Comparison between the Heun's and Haar Wavelet Methods for solution Differential-Algebraic Equations (DAEs)

صخلملا ايددع دحاولا ليلدلا تاذ ةيربجلا ةيلضافتلا تلاداعملا نم ماظن لح مت ثحبلا اذه يف ةقيرط مادختساب Heun تاجومل لماوعلا تافوفصم ةقيرط كلذكو Haar ةنراقم دعبو ةريصقلا ةجيومل لماوعلا تافوفصم ةقيرط نأ نيبت طوبضملا لحلا عم نيتقيرطلا جئاتن Haar ةءافك تاذ ةيلاع ةقيرط نم طوبضملا لحلا ىلإ برقا ةيددعلا اهجئاتنو Heun ، ةقيرطلا هذهل لحلا ةقد ناو ةفوفصملا ةعس وأ ةكبشلا طاقن ددع تدادزا املك صقانتي أطخلاو دادزت (m) .


Introduction:
In this paper we consider implicit differential equations    is singular are referred to as differential algebraic equations (DAEs) [6].
In this paper we take the special case of eq.(1) which is a semiexplicit DAE The index is one if 2 y g   is non singular, because then on differentiation of (2b) yields 2 y in principle. [7].
The index is 2 because to differentiation of ) t ( q where needed [1]. Let equation (1) is DAEs, the index along a solution y(t) is the minimum number of differentiations of the system which required to solve for y uniquely in terms of y and t (i.e. to define an ODE for y ). Thus, the index is defined in terms of the over determined system to be the smallest integer p so that y in (3) can be solved for in terms of y and t .
Haar wavelets have become an increasingly popular tool in the computational sciences. They have had numerous applications in a wide rang of areas such as signal analysis, data compression and many others [8]. Wu and Chen (2003) [8] studied the numerical solution for partial differential equations of first order via operational matrices , they used the Haar wavelets in the solution with constant initial and boundary conditions. Wu and Chen (2004) [9] studied the numerical solution for fractional calculus and the fractional differential equation by using the operational matrices of orthogonal functions. The fractional derivatives of the four typical functions and two classical fractional differential equations solved by the new method and they are compared the results with the exact solutions, they are found the solutions by this method is simple and computer oriented. Lepik and Tamme (2007) [3] derived the solution of nonlinear Fredholm integral equations via the Haar wavelet method, they are find that the main benefits of the Haar wavelet method are sparse representation, fast transformation, and possibility of implementation of fast algorithms especially if matrix representation is used.
Lepik Uio (2007) [4] studied the application of the Haar wavelet transform to solve integral and differential equations, he demonstrated that the Haar wavelet method is a powerful tool for solving different types of integral equations and partial differential equations. The method with far less degrees of freedom and with smaller CPU time provides better solutions then classical ones.
Numerical approaches for the solution of DAEs can be divided roughly into two classes: direct discretizations of the given system and methods which involve a reformulation (i.e. index reduction), combined with a discretization [1].
In this paper, we will study the numerical solution for Differential-Algebraic equations (DAEs) by using Heun's method and Haar wavelets method and we will compare the results of these methods with the exact solution.

Heun's Method:
We will use the Heun's Method to solve Eq.(2a) and (2b) [ where h is step size and h t t k 1 k + = + . then we illustrate this method in an example in numerical results

Review of the operational matrices and Haar wavelets:
The main characteristic of the operational method is to convert a differential equation into an algebraic one, and the core is the operational matrix for integration. The integral property of the basic orthonormal matrix, ( ) t  . we write the following approximation: are the discrete representation of the basis functions which are orthogonal on the interval [0,1) and  Q is the operational matrix for integration of ( ) t  [8,9].
The operational matrix of an orthogonal matrix ( ) t  ,  Q can be expressed by: where   B Q is the operational matrix of the block pulse function: , then the equation (7) can be rewritten as [8,9]: The Haar functions are an orthogonal family of switched rectangular waveforms where amplitudes can differ from one function to another. They are defined in the interval [0,1] by [8,9]: and  is a positive integer. J and k represent the integer decomposition of the index i , i.e.
can be expanded into Haar series by: Usually the series expansion of equation (11 ) contains infinite terms for a general smooth function y(t). However, if y(t) is approximated as piecewise constant during each subinterval, equation (11) will be terminated at finite terms, i. e. : The equation (11) can be written into the discrete form by: is the discrete form of the continuous function y(t), and m is the dimension and usually For deriving the operational matrix of Haar wavelets, we let     H =  in the equation (9), and obtain: For example, the operational matrix of the Haar wavelet in the case of m=4 is given by:

Haar wavelet method:
We will use the operational matrices of the Haar wavelets to solve the differential-algebraic equations (1) numerically. By using the equation (6), the integration of equation (12) with respect to variable t yields [8]: Further the double integration of y(t) with respect to variable (t) and by using equation (6), we get: where A1 and A2 are constants. by integrating equation (2a) with respect to (t), we get: we transform the equations (18a) and (18b) into the matrices forms by using equation (12), we get: such that by using the initial condition (17), we get: To find the coefficient matrix   T 1 C and   T 2 C which have m of the elements respectively, we solve the system (21a) and (21b) which given linear system of the equations such that the variables number are m 2  and we will can be solved this linear system by Gauss-Jordan method, after this we find the vectors solution   T 1 Y and   T 2 Y by using the equation (12) that is:

Conclusions:
The main goal of this paper was to demonstrate that the Haar wavelet method can be used to solve differential-algebraic equations (DAEs). The method is give results better then the classical (Heun's) method with small computation costs, As shown in table (1) and (2) and figure (1), when m=8, (m is size of matrices or mesh points).
When we increasing the values of (m) that obtained is more accuracy, i.e. when m=16, the results that obtained with Haar wavelet method it's show that in table (3) and (4) and figure (2) is more accurate and near to exact solution and the error is decrease as (m) is large.
The numerical solutions of these equations had been found using MATLAB which has the ability to approaches to the solution in high speed and accuracy and in less possible time.