Numerical solution of fractional partial differential equations by spectral methods

Fractional partial differential equations (FPDEs) have become essential tool for the modeling of physical models by using spectral methods. In the last few decades, spectral methods have been developed for the solution of time and space dimensional FPDEs. There are different types of spectral methods such as collocation methods, Tau methods and Galerkin methods. This research work focuses on the collocation and Tau methods to propose an efficient operational matrix methods via Genocchi polynomials and Legendre polynomials for the solution of two and three dimensional FPDEs. Moreover, in this study, Genocchi wavelet-like basis method and Genocchi polynomials based Ritz- Galerkin method have been derived to deal with FPDEs and variable- order FPDEs. The reason behind using the Genocchi polynomials is that, it helps to generate functional expansions with less degree and small coefficients values to derive the operational matrix of derivative with less computational complexity as compared to Chebyshev and Legendre Polynomials. The results have been compared with the existing methods such as Chebyshev wavelets method, Legendre wavelets method, Adomian decomposition method, Variational iteration method, Finite difference method and Finite element method. The numerical results have revealed that the proposed methods have provided the better results as compared to existing methods due to minimum computational complexity of derived operational matrices via Genocchi polynomials. Additionally, the significance of the proposed methods has been verified by finding the error bound, which shows that the proposed methods have provided better approximation values for under consideration FPDEs

Numerical solution of fractional diffusion wave equation and fractional klein–gordon equation via two-dimensional genocchi polynomials with a ritz–galerkin method

In this paper, two-dimensional Genocchi polynomials and the Ritz–Galerkin method were developed to investigate the Fractional Diffusion Wave Equation (FDWE) and the Fractional Klein–Gordon Equation (FKGE). A satisfier function that satisfies all the initial and boundary conditions was used. A linear system of algebraic equations was obtained for the considered equation with the help of two-dimensional Genocchi polynomials along with the Ritz–Galerkin method. The FDWE and FKGE, including the nonlinear case, were reduced to solve the linear system of the algebraic equation. Hence, the proposed method was able to greatly reduce the complexity of the problems and provide an accurate solution. The effectiveness of the proposed technique is demonstrated through several examples

Investigation of cutting temperature and cutting force from mist flow pattern in MQL technique

Minimum Quantity Lubrication (MQL) is an alternative method to supply the cutting fluid in the formation of mist. MQL has proven to reduce machining cost and increase machining performance. Previous research have stated that machining performance is affected by the lubricant type, flow rate, the distance between nozzle and tool tip, and the workpiece material. These important parameters are not reported in many research documents. MQL is known for its many benefits but no one was able to prove that the statement is true or ever suggested a systematic procedure to prove MQL’s efficiency. The effectiveness and the working principle of MQL are still questionable with very few explanations provided. The present study is about investigation of cutting temperature and cutting force from mist flow pattern in MQL technique The MQL nozzle distance and cutting fluid flow pattern are among the factors that can provide optimum machining performance in term of cutting force and cutting temperature. The objective of this study is to conduct machining process using MQL technique with different combination of spray parameters and to optimize spray parameters for minimum machining temperature and cutting forces. The four nozzle distances of 3, 6, 7 and 9 mm were selected based on the results obtained from Phase Doppler Anemometry (PDA). The machining performance was evaluated under three levels of cutting speed and two levels of feed rate at constant depth of cut. The cutting force was measured using a set of dynamometer and cutting temperature using thermal imager. The most suitable mist flow pattern during machining was the largest spray cone angle supplied under 0.4 MPa input air pressure. The results obtained from the machining process shows a significant reduction of cutting force and cutting temperature at the nozzle distance in the range of 6 to 9 mm under 0.4 MPa input air pressure for larger diameter OD30 nozzle

Fractal-fractional advection–diffusion–reaction equations by Ritz approximation approach

