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This paper makes a comparison study of the fifth-order weighted compact nonlinear scheme (WCNS) and weighted essentially non-oscillatory (WENO) scheme in the truncation error, dissipation and dispersion terms. Truncation error analysis on the WCNS shows that numerical discretization leads to dissipation and dispersion terms. The Fourier spectral analysis methods is used to analyze the dissipative and dispersive features of both schemes.

The Earth's gravity field is a fundamental physical field for research and analysis in Earth sciences. However, the limited degree of expansion in the gravity field model introduces truncation errors, which hinder the accurate representation of high-frequency information in Earth's gravity field model. To address this issue, this study refined the gravity field model in the spatial domain by constructing a residual terrain model. This study refined the XGM2019e-2159 gravity field model for the study area in Colorado, USA (108°W–104°W, 37°N–41°N). First, the residual terrain model (RTM) was constructed using the high-resolution terrain model SRTMV4.1 and the reference topography model Earth2014. Subsequently, the residual terrain model was discretized into regular grid prisms. Based on Newton's law of universal gravitation, the disturbance potential of each prism within a specified range at the computation point is calculated using the rectangular prism method in the spatial domain. Next, the disturbance potential is used to compute the RTM gravity anomalies and RTM vertical deflections. The results were verified using ground measured gravity anomaly data NGS99 and vertical deflection data GSVS17. The results show that, after RTM correction, the root mean square (RMS) of the difference between modeled and measured gravity anomalies decreased from 19.71 mGal to 13.80 mGal, and the effect of residual terrain correction improves as terrain undulation increases. The RMS of the North–South and East–West component differences between modeled and measured vertical deflections was 1.44″ and 1.82″ before correction, and decreased to 0.89″ and 0.93″ after RTM correction. Finally, a power spectral density analysis of the XGM2019e-2159 gravity anomaly and vertical deflection models before and after RTM correction showed a significant increase in short-wavelength energy after correction. These results indicate that RTM correction effectively compensated for truncation errors in the XGM2019e-2159 gravity anomaly and vertical deflection models, significantly improving data quality. Graphical Abstract

This study uses the cubic spline method to solve the one-dimensional (1D) (one spatial and one temporal dimension) heat problem (a parametric linear partial differential equation) numerically using both explicit and implicit strategies. The set of simultaneous equations acquired in both the explicit and implicit method may be solved using the Thomas algorithm from the tridiagonal dominating matrix, and the spline offers a continuous solution. The results are implemented with very fine meshes and with relatively small-time steps. Using mesh refinement, it was possible to find better temperature distribution in the thin bar. Five numerical examples are used to support the efficiency and accuracy of the current scheme. The findings are also compared with analytical results and other results in terms of error and error norms L 2 and L ∞ . The Von-Neuman technique is used to analyse stability. The truncation error of both systems is calculated and determined to have a convergence of order O h + Δ t 2 .

Integral transformations represent an important mathematical tool for gravitational field modelling. A basic assumption of integral transformations is the global data coverage, but availability of high-resolution and accurate gravitational data may be restricted. Therefore, we decompose the global integration into two parts: (1) the effect of the near zone calculated by the numerical integration of data within a spherical cap and (2) the effect of the far zone due to data beyond the spherical cap synthesised by harmonic expansions. Theoretical and numerical aspects of this decomposition have frequently been studied for isotropic integral transformations on the sphere, such as Hotine’s, Poisson’s, and Stokes’s integral formulas. In this article, we systematically review the mathematical theory of the far-zone effects for the spherical integral formulas, which transform the disturbing gravitational potential or its purely radial derivatives into observable quantities of the gravitational field, i.e. the disturbing gravitational potential and its radial, horizontal, or mixed derivatives of the first, second, or third order. These formulas are implemented in a MATLAB software and validated in a closed-loop simulation. Selected properties of the harmonic expansions are investigated by examining the behaviour of the truncation error coefficients. The mathematical formulations presented here are indispensable for practical solutions of direct or inverse problems in an accurate gravitational field modelling or when studying statistical properties of integral transformations.

