Shifted Legendre Functions Research Articles

An effective QLM-based Legendre matrix algorithm to solve the coupled system of fractional-order Lane-Emden equations

The purpose of this study is to propose a computationally effective algorithm for the numerical evaluation of a fractional-order system of singular Lane-Emden type equations arising in physical problems. The fractional operator considered is in the sense of the Liouville-Caputo derivative. The presented matrix collocation method is based upon a combination of the quasilinearization method (QLM) and the shifted Legendre functions (SLFs) and is called QLM-SLFs method. By applying first the QLM to the nonlinear underlying system, we get a family of linear equations. Hence, a spectral matrix collocation scheme relied on the SLFs is designed to solve the resulting sequence of linear system of equations at very few iterations. The uniform convergence of the shifted Legendre expansion series solution is established. To illustrate the effectiveness of the proposed QLM-SLFs technique in the present paper, three test examples are carried out. The applicability and validity of the proposed method are testified through comparisons with the outcomes of other existing procedures in the literature. The proposed QLM-SLFs method is efficient and easy to implement. The approximation obtained by the method also converges quickly to the solutions of the underlying model problem. In comparison with available existing computational procedures, the QLM-SLFs approach shows that the use of Legendre functions together with QLM provides solutions with high accuracy and exponential convergence rate.

Fractional shifted legendre tau method to solve linear and nonlinear variable-order fractional partial differential equations

Here, we shed light on the fractional linear and nonlinear Klein–Gorden partial differential equations via Fractional Shifted Legendre Tau Method. With this objective, the operational matrices of fractional-order shifted Legendre functions (FSLFs) are derived and combined with the Tau method to convert the fractional-order differential equations to a system of solvable algebraic equations. The validity and the efficiency of the operational matrices are tested. Our findings yield an affirmative consequence, indicating applicability of the proposed method for nonlinear equations appearing in science and engineering.

Numerical solution of a class of two-dimensional nonlinear Volterra integral equations using Legendre polynomials

In this paper, a method for finding an approximate solution of a class of two-dimensional nonlinear Volterra integral equations is discussed. The properties of two-dimensional shifted Legendre functions are presented. The operational matrices of integration and product together with the collocation points are utilized to reduce the solution of the integral equation to the solution of a system of nonlinear algebraic equations. Some results concerning the error analysis are obtained. We also consider the application of the method to the solution of certain partial differential equations. Illustrative examples are included to demonstrate the validity and applicability of the technique.

Determination of the heat transfer coefficient and thermal conductivity for coconut kernels using an inverse method with a developed hemispherical shell model

Simulation studies for improved efficiency of thermal food processes (dryers) require knowledge of the heat transfer coefficient and thermal conductivity of the food. A novel analytical solution to the heat transfer problem in a hemispherical shell has been developed using the shifted Legendre functions. This was fitted to experimental data to determine the heat transfer coefficient and thermal conductivity of coconut by an inverse method. To minimize errors the natural shape of the food was taken into account. Neither the values nor the solution have been reported to-date. The values of heat transfer coefficients ( h) and thermal conductivities ( k) were found to be 17 ⩽ h 2 ⩽ 36, 4.0 ⩽ h 2 ∗ ⩽11 and 7.0 ⩽ h 1 ⩽ 15, 1.2 ⩽ h 1 ∗ ⩽3.0 W/m 2 °C for the outer and inner surfaces, and 0.125 ⩽ k ⩽ 0.127 and 0.030 ⩽ k ∗⩽0.020 W/m °C, respectively. The maximum temperature occurred at the outer equatorial surface and the minimum at the middle near to the pole. In the future these thermal parameters can be used to develop coconut dryers with improved energy efficiency and the method can be used to determine parameters for foods of similar shape.

The Legendre‐tau technique for the determination of a source parameter in a semilinear parabolic equation

A numerical procedure for an inverse problem concerning diffusion equation with source control parameter is considered. The proposed method is based on shifted Legendre‐tau technique. Our approach consists of reducing the problem to a set of algebraic equations by expanding the approximate solution as a shifted Legendre function with unknown coefficients. The operational matrices of integral and derivative together with the tau method are then utilized to evaluate the unknown coefficients of shifted Legendre functions. Illustrative examples are included to demonstrate the validity and applicability of the presented technique.

