Tension Spline Research Articles

A High Order Numerical Scheme for Time-Fractional Telegraph Equation Via Cubic Spline in Tension

A High Order Numerical Scheme for Time-Fractional Telegraph Equation Via Cubic Spline in Tension

Fitted mesh methods based on non-polynomial splines for singularly perturbed boundary value problems with mixed shifts

<p>In this paper, numerical schemes based on non-polynomial splines, namely, spline in compression, tension, and adaptive spline, are constructed for singularly perturbed boundary value problems with mixed shifts. A convergence analysis is carried out on the proposed methods. A comparitive study of the results is performed on test problems and presented in the form of tables. Graphs are drawn to illustrate the behavior of the solution to the problems.</p>

Fitted Tension Spline Scheme for a Singularly Perturbed Parabolic Problem With Time Delay

A fitted tension spline numerical scheme for a singularly perturbed parabolic problem (SPPP) with time delay is proposed. The presence of a small parameter ε as a multiple of the diffusion term leads to the suddenly changing behaviors of the solution in the boundary layer region. This results in a challenging duty to solve the problem analytically. Classical numerical methods cause spurious nonphysical oscillations unless an unacceptable number of mesh points is considered, which requires a large computational cost. To overcome this drawback, a numerical method comprising the backward Euler scheme in the time direction and the fitted spline scheme in the space direction on uniform meshes is proposed. To establish the stability and uniform convergence of the proposed method, an extensive amount of analysis is carried out. Three numerical examples are considered to validate the efficiency and applicability of the proposed scheme. It is proved that the proposed scheme is uniformly convergent of order one in both space and time. Further, the boundary layer behaviors of the solutions are given graphically.

An Efficient Method for Solving Nonlinear Boundary Value Problems Via the Non-Polynomial Tension Spline Function

In this research, a nonlinear system of second-order boundary value problems is solved, by using non-polynomial tension spline function. The suggested method's validity and applicability are proved by a few numerical findings. The method's results show that it is both easy to use and efficient.

Tension spline approach to singularly perturbed delay differential equations involving large delay

To solve the singlularly perturbed delay differential equations of convection type, spline in tension technique is used. Taylor series approximation is used to tackle the first order derivative term involved in the problem and finally a tridiagonal scheme is obtained, which is then solved by Thomas algorithm. An example is illustrated to show the method's applicability and it is solved numerically for different values of perturbation parameter. The maximum absolute errors and rate of convergence are given in tables and it is seen that our method is of first order convergent.

An efficient numerical approach for solving singularly perturbed parabolic differential equations with large negative shift and integral boundary condition

In this study, we consider singularly perturbed large negative shift parabolic reaction–diffusion with integral boundary condition. The continuous solution's properties are discussed. On a non-uniform Shishkin mesh, the spatial derivative is discretized using the tension spline method, and the temporal derivative is discretized using the Crank–Nicolson method. In order to handle the integral boundary condition, Simpson's rule is used. After conducting an error analysis, it was determined that the method was uniformly convergent. To support the theoretical findings, numerical examples are taken into account and solved for different values of the perturbation parameter and mesh sizes. It is proved to be uniformly convergent to almost second order.

A tension spline fitted numerical scheme for singularly perturbed reaction-diffusion problem with negative shift

ObjectiveThe paper is focused on developing and analyzing a uniformly convergent numerical scheme for a singularly perturbed reaction-diffusion problem with a negative shift. The solution of such problem exhibits strong boundary layers at the two ends of the domain due to the influence of the perturbation parameter, and the term with negative shift causes interior layer. The rapidly changing behavior of the solution in the layers brings significant difficulties in solving the problem analytically. We have treated the problem by proposing a numerical scheme using the implicit Euler method in the temporal direction and a fitted tension spline method in the spatial direction with uniform meshes.ResultStability and uniform error estimates are investigated for the developed numerical scheme. The theoretical finding is demonstrated by numerical examples. It is obtained that the developed numerical scheme is uniformly convergent of order one in time and order two in space.

