International Journal of Applied Engineering Research ISSN 0973-4562 Volume 14, Number 10 (2019) pp. 2507-2521 © Research India Publications. http://www.ripublication.com Haar Wavelet Based Numerical Method for the Solution of Multidimensional Stochastic Integral Equations S. C. Shiralashetti*, Lata Lamani1 P. G. Department of Studies in Mathematics, Karnatak University, Dharwad-580003, Karnataka, India. Abstract for numerical solution of multi-term fractional differential equations [5] and numerical solution of initial value problems In this paper, we develop an accurate and efficient Haar [6]. Solution of nonlinear oscillator equations, Stiff systems, wavelet based numerical method for the solution of Multi- regular Sturm-Liouville problems, etc. using Haar wavelets dimensional Stochastic Integral equations. Initially, we study were studied by Bujurke et al[14, 15]. In 2012, Maleknejad et the properties of stochastic integral equations and Haar al applied block-pulse functions for the solution of wavelets. Then, Haar wavelets operational matrix of multidimensional stochastic integral equations [13]. In this integration and Haar wavelets stochastic operational matrix of paper, we developed the numerical method for the solution of integration are developed. Convergence and error analysis of stochastic integral equations using Haar wavelets. Consider the Stochastic Haar wavelet method is presented for the the stochastic Ito-volterra integral equation, solution of Multi-dimensional Stochastic Integral equations. Efficiency of the proposed method is justified through the t U(t)=g(t) k s,t U(s)ds illustrative examples. 0 0 n t (1.1) Keywords: Haar wavelets, Multi-dimensional Stochastic k s, t U(s)dB (s), t [0,T) 0 ii Integral equations, stochastic operational matrix of i1 integration. where Ut, gt, k s,t and k s,t,i 1,2,...,n , for AMS Subject Classification: 65T60, 60H05. 0 i s,t [0,T) , are the stochastic processes defined on the same probability space , F, P and U(t) is unknown. Also 1. INTRODUCTION B(t) is a Brownian motion process and Wavelet is newly emerging area in the field of mathematics. t Wavelets are extensively used for signal processing in ki s,tU(s)dBi (s), i 1,2,...,n ares the Ito integrals [3]. 0 communications and physics, and it is one of the best mathematical tools [4]. Integral equations are important tools This paper is presented as follows. In section 2, we study in applied mathematics for describing knowledge models. some basic definitions arising in Stochastic calculus, Since in many cases, the exact solution of integral equations properties of wavelets, Haar wavelets, Haar wavelets does not exist, the numerical approximation of integral operational matrix of Integration and Haar wavelets stochastic equations become necessary, since in many cases the exact operational matrix of integration. In section 3, Proposed solution to these equations does not exist. There are various method of solution for multi-dimensional stochastic integral methods for approximating these equations and different basis equations is given. In section 4, we study the convergence and functions [10] are used. error analysis of the proposed method. In section 5, some examples are presented to show the efficiency of the presented Stochastic integrals are used in modeling various phenomena method. Finally, in section 6, conclusion is drawn. in science, engineering and physics [12]. Many authors have studied the numerical approximation of stochastic integral 2. PROPERTIES OF STOCHASTIC CALCULUS AND equations. Some of them are Platen [1], Oksendal [3], WAVELETS Maleknejad et al [7], [13], Cortes et al [9], Douglas et al [12] and Zhang [8]. In this section, we discuss some basic definitions arising in Stochastic calculus, properties of Wavelets, Haar wavelets, Haar wavelet is one of the most important families of Haar wavelets operational matrix of Integration and Haar Wavelets. It has become an important tool for solving wavelets stochastic operational matrix of integration. Finally, differential equations, integral equations, integro-differential some results which will be used in further sections are equations and much other class of equations. Lepik applied mentioned. Haar wavelet method for solving integral equations [10, 11]. Shiralashetti et al. applied Haar wavelet collocation method * Corresponding author e-mail id: [email protected]; 2507 International Journal