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In this study, two reliable approaches to solving the nonlinear stochastic It\^o-Volterra integral equation are provided. These equations have been evaluated using the orthonormal Chelyshkov spectral collocation technique and the orthonormal Chelyshkov spectral Galerkin method. The techniques presented here transform this problem into a collection of nonlinear algebraic equations that have been numerically solved using the Newton method. Also, the convergence analysis has been studied for both approaches. Two illustrative examples have been provided to show the efficacy, plausibility, proficiency, and applicability of the current approaches.

In this paper, a two-dimensional operational matrix method based on Chelyshkov polynomials is implemented to numerically solve the two-dimensional stochastic It\^o-Volterra Fredholm integral equations. These equations arise in several problems such as an exponential population growth model with several independent white noise sources. In this paper a new stochastic operational matrix has been derived first time ever by using Chelyshkov polynomials. After that, the operational matrices are used to transform the It\^o-Volterra Fredholm integral equation into a system of linear algebraic equations by using Newton cotes nodes as collocation point that can be easily solved. Furthermore, the convergence and error bound of the suggested method are well established. In order to illustrate the effectiveness, plausibility, reliability, and applicability of the existing technique, two typical examples have been presented.

This article is concerned with an approximate analytical solution for the time fractional Kudryashov Sinelshchikov equation by using the reproducing kernel Hilbert space method. The main tools of this method are reproducing kernel theory, some important Hilbert spaces, the normal basis, orthogonalisation process, and homogenization. The effectiveness of reproduc ing kernel Hilbert space method is presented through the tables and graphs. These computa tional results indicate that this method is highly accurate and efficient for the time fractional Kudryashov Sinelshchikov equation. Also, it is demonstrated that the approximate solution uniformly converges to exact solution by using reproducing kernel Hilbert space method., Comment: arXiv admin note: substantial text overlap with arXiv:2310.07375

This article is concerned with solving the time fractional Vakhnenko Parkes equation using the reproducing kernels. Reproducing kernel theory, the normal basis, some important Hilbert spaces, homogenization of constraints, and the orthogonalization process are the main tools of this technique. The main advantage of reproducing kernel method is it is truly meshless. The solutions obtained by the implementation reproducing kernels Hilbert space method on the time-fractional Vakhnenko Parkes equation is in the form of a series. The obtained solution converges to the exact solution uniquely. It is observed that the implemented method is highly effective. The effectiveness of reproducing kernel Hilbert space method is presented through the tables and graphs. The perfectness of this method is tested by taking different error norms and the order of convergence of the errors.

This manuscript studies the numerical solution of the time-fractional Burgers-Huxley equation in a reproducing kernel Hilbert space. The analytical solution of the equation is obtained in terms of a convergent series with easily computable components. It is observed that the approximate solution uniformly converges to the exact solution for the aforementioned equation. Also, the convergence of the proposed method is investigated. Numerical examples are given to demonstrate the validity and applicability of the presented method. The numerical results indicate that the proposed method is powerful and effective with a small computational overhead.

This paper presents an efficient numerical technique for solving multi-dimensional fractional optimal control problems using fractional-order generalized Bernoulli wavelets. The numerical results obtained by this method have been compared with the results obtained by the method using orthonormal Bernoulli wavelets. Using fractional-order generalized Bernoulli wavelets, product and integral operational matrices have been obtained. By using these operational matrices, the multi-dimensional fractional optimal control problems have been reduced into a system of algebraic equations. To confirm the efficiency of the proposed numerical technique involving fractional-order generalized Bernoulli wavelets, numerical problems have been solved by using both orthonormal Bernoulli wavelets and fractional-order generalized Bernoulli wavelets and obtaining an approximate cost function value, which has been found by approximating state and control functions. In addition, the convergence rate and error bound of the proposed numerical method have also been derived., Comment: arXiv admin note: substantial text overlap with arXiv:2310.06570

This paper presents an efficient numerical method for solving fractional optimal control problems using an operational matrix for a fractional wavelet. Using well-known formulae such as Caputo and Riemann-Liouville operators to determine fractional derivatives and integral fractional wavelets, operational matrices were devised and utilised to solve fractional optimal control problems. The proposed method reduced the fractional optimal control problems into a system of algebraic equations. To validate the effectiveness of the presented numerical approach, some illustrative problems were solved using fractional Taylor and Taylor wavelets, and the approximate cost function value derived by approximating state and control functions was compared. In addition, convergence rate and error bound of the proposed method have been derived.

The article presents the formulation and a new approach to find analytic solutions for fractional continuously variable order dynamic models viz. Fractional continuously variable order mass-spring damper systems. Here, we use the viscoelastic and viscous-viscoelastic dampers for describing the damping nature of the oscillating systems, where the order of fractional derivative varies continuously. Here, we handle the continuous changing nature of fractional order derivative for dynamic systems, which has not been studied yet. By successive iteration method, here we find the solution of fractional continuously variable order mass-spring damper systems, and then give a close form solution. We then present and discuss the solutions obtained in the cases with continuously variable order of damping for this oscillator with graphical plots., Comment: 17 pages, 7 figures, submitted to Nonlinear Dynamics