Euler wavelets method for solving fractional-order linear Volterra–Fredholm integro-differential equations with weakly singular kernels

The solutions of weakly singular fractional Volterra integro-differential equations involving the Caputo derivative typically have solutions whose derivatives are unbounded at the left end-point of the interval of integration. In this paper, we design an algorithm to prevail on this non-smooth behavior of solutions of the nonlinear fractional Volterra integro-differential equations with a weakly singular kernel. The convergence of the proposed method is investigated.The proposed scheme is employed to solve four numerical examples in order to test its efficiency and accuracy.

Section: Numerical Examplesmentioning

confidence: 99%

“…8 .59e − 02 3.53e − 02 2 2 3.52e − 09 4.82e − 03 2 3 1.40e − 14 3.11e − 04 2 4 1.51e − 14 2.80e − 05 2 5 − 1.91e − 06 where…”

Section: Example 1 Consider the Following Equationunclassified

Section: Example 1 Consider the Following Equationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A numerical study on the non‐smooth solutions of the nonlinear weakly singular fractional Volterra integro‐differential equations

Sajjadi

Najafi

Aminikhah

2022

“…There are many researchers in literature, who utilized the different types of wavelets for the solution of fractional mathematical differential models. For example, Euler wavelets, 23,24 CAS wavelets, 19,25,26 Legendre wavelets, [27][28][29] Chebyshev wavelets, [30][31][32] Daubechies wavelets, 16,33 Taylor wavelets, 34,35 Gegenbauer wavelets, 20,36 Sine-Cosine wavelets, 37 and B-Spline wavelets 38 methods.…”

Section: Introductionmentioning

confidence: 99%

Krawtchouk wavelets method for solving Caputo and Caputo–Hadamard fractional differential equations

Saeed

Idrees

Javid

et al. 2022

The purpose of the present work is to develop a new wavelet method, named as Krawtchouk wavelets method, for solving both Caputo fractional and Caputo-Hadamard fractional differential equations on a semi-infinite domain.Design/methodology/approach: We have utilized the discrete Krawtchouk orthogonal polynomial for the construction of Krawtchouk wavelets method. The supporting analysis of the method such as construction of operational matrices, procedure of implementation, and convergence analysis of the method are being provided. We have also proposed a method by combining the Krawtchouk wavelets method with the method of step for the solution of Caputo-Hadamard fractional delay differential equations. Findings:We have provided the orthogonality condition for the Krawtchouk wavelets. We have derived and constructed the Krawtchouk wavelets matrix, Krawtchouk wavelets operational matrix of Riemann-Liouville and Hadamard-type fractional-order integration, and Krawtchouk wavelets operational matrix of Riemann-Liouville and Hadamard-type fractional-order integration for boundary value problems. These matrices are successfully utilized for the solution of Caputo and Caputo-Hadamard fractional differential equations. Operational matrices contains many zero entries, which makes the present method more efficient. Furthermore, we workout on the procedure of implementation of the method for the Caputo fractional differential equations as well as for the Caputo-Hadamard fractional differential equations. We also derived the convergence analysis of the Krawtchouk wavelets method, which completes the theoretical analysis of the proposed method.We have applied the Krawtchouk wavelets method for the numerical solutions of several Caputo fractional differential equations and Caputo-Hadamard fractional differential equations and compare the obtained results with the analytical solutions. The comparison shows the effectiveness of the present numerical method. In this paper, we have considered initial value problem, boundary value problem, and delay problem. According to the numerical results, the present method is more efficient and accurate.Originality/value: Since fractional differential equation is a latest and emerging field, many engineers and scientists can utilize the present method for solving their fractional models. To the best of the author's knowledge, the present

An efficient numerical method based on ultraspherical polynomials for linear weakly singular fractional Volterra integro‐differential equations

Sajjadi,

Saberi Najafi,

Aminikhah

2024

The linear weakly singular fractional Volterra integro‐differential equations involving the Caputo derivative have solutions whose derivatives are unbounded at the left endpoint of the integration interval. In this paper, we use suitable transformations to prevail on this nonsmooth behavior. We used the product integration method based on the new fractional basis function to solve these equations, which led to the production of the new interpolation formula and weights for the method. We investigate the convergence of the presented method. The proposed scheme is employed to solve some numerical examples to test its efficiency and accuracy.