SPLINE MODEL OF GRIDDED DATA OPERATION AS A PRINCIPLE OF ELECTRONIC MAPPING SEABED TOPOGRAPHY

The influence of technologies of automated processing of hydrographic survey results on the potential of the human factor in the transformation of polygraphic cartographic products into digital ones is analyzed. When configuring the digital bathymetric model, the concept of the navigation surface is used as a perspective principle of automated mapping. The approach to the problem of generating bathymetric contours from measurement results in the navigation surface paradigm is construed on continuous spline interpolation of geospatial data for reasonable cartographic generalization when creating electronic cartographic products. From the theoretical positions of the spline approach, the subjectivity of the method of artificial displacement of generalized isobaths to deep-water areas is excluded when creating a safe digital model of the bottom relief, interpreted in a mathematical sense as a navigational isosurface. The principle of electronic mapping based on the use of a spline in tension as an effective approach for the process of generalizing isobaths in order to obtain a wide range of morphometric characteristics of underwater topography has been developed. The generalized line of the active depth contour is estimated in the form of algorithmic reproduction on electronic charts of the safe convexity of the isobate towards the deep-sea area due to the practical implementation of the B-spline “snake model” by analogy with the serpentine configuration of the bathymetric isoline in the form of a piecewise polynomial function. When using the spline approach, an innovative principle of electronic mapping of the underwater landscape based on operating with a set of gridded data is implemented. The latter are interpreted as the results of depth measurements with the formalization of a two-dimensional frame of fixed values of bathymetric measurements for their representation as a navigational isosurface in three-dimensional Euclidean space. The actual synthesis of the seabed topography is implemented on the basis of a proven hybrid spline model for a specific indicative test case based on the processing of experimental gridded data. Hypothetically, the possibility of intellectual assistance to the watch officer in the strategy of instant orientation in conditions of a minimum depth reserve under the keel is organized when using computer three-dimensional visualization of the topography of the underwater relief in an unaffiliated graphic environment with foreign software.

Fitted Tension Spline Method for Singularly Perturbed Time Delay Reaction Diffusion Problems

A uniformly convergent numerical method is presented for solving singularly perturbed time delay reaction-diffusion problems. Properties of the continuous solution are discussed. The Crank–Nicolson method is used for discretizing the temporal derivative, and an exponentially fitted tension spline method is applied for the spatial derivative. Using the comparison principle and solution bound, the stability of the method is analyzed. The proposed numerical method is second-order uniformly convergent. The theoretical analysis is supported by numerical test examples for various values of perturbation parameters and mesh size.

Evaluation of the Performance of Spline Interpolation Method in Mapping and Estimating the Total Suspended Solids over the Coastal Water of Pulau Tuba, Kedah

The purpose of this paper is to evaluate the performance of the spline interpolation method in predicting and mapping the concentration of Total Suspended Solids (TSS) in the surface water of Pulau Tuba, Kedah. Thirty sampling points were set up and geolocated using the Geographic Positioning System (GPS). Gravimetric analyses were used to determine the TSS level. Fifty percent of the total sampling points were randomly chosen for developing spatial models using regularised and tension spline methods. The research found that the tension spline methods outperform the regularised spline method. The Mean Squared Error (MSE), Root Mean Squared Error (RMSE), Mean Absolute Error (MAE), and Mean Absolute Error Percentage (MAPE) were reported at 351.641, 18.752, 15.81, and 21.51%, respectively. This study’s findings are critical in the domains of spatial statistics and interpolation for creating a precise map of water properties.

ASSESSMENT OF THE SPLINE INTERPOLATION METHOD'S PERFORMANCE IN PREDICTING AND MAPPING THE PHOSPHATE OVER THE COASTAL WATERS OF PULAU TUBA, LANGKAWI, KEDAH

