The legendre wavelets operational matrix of integration pdf

Remark 2. On the other hand, when p t is known, a priori, there exist a unique solution for the problem 2. In the problem 2. In this case, we say that the pair of functions u, p t provides a solution to the inverse problem 2. Integrating equation 2. The Tikhonov regularized solution see Refs. M 3 Therefore, the Haar basis method will be convergent, i. Legendre Wavelet Method Wavelets constitute a family of functions constructed from dilation and translation of a single function called the mother wavelet.

Integrating from equation 3. Substituting equation 3. Numerical Results and Discussion In this section, we are going to study numerically the inverse problem 2. The main aim here is to show the applicability of the present method, described in Secs. As expected, the inverse problems are ill-posed and therefore it is necessary to investigate the stability of the present method by giving a test problem. Table 1. The comparison between the exact results and 0th, 1st order Tikhonov of problem 2.

Table 2. Remark 4. Table 3. In this example we solve problem 2. Table 5. The following results are obtained. References 1. Abtahi, R. Pourgholi and A. Beck, B. Blackwell and C. Cabeza, J. Garcia and A. Rodriguez, A sequential algorithm of inverse heat conduction problems using singular value decomposition, Int.

Chen and C. Elden, A Note on the computation of the generalized cross-validation function for ill-conditioned least squares problems, BIT 24 — Golub, M. Heath and G. Wahba, Generalized cross-validation as a method for choosing a good ridge parameter, Technometrics 21 — Haar, Zur theorie der orthogonalen Funktionsysteme, Math. Hariharan, K. Kannan and K. Hsiao and W. Huang and Y. Huang, C. Yeha and H. Kalpana and S. Balachandar, Haar wavelet method for the analysis of transistor circuits, Int.

AEU 61 — Lawson and R. Martin, L. Elliott, P. Heggs, D. Ingham, D. Lesnic and X. Sound Vib. Molhem and R. Pourgholi, A numerical algorithm for solving a one-dimensional inverse heat conduction problem, J.

Section 5 deals with function approximation by Legendre wavelets and block-pulse functions. An operational matrix of fractional order integration for Legendre wavelets is obtained.

In Section 6 we apply the Legendre wavelet method to initial and boundary value problems. To demonstrate the simplicity and validity of the numerical scheme, the numerical solutions by Legendre wavelets are compared with the solutions obtained by some other numerical techniques and with exact solutions as well.

Frac- tional calculus is years old topic. The Caputo frac- tional derivative allows the utilization of initial and boundary conditions involving integer order derivatives, which have clear physically interpretations.

Therefor, in this work we shall use the Caputo fractional derivative Da proposed by Caputo [22] in his work on the theory of viscoelasticity. Existence and uniqueness of solutions Lemma 3.

R is continuous. Then y 2 C[0, 1] is a solution of the boundary value problem 2. The space B is a Banach space [25]. Theorem 3. R be continuous and there exists a function l:[0, 1]? Then, the boundary value problem 2. For s 6 t, from 3. For y 2 W, using 3. Using 3. W: Finally, it remains to show that A is completely continuous. By the Arzela—Ascoli theorem, A : W! W is completely contin- uous.

R be continuous. From 3. Therefore, by the contraction mapping Theorem, the boundary value problem 2. Assume the hypothesis of Theorem 3. Then the initial value problem 2. Under the hypothesis of Theorem 3. The proofs of the Theorems 3. Legendre wavelets Wavelets constitute a family of functions constructed from dilation and translation of single function called the mother wavelet w t.

The functions bi are disjoint and orthogonal. This phenomena makes calculations fast. Numerical examples In order to show the effectiveness of Legendre wavelet method for solving fractional order differential equations, we pres- ent some numerical examples.

Example 6. The exact solution of the Eqs. The corresponding integral representation for Eqs. Numerical solutions for 1 6 a 6 2. Numerical results are obtained for different values of k and a. It is evident from the Fig. Consider the following initial value problem [34]. The corresponding integral equation for fractional order differential Eq.

Numerical results for different values of a and b are shown in Figs. From Figs. The equation has been thoroughly dis- cussed in [29,30,5,31]. The exact solution refers to the closed form series solution given in [36]. Clearly, the approximations obtained by the Legendre wavelet method are in agreement with those obtained with above mentioned numerical methods. In Example 6. Computer simulations are carried out for t 2 [0, 1] and the maximum absolute errors by Haar wavelet and the Legendre wavelet are shown in Table 3.

But the price here we have to pay is that the calculations with the Legendre wavelet are complicated, which increases the computational complexity and storage requirements. Table 2 Numerical results with comparison to Ref. The properties of fractional derivatives and integrals allow us to reduce the differential Eq. Conclusion In this work a Legendre wavelet operational matrix of fractional order integration is obtained and is used to solve frac- tional differential equations numerically.

The achieved results are compared with exact solutions and with the solutions ob- tained by some other numerical methods including Haar wavelet method. The method gives almost the same results as the Haar wavelet method. It is worth mentioning that obtained results agree well with exact solutions even for small values of k and M.

The method is very convenient for solving linear initial value problems as well as boundary value problems, science the boundary conditions are taken care of automatically. A disadvantage of this approach is the large computational complexity for nonlinear problems. Acknowledgements We are very grateful to the reviewer for his useful comments that led to improvement of our manuscript. References [1] Hilfer R, editor. Applications of Fractional Calculus in Physics. J Fluids Eng ; Fractional differential equations in electrochemistry.

Adv Eng Soft ;— Fractional Differential Equations. San Diego: Academic Press; A predictor corrector approach for the numerical solution of fractional differential equation. Nonlin Dyn ;— Homotopy analysis method for fractional IVPs. Commun Nonlin Sci Num Simul ;— Solving fractional integral equations by the Haar wavelet method. Appl Math Comp ;— Haar wavelet method for solving lumped and distributed-parameter systems.

Kronecker operational matrices for fractional calculus and some applications. A new operational matrix for solving fractional-order differential equations. Comp Math Appl ;— Haar wavelet operational matrix of fractional order integration and its applications in solving the fractional order differential equations.