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In this paper, Computing the eigenvalues of the Conformable Sturm-Liouville Problem (CSLP) of order 2α, 1/2 < α; ≤ 1, and dirichlet boundary conditions is considered. For this aim, CSLP is discretized to obtain a matrix eigenvalue problem (MEP) using finite element method with fractional shape functions. Then by a method based on the asymptotic form of the eigenvalues, we correct the eigenvalues of MEP to obtain efficient approximations for the eigenvalues of CSLP. Finally, some numerical examples to show the efficiency of the proposed method are given. Numerical results show that for the nth eigenvalue, the correction technique reduces the error order from O(n4h²) to O(n²h²). [ABSTRACT FROM AUTHOR]

In this paper, we focus on investigating a post-processing technique designed for one-dimensional singularly perturbed parabolic convection-diffusion problems that demonstrate a regular boundary layer. We use a backward Euler numerical approach for time derivatives with uniform mesh in the temporal direction, and a simple upwind scheme is used for spatial derivatives with modified graded mesh in the spatial direction. In this study, we demonstrate the effectiveness of the Richardson extrapolation technique in enhancing the ε-uniform accuracy of simple upwinding within the discrete supremum norm, as evidenced by an improvement from O(N-1 ln(1/ε) +Δθ) to O(N-2 ln²(1/ε) +Δθ²). Furthermore, to validate the theoretical findings, computational experiments are conducted for two test examples by applying the proposed technique. [ABSTRACT FROM AUTHOR]

If the matrix function f (A t) posses the properties of f (A t) = g f (t k A , then the recurrence formula f i − 1 = g f i , i = N , N − 1 , ⋯ , 1 , f (t A) = f 0 , can be established. Here, f N = f (A N) = ∑ j = 0 m a j A N j , A N = t k N A . This provides an algorithm for computing the matrix function f (A t) . By specifying the calculation accuracy p, a method is presented to determine m and N in a way that minimizes the time of the above algorithm, thus providing a fast algorithm for f (A t) . It is important to note that m only depends on the calculation accuracy p and is independent of the matrix A and t. Therefore, f (A N) has a fixed calculation format that is easily computed. On the other hand, N depends not only on A, but also on t. This provides a means to select t such that N is equal to 0, a property of significance. In summary, the algorithm proposed in this article enables users to establish a desired level of accuracy and then utilize it to select the appropriate values for m and N to minimize computation time. This approach ensures that both accuracy and efficiency are addressed concurrently. We develop a general algorithm, then apply it to the exponential, trigonometric, and logarithmic matrix functions, and compare the performance with that of the internal system functions of Mathematica and Pade approximation. In the last section, an example is provided to illustrate the rapid computation of numerical solutions for linear differential equations. [ABSTRACT FROM AUTHOR]

We analyze the recently introduced family of preconditioners in [M. M. Rana et al., SIAM J. Sci. Comput., 43 (2021), pp. S475-S495] for the stage equations of implicit Runge--Kutta methods for s-stage methods. We simplify the formulas for the eigenvalues and eigenvectors of the preconditioned systems for a general s-stage method and use these to obtain convergence rate estimates for preconditioned GMRES for some common choices of the implicit Runge-Kutta methods. This analysis is based on understanding the inherent matrix structure of these problems and exploiting it to qualitatively predict and explain the main observed features of the GMRES convergence behavior, using tools from approximation and potential theory based on Schwarz--Christoffel maps for curves and close, connected domains in the complex plane. We illustrate our analysis with numerical experiments showing very close correspondence of the estimates and the observed behavior, suggesting the analysis reliably captures the essence of these preconditioners. [ABSTRACT FROM AUTHOR]

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A numerical-analytical solution of a problem concerning the current distribution and input characteristics of asymmetric biconical dipole with distributed surface impedance and arbitrary excitation and derived in the thin-wire approximation. Solution correctness is confirmed by satisfactory agreement of numerical and experimental results from literary sources. Numerical results are given for the input characteristics of the dipole in the case of its asymmetric excitation by a point source. [ABSTRACT FROM AUTHOR]

This article presents a novel numerical approach to the solution of the nonlinear Kawahara equation. The desired approximations are obtained from the combination of Dickson polynomials and Taylor's expansion. The combined approach is based on Taylor's expansion for discretizing the time derivative and Dickson polynomials for space derivatives. The problem will be converted into a system of linear algebraic equations for each time step via some suitable collocation points. Error estimation is presented after obtaining the approximate solution. The newly proposed technique is compared with some existing numerical methods to show the method's applicability, accuracy, and efficacy. Two problems are solved to demonstrate the method's power and effect, and the results are presented as a table and graphics. [ABSTRACT FROM AUTHOR]