Solving ordinary differential equations by the Dormand Prince method

In general, differential equations in mathematics can be defined as an equation that comprises of one or more functions and its derivatives. Meanwhile ordinary differential equation in mathematics is declared as differential equations that contains one or more functions of one independent variable and its ordinary derivatives. Unlike partial differential equations, ordinary differential equations involve only the ordinary derivatives with respect to one independent variable. This research was conducted to solve ordinary differential equations by a numerical method called the Dormand Prince method. Consequently the solutions obtained are compared with the other numerical method in terms of accuracy. Dormand Prince method is one of the similar methods as RungeKutta method. It is used to solve an ordinary differential equation explicitly by six function evaluations. Throughout this research, the accuracy of the Dormand Prince method in solving ordinary differential equations was examined by comparing it with the other numerical method, which is Runge Kutta Fehlberg method

Fuzzy finite switchboard automata with complete residuated lattices

The theory of fuzzy finite switchboard automata (FFSA) is introduced by the use of general algebraic structures such as complete residuated lattices in order to enhance the process ability of FFSA. We established the notion of homomorphism, strong homomorphism and reverse homomorphism and shows some of its properties. The subsystem of FFSA is studied and the set of switchboard subsystemforms a complete ℒ -sublattices is shown. The algorithm of FFSA with complete residuated lattices is given and an example is provided

Barzilai-Borwein gradient method for solving fuzzy nonlinear

In this paper, we employ a two-step gradient method for solving fuzzy nonlinear equations. This method is Jacobian free and only requires a line search for . The fuzzy coefficients are presented in parametric form. Numerical experiments on well-known benchmark problems have been presented to illustrate the efficiency of the proposed method

Fuzzification of quantitative data to predict tumour size of colorectal cancer

Regression analysis has become more popular among researchers as a standard tool in analyzing data. This paper used fuzzy linear regression model (FLRM) to predict tumour size of colorectal cancer (CRC) data in Malaysia. 180 patients with colorectal cancer received treatment in hospital were recorded by nurses and doctors. Based on the patient records, a triangular fuzzy data will be built toward the size of the tumour. Mean square error (MSE) and root mean square error (RMSE) will be measured as a part of the process for predicting the size of the tumour. The degree of fitting adjusted is set between 0 and 1 in order to find the least error. It was found that the combination of FLRM model with fuzzy data provided a better prediction compared to the FLRM model alone. Hence, this study concluded that the tumour size is directly proportional to several factors such as gender, ethnic, icd 10, TNM staging, diabetes mellitus, Crohn’s disease

An impulsive approach for numerical investigation of hybrid fuzzy differential equations and intuitionistic treatment for fuzzy ordinary and partial differential equations

Many evolution processes are characterized by the fact that at certain moments of time, they experience a change of state abruptly. It is assume naturally, that those perturbations act instantaneously, in the form of impulses. The impulsive differential equations, by means differential equations involving impulse effects, are seen as a natural description of observed evolution phenomenon of several real world problems. For example, systems with impulse effect have applications in physics, biotechnolagy, industrial robotics, pharmacokinetics, population dynamics, ecology, optimal control production theory and many others. Therefore, it is beneficial to study the theory of impulsive differential equations as a well deserved discipline, due to the increase applications of impulsive differential equations in various fields in the future. However, in many mathematical modelling of the real world problems, fuzziness and impulsiveness occurs simultaneously. This problem would be better modelled by impulsive fuzzy differential equations. Therefore, this research applies the theory of impulsive fuzzy differential equations by combining the theories of impulsive differential equations and fuzzy differential equations. The numerical algorithms are developed and the solutions are verified by comparing the results with the analytical solutions. The novel method for the first order linear impulsive hzzy differential equations under generalized differentiability is also proposed analytically and numerically, The convergence theor~m for the impulsive fuzzy differential equations (FDE) under generalized differentiability is defined. In this study, Ant Colony Programming (ACP) was used to find the optimal solution of FDE. Results obtained show that the method is effective in solving fuzzy differential equation. The solution in this method is equivaIent to the exact solution of the problem. Modified Romberg's method and Modified Two-step Simpson's 318 method are used to solve FDE with hzzy IVP has been successfully derived. The result has been shown that Modified Rornberg's method gave smaller error than the Standard Euler's method. Therefore Modified Romberg's method can estimate the solution of fizzy differential equation more effectively than the Euler's method in solving fuzzy differential equation. Meanwhile, by using the modified wo-step Simpson's 318 methods, it has been shown that the solution of FDE provide more accurate approximation to the exact solution and it also gives better results than the Runge-Kutta method. In other words, Modified Twostep Simpson's 318 method is an effective method to solve fuzzy differential equation compared to the Runge-Kutta method

Effect in positioning gold nanoparticle inside Plasmonic Solar Cell on absorption, reflection and transmission

Gold nanoparticle has been explored in different ways to enhance the absorption of light and improve the efficiency of plasmonic solar cell. In this study, various positions of a gold nanoparticle which are at 115 nm, 230 nm and 305 nm measured vertically from the bottom of the solar cell to the centre of gold nanoparticle embedded into silicon layer of plasmonic solar cell is demonstrated using numerical simulation. The aim is to investigate the absorption, reflection and transmission percentage with different wavelength in different position of gold nanoparticle in plasmonic solar cell. The numerical results showed that the highest absorption and lowest reflection and transmission occurred at position 305 nm in the range 100 nm to 1000 nm compared to the simulation without nanoparticle and other position. The overall simulation results proved that at position 305 nm of gold nanoparticle which is near to the top layer is more efficient because this position has high electric field intensity in visible range

Analysis of plasmonic structure using finite element method

Plasmonic structure was investigated by modelling the different shape of v-groove shape and curve shape in three dimension (3D) using Finite Element Method (FEM). We simulate and compare the shape with electric and magnetic wave propagation in term of total electric energy and total magnetic energy using gold as a metal material and glass as a dielectric material. The result of the reflection and transmission of transverse electric (TE) wave and transverse magnetic (TM) wave with the angle of incidence are shown depending on the shape of plasmonic structure by using analytic solution based on Fresnel equation

Optical absorption of plasmonic cylindrical gold nanoparticle in hexagonal geometry

A high quality solar cell depends on how good the design of the solar cell can absorb light. In this study, cylindrical gold nanoparticles were embedded into indium tin oxide (ITO) layer and silicon layer arranged in hexagonal geometry on plasmonic solar cell simulation design. The aim is to investigate the optical absorption percentage in terms of wavelength and angle of incidence for the solar cell design. The numerical results showed that the highest absorption has occurred in 480 nm in the range of visible spectrum. In this wavelength, the highest absorption occurred at the incidence angle of 48 degree