In this work, we propose the Ritz approximation approach with a satisfier function to solve fractalfractional advection–diffusion–reaction equations. The approach reduces fractal-fractional advection–diffusion– reaction equations to a system of algebraic equations; hence, the system can be solved easily to obtain the numerical solution for fractal-fractional advection–diffusion–reaction equations. With only a few terms of two variables-shifted Legendre polynomials, this method is capable of providing high-accuracy solution for fractal-fractional advection–diffusion–reaction equations. Numerical examples show that this approach is comparable with the existing numerical method. The proposed approach can reduce the number of terms of polynomials needed for numerical simulation to obtain the solution for fractal-fractional advection–diffusion–reaction equations

Fractal-fractional advection–diffusion–reaction equations by Ritz approximation approach

: In this work, we propose the Ritz approximation approach with a satisfier function to solve fractalfractional advection–diffusion–reaction equations. The approach reduces fractal-fractional advection–diffusion– reaction equations to a system of algebraic equations; hence, the system can be solved easily to obtain the numerical solution for fractal-fractional advection–diffusion–reaction equations. With only a few terms of two variables-shifted Legendre polynomials, this method is capable of providing high-accuracy solution for fractal-fractional advection–diffusion–reaction equations. Numerical examples show that this approach is comparable with the existing numerical method. The proposed approach can reduce the number of terms of polynomials needed for numerical simulation to obtain the solution for fractal-fractional advection–diffusion–reaction equations

Numerical Solution of Fractional Diffusion Wave Equation and Fractional Klein–Gordon Equation via Two-Dimensional Genocchi Polynomials with a Ritz–Galerkin Method

In this paper, two-dimensional Genocchi polynomials and the Ritz–Galerkin method were developed to investigate the Fractional Diffusion Wave Equation (FDWE) and the Fractional Klein–Gordon Equation (FKGE). A satisfier function that satisfies all the initial and boundary conditions was used. A linear system of algebraic equations was obtained for the considered equation with the help of two-dimensional Genocchi polynomials along with the Ritz–Galerkin method. The FDWE and FKGE, including the nonlinear case, were reduced to solve the linear system of the algebraic equation. Hence, the proposed method was able to greatly reduce the complexity of the problems and provide an accurate solution. The effectiveness of the proposed technique is demonstrated through several examples

Fractal-fractional advection–diffusion–reaction equations by Ritz approximation approach

: In this work, we propose the Ritz approximation approach with a satisfier function to solve fractalfractional advection–diffusion–reaction equations. The approach reduces fractal-fractional advection–diffusion–reaction equations to a system of algebraic equations; hence, the system can be solved easily to obtain the numerical solution for fractal-fractional advection–diffusion–reaction equations. With only a few terms of two variables-shifted Legendre polynomials, this method is capable of providing high-accuracy solution for fractal-fractional advection–diffusion–reaction equations. Numerical examples show that this approach is comparable with the existing numerical method. The proposed approach can reduce the number of terms of polynomials needed for numerical simulation to obtain the solution for fractal-fractional advection–diffusion–reaction equations

کئی چاند تھے سرِآسماں ،حلیہ وسراپا نگاری کا مرقع: KAI CHAND THE SAR-E-AASMAN , A MAGNUM OPUS OF DESCRIPTIVE NARRATION AND THE ART OF CHARACTERIZATION

Shams ur Rahman Farooqi was an epoch-making literary figure. His literary prowess is highly praised in all academic and literary circles. His novel "Kai Chand The Sar-e-Aasman" is considered a portrait gallery of Delhi's rich culture, civilization and social structure. In this novel, he presents the events of Delhi's final gathering. Farooqi has tried to sketch characters who are very close to reality. His observation is very unique and he has drawn numerous characters who look lively creature to his readers. The protagonist lady, Wazir Khanum, in novel "Kai Chand The Sar-e-Aasman," is also a very minutely depicted character, and it is a great example of Farooqi's mastery and proficiency in his art of descriptive writing. Upon aesthetics examination, the novel is a masterpiece of aesthetics work of art. Shams ur Rahman Farooqi's prose and narration style reflects refined and exquisite taste of Delhi's people. So "Kai Chand the Sar e Asman" can rightly be said a magnum opus of descriptive narration and art of characterization.&nbsp