Centroidal Mean Research Articles

Runge-Kutta Method Based on the Linear Combination of Arithmetic Mean, Geometric Mean, and Centroidal Mean: A Numerical Solution for the Hybrid Fuzzy Differential Equation

Objectives: To find the absolute error of the Hybrid fuzzy differential equation with triangular fuzzy number since its membership function has a triangular form. Methods: The third order runge-kuttta method based on the linear combination of Arithmetic mean, Geometric mean and Centroidal mean is introduced to solve Hybrid fuzzy differential equation. Seikkala’s derivatives are considered, and Numerical example is given to demonstrate the effectiveness of the proposed method. Findings: The comparative study have been done between the proposed method and the existing third order runge-kutta method based on Arithmetic mean. The proposed method gives better error tolerance than the third order runge-kutta method based on Arithmetic mean. Novelty: In this study a new formula has been developed by combining three means Arithmetic Mean, Geometric Mean and Centroidal Mean using Khattri’s formula and it is solved by the third order runge-kutta method for the first order hybrid fuzzy differential equation. The error tolerance has been calculated for the proposed method and the Arithmetic mean and it is seen that the error tolerance is better for the proposed method. Keywords: Hybrid fuzzy differential equations, Triangular fuzzy number, Seikkala’s derivative, Third order runge-kutta method, Initial value problem

Numerical Solution of Hybrid Fuzzy Differential Equation by using Third Order Runge-Kutta Method Based on Linear Combination of Arithmetic Mean, Root Mean Square and Centroidal Mean

Objectives: This article develops the third order Runge-Kutta method, which uses a linear combination of the arithmetic mean, root mean square, and centroidal mean, to solve hybrid fuzzy differential equations. Methods: Seikkala's derivative is taken into account, and a numerical example is provided to show the efficacy of the proposed method. The outcomes demonstrate that the suggested approach is an effective tool for approximating the solution of hybrid fuzzy differential equations. Findings: The comparative analysis was carried out using the third order Runge-Kutta method that is currently in use and is based on arithmetic mean, root mean square, and centroidal mean. Compared to other methods, the suggested method offers a more accurate approximation. Novelty: In this study a new formula has been developed by combining three means Arithmetic Mean, Root Mean Square, and Centroidal Mean using Khattri's formula. And the developed formula is used to solve the third order Runge-Kutta method for the first order hybrid fuzzy differential equation. All real life problems which can be modeled in to an initial value problem can be solved using this formula. Keywords: Hybrid fuzzy differential equations, Triangular fuzzy number, Seikkala's derivative, third order Runge-Kutta method, Arithmetic mean, Root mean square, Centroidal mean, Initial value problem

Solving fuzzy fractional differential equations using fuzzy fractional fourth order Runge-Kutta method based on centroidal mean

Solving fuzzy fractional differential equations using fuzzy fractional fourth order Runge-Kutta method based on centroidal mean

Computation of square root of a number using centroidal mean and its invariant

Computation of square root of a number using centroidal mean and its invariant

Centroidal Polygon: A New Enhancement of Euler to Improve Accuracy of First Order Non-Linear Ordinary Differential Equation

The Euler method is one of the oldest methods to solve differential equation problems. The Euler method produces the simplest solution. However, although is not computationally expensive, the Eulermethod is lack of accuracy. To improve the Euler method, the researcher proposed a new scheme for better accuracy. The Euler method equation and the mean method were combined to enhance this method. As the improvement basis, the researcher used the Centroidal mean and the midpoint method or Polygon to improve the Euler method. The combination of the Euler and Centroidal mean is known as Centroidal Polygon (CP) scheme. The CP scheme was used to solve first-order non-linear Ordinary Differential Equations (ODE). The researcher used SCILAB 6.0 software to solve the equation and the CP scheme was tested in three different step sizes (0.1,0.01, and 0.001). Aside from that, the researcher had compared the CP scheme with previous schemes such as ZulZamri's Polygon (P) scheme, Nurhafizah's Harmoni-Polygon (HP) scheme, and Nooraida's Cube-Polygon (CuP) scheme to ensure that the CP scheme is more accurate than previous research. When the maximum error is calculated by subtracting the scheme and exact solution, the results show that the CP scheme delivers the highest accuracy results in the shortest amount of time. The new enhanced, modified Euler method is useful for other researchers to achieve good accuracy at low computational cost as an alternative to the more computationally expensive methods.

