-6 7.E-7 7.E E-7 8.E-7.3E-6.3E-6 8.E-7 8.E E-7 9.E-7.4E-6.4E-6 9.E-7 9.E-8.E-6.E-6.5E-6.5E-6.E-6 9.9E-8 Fig. Error estimation of Example 5. at Fig. Error estimation of Example 5. at Fig. 3 Error estimation of Example 5. at Fig. 4 Error estimation of Example 5. at Fig. 5 Error estimation of Example 5.3 at Fig. 6 Error estimation of Example 5.3 at VI. Conclusion In this paper, the Leapfrog method has been successfull emploed to obtain the approximate analtical solutions of the first order linear fuzz differential equations. Compare to STHWS method, Leapfrog method gives less error from the Table and Table. Also it is clear that from the Fig. to Fig. 6 the Leapfrog method introduced in Section performs better than STHWS method. Acknowledgements The authors gratefull acknowledge the Dr. A. Murugesan, Assistant Professor, Department of Mathematics, Government Arts College (Autonomous), Cherr Road, Salem , for encouragement and support. The authors also thank Dr. S. Mehar Banu, Assistant Professor, Department of Mathematics, Government Arts College for Women (Autonomous), Salem , Tamil Nadu, India, for her kind help and suggestions. Page

6 Numerical solution of first order linear fuzz differential equations using Leapfrog method References [] S.Sekar and S.Senthilkumar, Single-term Haar wavelet series for fuzz differential equations, International Journal of Mathematics Trends and Technolog, 4(9), 3, 8 88 [] S. Abbasband and T. Allahviranloo, Numerical solutions of fuzz differential equations b Talor method, Journal of Computational Methods in Applied Mathematics.,, 3-4. [3] S. S. L. Chang and L. A. Zadeh, On fuzz mapping and control, IEEE Transactions on Sstems, Man, and Cbernetics,, 97, [4] D. Dubois and H. Prade, Towards fuzz differential calculus.iii. Differentiation, Fuzz Sets and Sstems, 8( 3), 98, [5] M. L. Puri and D. A. Ralescu, Differentials of fuzz functions, Journal of Mathematical Analsis and Applications, 9(), 983, [6] R. Goetschel and Woxman, Elementar fuzz calculus, Fuzz Sets and Sstems, 8, 986, [7] O. Kaleva, Fuzz differential equations, Fuzz Sets and Sstems, 4(3), 987, [8] O. Kaleva, The Cauch problem for fuzz differential equations, Fuzz Sets and Sstems, 35(3), 99, [9] S. Seikkala, On the fuzz initial value problem, Fuzz Sets and Sstems, 4(3), 987, [] M. Ma, M. Friedman and A. kandel, Numerical Solutions of Fuzz Differential Equations, Fuzz Sets and Sstems, 5, 999, Page