To solve fractional differential equations, they are typically converted into their corresponding crisp problems through a process known as the embedding method. This paper introduces a novel direct approach to solving fuzzy differential equations using fuzzy calculations, bypassing the need for this transformation. In this study, we develop the fuzzy Adams–Bashforth (A-B) method and the fuzzy Adams–Moulton (A-M) method to find numerical solutions for fuzzy fractional differential equations (FFDEs) with fuzzy initial values. To demonstrate the accuracy and efficiency of the proposed methods, we determine both the local truncation error and the global truncation error. Additionally, we establish the convergence and stability of these methods in detail. Finally, numerical examples are provided to illustrate the flexibility and effectiveness of the proposed methods.

In this field of research, in order to solve fuzzy fractional differential equations, they are normally transformed to their corresponding crisp problems. This transformation is called the embedding method. The aim of this paper is to present a new direct method to solve the fuzzy fractional differential equations using fuzzy calculations and without this transformation. In this work, the fuzzy generalized Taylor expansion by using the sense of fuzzy Caputo fractional derivative for fuzzy-valued functions is presented. For solving fuzzy fractional differential equations, the fuzzy generalized Euler’s method is introduced and applied. In order to show the accuracy and efficiency of the presented method, the local and global truncation errors are determined. Moreover, the consistency, convergence, and stability of the generalized Euler’s method are proved in detail. Eventually, the numerical examples, especially in the switching point case, show the flexibility and the capability of the presented method.

We establish convergence analysis for Hermite-type interpolations for L 2 ( R ) -entire functions of exponential type whose linear canonical transforms (LCT) are compactly supported. The results bridges the theoretical gap in implementing the derivative sampling theorems for band-limited signals in the LCT domain. Both complex analysis and real analysis techniques are established to derive the convergence analysis. The truncation error is also investigated and rigorous estimates for it are given. Nevertheless, the convergence rate is O ( 1 / N ) , which is slow. Consequently the work on regularization techniques is required.

A new numerical approach for the time independent Helmholtz equation on irregular domains has been developed. Trivial Cartesian meshes and simple 9-point stencil equations with unknown coefficients are used for 2-D irregular domains. The calculation of the coefficients of the stencil equations is based on the minimization of the local truncation error of the stencil equations and yields the optimal order of accuracy. At similar 9-point stencils, the accuracy of the new approach is two orders higher for the Dirichlet boundary conditions and one order higher for the Neumann boundary conditions than that for the linear finite elements. The numerical results for irregular domains also show that at the same number of degrees of freedom, the new approach is even much more accurate than the quadratic and cubic finite elements with much wider stencils. The new approach can be equally applied to the Helmholtz and screened Poisson equations.

Ordinary differential equations (ODEs) are very basic when it comes to modeling dynamic systems in various fields of science and engineering. However, solving high‐dimensional, nonlinear, and stiff ODEs is still a major challenge given the limitations of existing numerical methods, which tend to have difficulties in terms of accuracy and efficiency. This paper presents a new numerical approach called the enhanced differential transform homotopy perturbation method (E‐DTHPM), dedicated to overcoming the above difficulties. E‐DTHPM presents a combination of the benefits of the differential transform method (DTM) and homotopy perturbation method (HPM) with an adaptive step sizing mechanism that is dynamic in step sizing based on local truncation errors (LTEs). The combination of these methods addresses an important research gap by offering an efficient and accurate solution to the IVP of ODEs with nonsmooth behaviors (singularities and discontinuity behaviors). The proposed method is an improvement over conventional techniques and will provide increased computational efficiency and precision for complicated high‐dimensional systems. Numerical experiments show that E‐DTHPM is better than existing methods and is useful for applications that need to be dynamic enough and accurate enough for fields such as physics, biology, and engineering.

The current research proposes a novel efficient quintic B-spline (QBS) numerical technique based on piecewise uniform mesh for the numerical solution of fourth-order singularly perturbed boundary value problems (SPBVP) with discontinuous source terms (DST). The fifth-degree basis spline functions are employed along with a new approximation for the fourth-order derivative without altering the order of differential equations in which they appear. The proposed method is verified based on two problems to corroborate the scheme. The comparison of computational results shows that the proposed approach outperforms the existing approach in the literature. The stability and truncation error of the proposed way is analyzed.