Optimal bounds of exponential type for arithmetic mean by Seiffert-like mean and centroidal mean

Optimal bounds of exponential type for arithmetic mean by Seiffert-like mean and centroidal mean

Centroidal-Polygon: Solving First Order Ordinary Differential Equation using Centroidal mean to improve Euler Method

Euler method is one of the oldest and simplest numerical techniques to solve mathematical problems. However, the Euler method lacks accuracy compared to other methods such as the Runge-Kutta and the Adam Bashforth. This research proposed a new scheme by enhancing the Euler method using centroidal mean, i.e. combining the Euler method and the mean concept. In this research, the proposed scheme would be used to improve the accuracy of the first order Ordinary Differential Equation (ODE). The enhanced Euler method was used based on the improved Euler method, which is a midpoint method. To analyse the accuracy of the proposed scheme and the existing schemes, SCILAB 6.0 software was used to compare the results of different step sizes. This research compared the proposed scheme with the Polygon scheme by Zul Zamri and the Harmonic-Polygon scheme by Nurhafizah. By using Centroidal-Polygon as the proposed scheme, more accurate results were yielded. However, the scheme was limited to some of the linear and non-linear equations only.

EFFICIENT DERIVATIVE-BASED SIMPSON’S 1/3-TYPE SCHEME USING CENTROIDAL MEAN FOR RIEMANN-STIELTJES INTEGRAL

In this paper, a new efficient derivative-based quadrature scheme of Simpson’s 1/3-type is proposed using the centroidal mean for the approximation of Riemann-Stieltjes integral (RS-integral). Theorems are proved related to the basic form, composite form, local and global errors of the new scheme for the RS-integral. The reduction of the new proposed scheme is verified using g(t) = t for Riemann integral. The theoretical results of new proposed scheme have been proved by experimental work using programming in MATLAB against existing schemes. The order of accuracy, computational cost and average CPU time (in seconds) of the new proposed scheme are determined. The results obtained show the effectiveness of the proposed scheme compared to the existing schemes.

A NEW AND EFFICIENT CENTROIDAL MEAN DERIVATIVE-BASED TRAPEZOIDAL SCHEME FOR NUMERICAL CUBATURE

This research presents a new and efficient Centroidal mean derivative-based numerical cubature scheme which has been proposed for the accurate evaluation of double integrals under finite range. The proposed modification is based on the Trapezoidal-type quadrature and cubature rules. The approximate values can only be obtained for some important applications to evaluate the complex double integrals. Higher precision and order of accuracy could be achieved by the proposed scheme. The schemes, in basic and composite forms, with local and global error terms are presented with necessary supporting arguments with their performance evaluation against conventional Trapezoid rule through some numerical experiments. The simultaneously observed error distributions of the proposed schemes are found to be lower than the conventional Trapezoidal cubature scheme in composite form

Heron and Alpha-Centroidal Mean Filters with Exponential factors for Deep Space Images

In this article, we discuss empirical methods that can be add-on to enhance spatial image filters. Discussing on the implementation of such methods for Deep-Sky images from space telescopes. Spatial image filtering houses mean filters that use kernel method to filter a digital image in the spatial domain. To improve the performance of the existing filters and aid the new upcoming filter, a technique is developed that can be added to the filter to improve their performance. This study gives a detailed analysis of an empirical weight factor and exponential factoring methods for the mean filters that are proposed and implemented. The exponential factoring is applied to the recent mean filters Heron, Centroidal and Inverse-Contraharmonic. Performance analysis of both methods and comparison to existing mean filtering performance are represented. The images are classified into white or black based background and analyzed for performance parameters, comparing the results from the filters and filters with the exponential factoring. The empirical methods were proven to improve the efficiency in denoising of images of the existing mean filters and the new Heron and Centroidal mean filters, hence can be used as add-ons to existing spatial mean filters. One of the empirical techniques, exponential factoring is promising in denoising high density noised images. This novel method is implemented to filter noise from Deep Space images taken by space telescopes. Ultra-high-resolution images are filtered and implemented on Single Board Computers for remote handling, with testing done on MATLAB.

On Two Double Inequalities (Optimal Bounds and Sharps Bounds) for Centroidal Mean in Terms of Contraharmonic and Arithmetic Means

This research work considers the following inequalities: λA(a,b) + (1-λ)C(a,b) ≤ C(a,b) ≤ μA(a,b) + (1-μ)C(a,b) and C[λa + (1-λ)b, λb + (1-λ)a] ≤ C(a,b) ≤ C[μa + (1-μ)b, μb + (1-μ)a] with . The researchers attempt to find an answer as to what are the best possible parameters λ, μ that (1.1) and (1.2) can be hold? The main tool is the optimization of some suitable functions that we seek to find out. By searching the best possible parameters such that (1.1) and (1.2) can be held. Firstly, we insert f(t) = λA(a,b) + (1-λ)C(a,b) - C(a,b) without the loss of generality. We assume that a>b and let to determine the condition for λ and μ to become f (t) ≤ 0. Secondly, we insert g(t) = μA(a,b) + (1-μ)C(a,b) - C(a,b) without the loss of generality. We assume that a>b and let to determine the condition for λ and μ to become g(t) ≥ 0.

Sharp inequalities for bounding Seiffert mean in terms of the arithmetic, centroidal, and contra-harmonic means

AbstractIn the paper, the authors find two sharp and double inequalities for bounding the second Seiffert mean either by a one-parameter family of means derived from the centroidal mean or by a convex combination of the arithmetic and contra-harmonic means.