To date, there has been limited knowledge on the applicability of the spline interpolation method to estimate coastal water quality parameters. This study aims to assess the performance of the regularised and tension spline interpolation techniques for predicting and mapping the phosphate concentration in the surface water of Pulau Tuba, Langkawi, Kedah. Sampling points were set up randomly along the coastal water of Pulau Tuba in November 2018. Samples were obtained using the Niskin Water Sampler at 1 meter below the surface and immediately transported into the marine technology laboratory. Phosphate levels were determined using an Ultraviolet Visible spectrophotometer and the ascorbic acid technique. The Geographic Positioning System (GPS) was used to record the geolocation of each sampling point. The training set (50%) and the testing set (50%) were chosen randomly based on 20 sampling points. The regularized and tension spline interpolation methods were used for this study. Evaluation of prediction of models was carried out using the Root Mean Square Error (RMSE) statistics. The study found that the tension spline interpolation method better predicted phosphate concentration. The Root Mean Square Error (RMSE) was reported at 0.106 and 0.094 for the regularized and tension spline interpolation methods, respectively. Concerned parties can utilize the study's findings as guidelines for monitoring, managing, and developing strategic strategies for the long-term development of the coastal waters of Pulau Tuba, Langkawi, Kedah.

A Fitted Mesh Cubic Spline in Tension Method for Singularly Perturbed Problems with Two Parameters

A numerical treatment via a difference scheme constructed by the Crank–Nicolson scheme for the time derivative and cubic spline in tension for the spatial derivatives on a layer resolving nonuniform Bakhvalov-type mesh for a singularly perturbed unsteady-state initial-boundary-value problem with two small parameters is presented. Error analysis of the constructed scheme is discussed and shown to be parameter-uniformly convergent with second-order convergence. Numerical experimentation is taken to confirm the theoretical findings.

A Comparative Accuracy of Regularized and Tension Spline Methods to Estimate and Model the Surface Water pH of Pulau Tuba, Langkawi, Kedah

This research is conducted to assess the accuracy of spline interpolation methods to predict and model the surface water pH of Pulau Tuba, Langkawi, Kedah, Malaysia. In-situ sampling activities using pH-meter and Geographic Positioning Systems (GPS) were carried out during high tides and at noon in November 2018. The development of spatial models was constructed using Regularized and Tension spline methods. Then, validation of models was carried out to compare the observed and predicted values of pH using correlation analysis, regression analysis, and error analysis. The accuracy of the developed map was calculated using the overall accuracy equation. This research found that the regularized spline method had more accuracy in estimating surface water pH variability than the tension spline method. Pearson correlation coefficient (r), Coefficient of determination (R2), Mean Absolute Error (MAE) and Root Mean Square Error (RMSE) were reported at 0.896, 0.803, 0.0265 and 0.0344 for the regularized spline method,respectively. The developed spatial model was then transformed into a map by adding map elements such as legend, title, north arrow, and scales for effective visualization. The developed map has an accuracy of 87.50%. The surface water pH was found at the range of 7-8. Low reading of pH is expected due to the addition of rainwater to the coastal water of Pulau Tuba, Langkawi, Kedah. The research outcomes would benefit government and non-government agencies to monitor the coastal and ocean acidification and the development of strategic policies and rules to reduce the impact of anthropogenic activities and climate changes for this area.

$$C^2$$ Tension Splines Construction Based on a Class of Sixth-Order Ordinary Differential Equations

$$C^2$$ Tension Splines Construction Based on a Class of Sixth-Order Ordinary Differential Equations

Sea Level Fusion of Satellite Altimetry and Tide Gauge Data by Deep Learning in the Mediterranean Sea

Satellite altimetry and tide gauges are the two main techniques used to measure sea level. Due to the limitations of satellite altimetry, a high-quality unified sea level model from coast to open ocean has traditionally been difficult to achieve. This study proposes a fusion approach of altimetry and tide gauge data based on a deep belief network (DBN) method. Taking the Mediterranean Sea as the case study area, a progressive three-step experiment was designed to compare the fused sea level anomalies from the DBN method with those from the inverse distance weighted (IDW) method, the kriging (KRG) method and the curvature continuous splines in tension (CCS) method for different cases. The results show that the fusion precision varies with the methods and the input measurements. The precision of the DBN method is better than that of the other three methods in most schemes and is reduced by approximately 20% when the limited altimetry along-track data and in-situ tide gauge data are used. In addition, the distribution of satellite altimetry data and tide gauge data has a large effect on the other three methods but less impact on the DBN model. Furthermore, the sea level anomalies in the Mediterranean Sea with a spatial resolution of 0.25° × 0.25° generated by the DBN model contain more spatial distribution information than others, which means the DBN can be applied as a more feasible and robust way to fuse these two kinds of sea levels.