of Applied Engineering Research ISSN 0973-4562 Volume 14, Number 10 (2019) pp. 2507-2521 © Research India Publications. http://www.ripublication.com 2.1. Stochastic calculus 2.2. Wavelets Definition 2.1: A standard Brownian motion defined on the Wavelets constitute a family of functions constructed from interval 0, T is a random variable Bt which depends on dilation and translation of a single function called the mother wavelet. When the dilation parameter a and the translation t 0,T and satisfies the following conditions: parameter b varies continuously, we have the following B(0) 0 with probability 1. family of continuous wavelets: 1 For 0 s < t T , the random variable given by t b 2 a,b t = a (2.1) increment Bt Bs is distributed normally with a mean zero and variance t s , equivalently, If we restrict the parameters a and b to discrete values as, Bt B0 t sN 0,1, where N0,1 is a k k random variable distributed normally with mean zero and a = a0 ,b = nb0a0 ,a0 >1,b0 > 0 variance 1. we have the following family of discrete wavelets, The increments Bt Bs and Bv Bu are 1 k independent for 0 s < t < u < v T 2 n,k t = a0 a0 t nb0 Let p 2 . Let us consider a random variable U with where form a wavelet basis for L2 R. In particular, distribution fU , so that n,k when a = 2 and b =1, then (x) forms an p p 0 0 n,k EU = U fU dY < orthonormal basis. Let Lp ,H be the collection of all strongly measurable, p - integrable and H-valued random variables. It is a routine 2.3. Haar wavelets, Haar Wavelets Operational matrix of p Integration and Stochastic operational matrix of to check that L -space is a Banach space with Haar wavelets 1 2.3.1. Haar Wavelets p V = E V p Lp ,H For the family of Haar wavelet the scaling function h0 t is for each V Lp ,H . Here we consider p = 2 i.e, defined as 2 1, t (0,1] L , H . h0 t = (2.2) 2 0, otherwise Definition 2.2.1[2] The sequence U n converge to U in L 2 The Haar Wavelet family for t 0,1 is defined as, if for each n , E|Un | < . Let us assume that 0 s T , let v = v(s,T) be the class of 1, t (, ] n functions that g(t,w):[0,] R , satisfy , hi t = 1, t (, ] (2.3) the function t,w gt,w is G measurable, 0, elsewhere where is Borel algebra. k k 0.5 k 1 where = , = ,and = g is adapted to Gt . m m m T 2 E gt, w dt < . m = 2 j , j = 0,1,..., J , J is the level of resolution and s k = 0,1,...,m1 is the translation parameter. Maximum Definition 2.3. (The Ito-integral [3]) Let g v(s,T) , then level of resolution is J . The index i in equation (2.3) is the Ito-integral of g is defined by calculated using i = m k 1. In case of minimal values, T T m =1, k = 0 then i = 2. The maximal value of i is g(t,w)dB(t)(w) = lim t,wdB(t)(w), J 1 s n s N = 2 . l 0.5 where, is the sequence of elementary functions such that Let us define the collocation points t = , l N T 2 E g n dt 0 a.s, n l = 1,2,..., N , then, the Haar coefficient matrix s 2508 International Journal of Applied Engineering Research ISSN 0973-4562 Volume 14, Number 10 (2019) pp. 2507-2521 © Research India Publications. http://www.ripublication.com Hi,l = h t has dimension N N . For instance, if 2.3.3. Haar Wavelets Stochastic Operational Matrix of i l Integration J = 2 N = 8, we have The Ito-Integral of Haar wavelets can be computed as follows: 1 1 1 1 1 1 1 1 t 1 1 1 1 1 1 1 1 H(s)dB(s) = Ps H(s) (2.6) 0 1 1 1 1 0 0 0 0 where Ps is called Haar Wavelets Stochastic Operational 0 0 0 0 1 1 1 1 H 8,8 = Matrix of integration and is given by, 1 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 k Bt , t (, ] 0 0 0 0 1 1 0 0 m k 1 0 0 0 0 0 0 1 1 (2.7) PS = B t , t (, ] m 0, elsewhere 2.3.2. Haar Wavelets Operational Matrix of Integration Now, we write the operational matrix of integration for Haar wavelets. For instance, for N = 8, using (2.7), we have 1 3 5 7 9 11 13 15 By integrating equation (2.3), we get the operational matrix of B B B B B B B B 16 16 16 16 16 16 16 16 integration as, 1 3 5 7 7 5 3 1 B B B B B B B B 16 16 16 16 16 16 16 16 t 1 3 3 1 H(s)ds PH (t) (2.4) B B B B B0 B0 B0 B0 0 16 16 16 16 1 3 3 1 B0 B0 B0 B0 B B B B 16 16 16 16 where H(t) is the Haar wavelet vector given by P (8,8) = s 1 1 B B B0 B0 B0 B0 B0 B0 [h (t),h (t),h (t),...,h (t)] 16 16 0 1 2 N1 1 1 B0 B0 B B B0 B0 B0 B0 By using equation (2.2) P can be evaluated as, 16 16 1 1 B0 B0 B0 B0 B B B0 B0 k 16 16 t , t (, ] 1 1 B0 B0 B0 B0 B0 B0 B B m 16 16 k 1 P = t, t (, ] (2.5) m where B(0) = 0 using definition (2.1).