On the centroidal mean Newton's method for simple and multiple roots of nonlinear equations

In this paper, the convergence behaviour of a variant of Newton's method based on the centroidal mean is considered. The convergence properties of this method for solving equations which have simple or multiple roots have been discussed. It is shown that it converges cubically to simple roots with efficiency index is 1.442 and linearly to multiple roots. Moreover, the values of the corresponding asymptotic error constants of convergence are determined. Theoretical results have been verified on the relevant numerical problems. The proposed new method has the advantage of evaluating only the first derivative and less number of iterations to achieve third order accuracy. A comparison of the efficiency of this method with other mean-based Newton's methods, based on the arithmetic, geometric and harmonic means, is also included. Convergences to the root and error propagation with iteration are exhibited graphically with iterations.

A double inequality for bounding Toader mean by the centroidal mean

In the paper, the authors find the best numbers $\alpha$ and $\beta$ such that $$ \overline{C}\bigl(\alpha a+(1-\alpha)b,\alpha b+(1-\alpha)a\bigr)<T(a,b) <\overline{C}\bigl(\beta a+(1-\beta)b,\beta b+(1-\beta)a\bigr) $$ for all $a,b>0$ with $a\ne b$, where $\overline{C}(a,b)={2\bigl(a^2+ab+b^2\bigr)}{3(a+b)}$ and $T(a,b)=\frac{2}{\pi}\int_{0}^{{\pi}/{2}}\sqrt{a^2{\cos^2{\theta}}+b^2{\sin^2{\theta}}}\,d\theta$ denote respectively the centroidal mean and Toader mean of two positive numbers $a$ and $b$.

Generalization of -Centroidal Mean and its Dual

In this paper, the generalized �-centroidal mean and its dual form in 2 variables are introduced. Also, we will study some properties and prove their monotonicity. Further, shown that various means are particular cases of generalized �-centroidal mean.

Numerical Solution of N th - Order Fuzzy Initial Value Problems by Fourth Order Runge-Kutta Method Basesd On Centroidal Mean

In this paper, a numerical method for N th - order fuzzy initial value problems (FIVP) based on Seikkala derivative of fuzzy process is studied. The fourth order Runge-Kutta method based on Centroidal Mean (RKCeM4) is used to find the numerical solution and the convergence and stability of the method is proved. This method is illustrated by solving second and third order FIVPs. The results show that the proposed method suits well to find the numerical solution of N th - order FIVPs. Keywords - Fuzzy numbers, N th - order Fuzzy Initial Value Problems, Runge-Kutta method, Centroidal Mean, Lipschitz condition.

Bounds for Combinations of Toader Mean and Arithmetic Mean in Terms of Centroidal Mean

The authors find the greatest value λ and the least value μ, such that the double inequality holds for all α ∈ (0,1) and a, b > 0 with a ≠ b, where , A(a, b) = (a + b)/2, and denote, respectively, the centroidal, arithmetic, and Toader means of the two positive numbers a and b.

Optimal bounds for logarithmic and identric means in terms of generalized centroidal mean

Optimal bounds for logarithmic and identric means in terms of generalized centroidal mean

A New Non-linear Multistep Method Based on Centroidal Mean in Solving Initial Value Problems

A new 2-step fourth order implicit non-linear multistep method based on centroidal mean is considered in this paper. The new method is tested on some test problems; and numerical results show that the new method is able to produce acceptable numerical solutions for these test problems. Comparisons in terms of numerical accuracy between the new method and the classical 2-step Adams-Moulton method are carried out as well. Numerical experiments show that our new method performs better than the classical 2-step Adams-Moulton method in solving these test problems. Keywords: Initial value problems; non-linear multistep method; centroidal mean; Adams-Moulton method; 3-stage fourth order Lobatto IIIC method

A new fourth order embedded RKAHeM(4,4) method with error control on multilayer raster cellular neural network

We introduce a new technique for solving initial value problems (IVPs) with error control by formulating an embedded method involving RK methods based on arithmetic mean (AM) and Heronian mean (HeM). The function of the simulator is that it is capable of performing raster simulation for any kind as well as any size of input image. It is a powerful tool for researchers to examine the potential applications of CNN. By using the newly proposed embedded method, a versatile algorithm for simulating multilayer CNN arrays is implemented. This article proposes an efficient pseudo code for exploiting the latency properties of CNN along with well known RK-fourth order embedded numerical integration algorithms. Simulation results and comparison have also been presented to show the efficiency of the numerical integration algorithms. It is found that the RK-embedded Heronian mean outperforms well in comparison with the RK-embedded centroidal mean, harmonic mean and contra-harmonic mean. A more quantitative analysis has been carried out to clearly visualize the goodness and robustness of the proposed algorithm.