Numerical methods for fractional differential equations by new caputo and hadamard types operators

Fractional ordinary differential equation (FODE) and fractional partial differential equation (FPDE) emerges in various modelling of physics phenomena. Over past decades, several fractional derivative and operator has been introduced such as Caputo-Fabrizio operator, Caputo-Hadamard fractional derivative, Marchaud fractional derivative or Caputo fractional derivative. The fractional differential equations defined in these fractional derivatives and operators definition are difficult or impossible to solve analytically. Therefore, we seek after highly accurate numerical scheme in efficient ways such as predictor-corrector method, finite difference scheme and spectral collocation method in this research for FODE and FPDE. Caputo-Fabrizo operator is a definition which is verified that does not fit the usual concept neither for fractional nor for integer derivative integral. The main interest of this operator is having regular kernel and which is a necessity of using a model describing the behavior of classical viscoelastic materials, electromagnetic system and viscoelastic materials. Furthermore, associated integral for Caputo�Fabrizio operator is also presented using Laplace transform and Inverse Laplace transform. Hence, we first introduce predictor-corrector scheme involving Caputo�Fabrizo operator, α > 0 which represents higher order of approximatio

On the new properties of Caputo–Fabrizio operator and its application in deriving shifted Legendre operational matrix

In this paper, we study the recently introduced Caputo and Fabrizio operator, which this new operator was derived by replacing the singular kernel in the classical Caputo derivative with the regular kernel. We introduce some useful properties based on the definition by Caputo and Fabrizio for a general order n < α < n + 1, n ∈ N. Here, we extend the associated integral of Caputo–Fabrizio sense to n < α < n + 1, n ∈ N. We also find the general formula for the Caputo–Fabrizio operator of (t − a)β . Then, we derive Legendre operational matrix based on this new operator and together with Tau method, we use it to solve the differential equations defined in the Caputo–Fabrizio sense. As far as we know, the operational matrix method has yet been derived or attempted for solving the differential equations in Caputo–Fabrizio sense, while it has been successfully used to solve fractional calculus problems involving the classical Caputo sense. Some numerical examples are given to display the simplicity and accuracy of the proposed technique

New predictor-corrector scheme for solving nonlinear differential equations with Caputo-Fabrizio operator

In this paper, we develop a new, simple, and accurate scheme to obtain approximate solution for nonlinear differential equation in the sense of Caputo-Fabrizio operator. To derive this new predictor-corrector scheme, which suits on Caputo-Fabrizio operator, firstly, we obtain the corresponding initial value problem for the differential equation in the Caputo-Fabrizio sense. Hence, by fractional Euler method and fractional trapeziodal rule, we obtain the predictor formula as well as corrector formula. Error analysis for this new method is derived. To test the validity and simplicity of this method, some illustrative examples for nonlinear differential equations are solved

Assessment of Urban Noise in School Environments - Case Study in Batu Pahat, Johor

In recent decades, urban noise has become well-known as one of the critical problems affecting the quality of life in urban areas around the world. Noise assessment is becoming more common in Malaysia as many cities in this country become highly populated and industrialized with serious noise pollution issues, particularly in schools. The aims of this study are to identify the factors and effects of noise in the school environments and evaluate the acoustic comfort of the teachers during the teaching lesson in the school environment. A total of 3 schools located in the urban area of Batu Pahat, Johor, which were Sekolah Menengah Kebangsaan (SMK), Tinggi Batu Pahat, SMK Tun Aminah and SMK Semerah were chosen as study areas. In the present study, questionnaires survey using Google Form were distributed to teachers working in the selected schools. Expert reviews and a pilot study were carried out before the actual survey. The results indicated that the noise factors were coming from inside and outside the classroom, mainly from traffic noise and heavy vehicles near the school environment and noise from student activities and chattering. Teachers need to raise their voice during the teaching and learning process, and some of them had a sore throat.&nbsp; SMK Tinggi Batu Pahat and SMK Tun Aminah teachers were uncomfortable with the existing acoustic comfort of the school environments

Assessment of Urban Noise in School Environments - Case Study in Batu Pahat, Johor

In recent decades, urban noise has become well-known as one of the critical problems affecting the quality of life in urban areas around the world. Noise assessment is becoming more common in Malaysia as many cities in this country become highly populated and industrialized with serious noise pollution issues, particularly in schools. The aims of this study are to identify the factors and effects of noise in the school environments and evaluate the acoustic comfort of the teachers during the teaching lesson in the school environment. A total of 3 schools located in the urban area of Batu Pahat, Johor, which were Sekolah Menengah Kebangsaan (SMK), Tinggi Batu Pahat, SMK Tun Aminah and SMK Semerah were chosen as study areas. In the present study, questionnaires survey using Google Form were distributed to teachers working in the selected schools. Expert reviews and a pilot study were carried out before the actual survey. The results indicated that the noise factors were coming from inside and outside the classroom, mainly from traffic noise and heavy vehicles near the school environment and noise from student activities and chattering. Teachers need to raise their voice during the teaching and learning process, and some of them had a sore throat.&nbsp; SMK Tinggi Batu Pahat and SMK Tun Aminah teachers were uncomfortable with the existing acoustic comfort of the school environments