Numerical Study of a Singularly Perturbed Two Parameter Problems on a Modified Bakhvalov Mesh

In this article, we consider a two parameter singular perturbation problem (TPSPP) and the numerical solution is obtained by using tension spline method on a layer adapted Bakhvalov type mesh. The method is analyzed for convergence. Numerical results are presented to support the theory.

A quartic trigonometric tension b-spline algorithm for nonlinear partial differential equation system

Purpose The purpose of this study is to construct quartic trigonometric tension (QTT) B-spline collocation algorithms for the numerical solutions of the Coupled Burgers’ equation. Design/methodology/approach The finite elements method (FEM) is a numerical method for obtaining an approximate solution of partial differential equations (PDEs). The development of high-speed computers enables to development FEM to solve PDEs on both complex domain and complicated boundary conditions. It also provides higher-order approximation which consists of a vector of coefficients multiplied by a set of basis functions. FEM with the B-splines is efficient due both to giving a smaller system of algebraic equations that has lower computational complexity and providing higher-order continuous approximation depending on using the B-splines of high degree. Findings The result of the test problems indicates the reliability of the method to get solutions to the CBE. QTT B-spline collocation approach has convergence order 3 in space and order 1 in time. So that nonpolynomial splines provide smooth solutions during the run of the program. Originality/value There are few numerical methods build-up using the trigonometric tension spline for solving differential equations. The tension B-spline collocation method is used for finding the solution of Coupled Burgers’ equation.

An Iterative Approach for Solving Nonlinear Volterra–Fredholm Integral Equation Using Tension Spline

According to mathematical literature, the parametric spline functions have not been used directly for solving the integral equations so far. In this work, we develop the tension spline approximation to obtain the numerical solution of Volterra–Fredholm integral equation. First, the tension spline method was obtained and then the combination of this method with the quasi-linearization method has been used for solving the nonlinear Volterra–Fredholm integral equation. The error analysis of the method is given. Illustrative examples are included to verify the effectiveness and applicability of the presented approach.

Strain rate field on the Nigeria lithosphere derived from GNSS velocity

The Global Navigation Satellite System (GNSS) over the years has become an essential instrument in modelling geodynamic processes, because of its high spatio-temporal characteristics. Accurate velocity modelling of GNSS stations and strain rate analysis can provide an accurate mechanism for studying the deformation of the earth crust. Therefore, this study presents the strain rate tensor of the Nigeria crust from GNSS network, but first, the GAMIT/GLOBK was used to process and obtain accurate velocity solution of the Nigerian Permanent GNSS Network (NigNET). The surface deformation of Nigeria deduced from strain analysis was computed using splines in tension. The results revealed that the highest extensional and compressional strain rates of up to 2.67 to 2.89μstrain/year were observed around parts of the north-west (Jigawa, Kaduna and Kano) and north central region (parts Abuja and Jos) and some parts of the north-east (parts of Bauchi and Gombe), while the lowest minimal or low extensional and compressional strain rates of 0.09 to 0.534μstrain/year were observed around some parts of the north central (parts of Kwara and Niger) and parts of north-west (parts of Kebbi, Zamfara and Sokoto). The highest maximum shear strain rate of about 0.4μstrain/year which has its peak around some parts of Kaduna, Abuja, Plateau and Bauchi was observed. The highest areal dilatation rate observed in Nigeria is about 1.02μstrain/year. The study serves as a benchmark for further studies.

Computational study for a class of time-dependent singularly perturbed parabolic partial differential equation through tension spline

This article contributes a numerical technique for a class of singularly perturbed time delayed parabolic partial differential equation. A priori results of maximum principle, stability and bounds are discussed. The continuous problem is semi-discretized by the Crank–Nicolson based scheme in the temporal direction and then discretized by the tension spline scheme on non-uniform Shishkin mesh. Error estimation for the discretized problem is derived. To validate the theoretical findings, the numerical outcomes for linear and nonlinear problems are tested.