Quantum Algorithms for Weighing Matrices and Quadratic Residues

In this article we investigate how we can employ the structure of combinatorial objects like Hadamard matrices and weighing matrices to device new quantum algorithms. We show how the properties of a weighing matrix can be used to construct a problem for which the quantum query complexity is ignificantly lower than the classical one. It is pointed out that this scheme captures both Bernstein & Vazirani's inner-product protocol, as well as Grover's search algorithm. In the second part of the article we consider Paley's construction of Hadamard matrices, which relies on the properties of quadratic characters over finite fields. We design a query problem that uses the Legendre symbol chi (which indicates if an element of a finite field F_q is a quadratic residue or not). It is shown how for a shifted Legendre function f_s(i)=chi(i+s), the unknown s in F_q can be obtained exactly with only two quantum calls to f_s. This is in sharp contrast with the observation that any classical, probabilistic procedure requires more than log(q) + log((1-e)/2) queries to solve the same problem.

Optimal control of lumped parameter systems via shifted Legendre polynomial approximation

Shifted Legendre polynomial functions are employed to solve the linear-quadratic optimal control problem for lumped parameter system. Using the characteristics of the shifted Legendre polynomials, the system equations and the adjoint equations of the optimal control problem are reduced to functional ordinary differential equations. The solution of the functional differential equations are obtained in a series of the shifted Legendre functions. The operational matrix for the integration of the shifted Legendre polynomial functions is also introduced in the simulation step in order to simplify the computational procedure. An illustrative example of an optimal control problem is given, and the computational results are compared with those of the exact solution. The proposed method is effective and accurate.

Analysis of stiff systems via method of shifted Legendre functions

The method of shifted Legendre functions is successfully applied to the analysis of stiff systems. The basic idea is that the state variables are expressed in terms of shifted Legendre series. A powerful calculation algorithm is proposed to integrate the stiff equation with very accurate results for any length of time. A very simple recursive formula is developed to calculate the state functions. Taking only the single term (or second term) of expansion coefficients is enough to calculate the state functions. Illustrative examples are given. The computational results are compared with that of other numerical methods. Very satisfactory results are obtained.

Analysis of stiff systems via method of shifted Legendre functions

The method of shifted Legendre functions is successfully applied to the analysis of stiff systems. The basic idea is that the state variables are expressed in terms of shifted Legendre series. A powerful calculation algorithm is proposed to integrate the stiff equation with very accurate results for any length of time. A very simple recursive formula is developed to calculate the state functions. Taking only the single term (or second term) of expansion coefficients is enough to calculate the state functions. Illustrative examples are given. The computational results are compared with that of other numerical methods. Very satisfactory results are obtained.

Solution of population balance equation of breakage model via shifted Legendre functions

The stretched functional ordinary differential equations of population balance which can be used to describe the crystallization process of breakage models are solved by shifted Legendre functions. The key method is that the population density function is expressed by a series of shifted Legendre functions. The solution of the population balance equation is obtained by first transforming the ordinary differential equation into a series of algebraic equations of expansion coefficients which can be solved easily. The calculation procedure is simple and straightforward owing to the development of the recursive algorithm for the integration of the triple product of the shifted Legendre functions. Very satisfactory results of population density functions are obtained when they are compared with the numerical values obtained by MWR and the block pulse function.

Shifted Legendre function approximation of differential equations; application to crystallization processes

An effective new method solves either the stretched functional ordinary differential equation describing a crystallization process using breakage models, or the partial differential equations for a population balance which represent the transient crystallization process. The population balance density function is first expanded into a series of shifted Legendre polynomial functions. The partial differential equation (or the ordinary differential equation) is first transformed into a series of ordinary differential equations (or of algebraic equations) for the expansion coefficients which are solved by numerical computation. The computational time is greatly reduced through a recursive algorithm for the integration of the triple product of the shifted Legendre polynomial functions. Illustrative examples are given and the results are compared with data available in the literature. The proposed method is powerful, accurate and more precise than previously documented ones.

Model Reduction and Control System Design by Shifted Legendre Polynomial Functions

A method of model reduction for reducing a higher order transfer function to its lower order model is developed based on the shifted Legendre function approximation. The expansion coefficients of the shifted Legendre series which represents the approximate responses of transfer functions are computed by the recursive formula, via operational matrix approach. The significances of applying shifted Legendre function to model reduction problem are that the method is simple, straightforward and the computational results obtained are accurate as well as the final time of the control system can be adjustable without any restriction. Based on the model reduction technique, a new algebraic method is proposed for the design of a feedback control system to satisfy specifications. Illustrative examples are given and satisfactory results are obtained.