Legendre wavelet operational matrix method for solving fractional differential equations in some special conditions

Abstract

This paper proposes a new technique which rests upon Legendre wavelets for solving linear and non-linear forms of fractional order initial and boundary value problems. In some particular circumstances, a new operational matrix of fractional derivative is generated by utilizing some significant properties of wavelets and orthogonal polynomials. We approached the solution in a finite series with respect to Legendre wavelets and then by using these operational matrices, we reduced the fractional differential equations into a system of algebraic equations. Finally, the introduced technique is tested on several illustrative examples. The obtained results demonstrate that this technique is a very impressive and applicable mathematical tool for solving fractional differential equations.

Solving Linear and Nonlinear Fractional Differential Equations Using Bees Algorithm

“…Numerical methods are a vital tool for finding the approximated solutions to FODEs [2]. the mother wavelets, for example, Haar wavelet, Legendre wavelet, Chebyshev wavelet, Jacobi wavelet and Bernoulli wavelet [3][4][5][6][7].…”

“…Consider the following linear FODE[15]: (𝐷 2 − 2𝐷 + 𝐷 0.5 + 1)𝑦(𝓉) = 𝑓(𝓉), where 𝑓(𝓉) = 𝓉 3 + 6𝓉 − 6𝓉 2 + with initial conditions (I.Cs) 𝑦(0) = 𝑦 ′ (0) = 0 and the exact solution 𝑦 = 𝓉3 .Let 𝑛𝑉𝑟 = 5 and 𝑦(𝓉) ≈ 𝑌(𝓉) = ∑ 𝛼 𝑖 𝐶 𝑖 (𝓉) 𝑛𝑉𝑟 𝑖=1, then the error function is obtained as follows: 𝐸𝑟𝑟𝑜𝑟(𝓉) = |(𝐷 2 − 2𝐷 + 𝐷 0.5 + 1)𝑌(𝓉) − 𝑓(𝓉)|, and the fitness function is calculated as follows:By applying the Bees algorithm, we get the following numerical solution:𝑌(𝓉) = 0.753423 𝐶 1 (𝓉) + 0.2501649 𝐶 3 (𝓉) + 𝑂(𝓉4 ) Example Suppose the linear FODE[16]:(𝐷 1.5 − 𝓉1.5 )𝑦(𝓉) = 𝑔(𝓉),where 𝑔(𝓉) = 4√𝓉 𝜋 − 𝓉 3.5 , with boundary conditions 𝑦(0) = 0 and 𝑦 (1) = 1 . The exact solution is 𝑦 = 𝓉 2 .…”

“…0)| + |𝑌′(0)| + |𝑌(1) − 1| + |𝑌′′′(0) − 6| By utilize the Bees algorithm, have 𝑌(𝓉) = −0.002024 + 0.75447 𝐶 1 (𝓉) − 0.00122 𝐶 2 (𝓉) + 0.2503 𝐶 3 (𝓉) + 𝑂(𝓉 4 ) Consider the nonlinear FODE [19]:(𝐷 3 +𝐷 2.5 )𝑦(𝓉) + 𝑦 2 (𝓉) = 𝓉4 . with the boundary conditions 𝑦(0) = 0, y ′′ (0) = 2, 𝑦(1) = 1 and exact solution 𝑦 = 𝓉2 .Let 𝑛𝑉𝑟 = 4 ,then we have: 𝐸𝑟𝑟𝑜𝑟(𝓉) = |(𝐷 3 + 𝐷2.5 )𝑌(𝓉) + 𝑌 2 (𝓉) − 𝓉4 |, so that𝐹𝐹 = ∑ √(𝐸𝑟𝑟𝑜𝑟(𝓉 𝑖 )) 2 𝑛 𝑛 𝑖=0 + |𝑌(0)| + |𝑌′′(0) − 2| + |𝑌(1) − 1|By utilize the Bees algorithm, we get 𝑌(𝓉) = 0.499804 − 0.00025 𝐶 1 (𝓉) + 0.499939 𝐶 2 (𝓉) − 0.00019 𝐶 3 (𝓉) + 𝑂(𝓉 4 )…”

A highly accurate artificial neural networks scheme for solving higher multi‐order fractal‐fractional differential equations based on generalized Caputo derivative

“…Thus, we have to perform substantial numerical calculations to solve them. A variety of well-known algorithms can solve FDEs, such as operational matrix, [22][23][24][25][26] Adomian decomposition, 27 Harr wavelet Method, 28 Homotopy perturbation, 29 variational iteration, 30 neural networks (NNs), [31][32][33] and finite difference. 34,35 In comparison to classic numerical approaches, the approximate calculation of ANN appears to be less sensitive to the spatial dimension.…”

“…Thus, we have to perform substantial numerical calculations to solve them. A variety of well‐known algorithms can solve FDEs, such as operational matrix, 22‐26 Adomian decomposition, 27 Harr wavelet Method, 28 Homotopy perturbation, 29 variational iteration, 30 neural networks (NNs), 31‐33 and finite difference 34,35 …”