Youssri H Youssri | Cairo University (original) (raw)

Papers by Youssri H Youssri

Contemporary mathematics, Jun 13, 2024

International Journal of Applied and Computational Mathematics, 2021

Arabian Journal of Mathematics, 2019

This work reports a collocation algorithm for the numerical solution of a Volterra–Fredholm integ... more This work reports a collocation algorithm for the numerical solution of a Volterra–Fredholm integral equation (V-FIE), using shifted Chebyshev collocation (SCC) method. Some properties of the shifted Chebyshev polynomials are presented. These properties together with the shifted Gauss–Chebyshev nodes were then used to reduce the Volterra–Fredholm integral equation to the solution of a matrix equation. Nextly, the error analysis of the proposed method is presented. We compared the results of this algorithm with others and showed the accuracy and potential applicability of the given method.

International Journal of Applied and Computational Mathematics, 2019

International Journal of Modern Physics C

The time-fractional diffusion equation is applied to a wide range of practical applications. We s... more The time-fractional diffusion equation is applied to a wide range of practical applications. We suggest using a potent spectral approach to solve this equation. These techniques’ main objective is to efficiently solve the linear time-fractional problem by transforming it into a system of linear algebraic equations in the expansion coefficients, together with the problem’s initial and boundary conditions. The main advantage of our technique is that the resulting linear systems have special structures which facilitate their computational solution. The numerical methods are supported by a thorough convergence study for the suggested Chebyshev expansion. Some test problems are offered to demonstrate the suggested methods’ broad applicability and a high degree of accuracy.

Computational and Applied Mathematics, 2018

We developed a numerical scheme to solve the variable-order fractional linear subdiffusion and no... more We developed a numerical scheme to solve the variable-order fractional linear subdiffusion and nonlinear reaction-subdiffusion equations using the shifted Jacobi collocation method. Basically, a time-space collocation approximation for temporal and spatial discretizations is employed efficiently to tackle these equations. The convergence and stability analyses of the suggested basis functions are presented in-depth. The validity and efficiency of the proposed method are investigated and verified through numerical examples.

Cmes-computer Modeling in Engineering & Sciences, 2015

In this paper, a new spectral algorithm for solving linear and nonlinear fractional-order initial... more In this paper, a new spectral algorithm for solving linear and nonlinear fractional-order initial value problems is established. The key idea for obtaining the suggested spectral numerical solutions for these equations is actually based on utilizing the ultraspherical wavelets along with applying the collocation method to reduce the fractional differential equation with its initial conditions into a system of linear or nonlinear algebraic equations in the unknown expansion coefficients. The convergence and error analysis of the suggested ultraspherical wavelets expansion are carefully discussed. For the sake of testing the proposed algorithm, some numerical examples are considered. The numerical results indicate that the resulting approximate solutions are close to the analytical solutions and they are more accurate than those obtained by some other existing techniques in literature.

International Journal of Applied and Computational Mathematics, 2021

This paper deals with the implementation and presentation of numerical solutions of fractional pa... more This paper deals with the implementation and presentation of numerical solutions of fractional pantograph differential equations (FPDEs) using generalized Lucas polynomials (GLPs). The derivation of our proposed algorithms is built on introducing an operational matrix of derivatives (OMDs) of the GLPs and after that employing it to convert the problem into an algebraic system of equations whose solution can be found through some suitable algorithms such as Gauss elimination and Newton–Raphson methods. Finally, by providing various illustrative examples, including comparisons with the results obtained by some other existing literature methods, the efficiency and applicability of our proposed algorithms are demonstrated.

International Journal of Computational Methods, 2018

In this paper, we analyze and implement a new efficient spectral Galerkin algorithm for handling ... more In this paper, we analyze and implement a new efficient spectral Galerkin algorithm for handling linear one-dimensional telegraph type equation. The principle idea behind this algorithm is to choose appropriate basis functions satisfying the underlying boundary conditions. This choice leads to systems with specially structured matrices which can be efficiently inverted. The proposed numerical algorithm is supported by a careful investigation for the convergence and error analysis of the suggested approximate double expansion. Some illustrative examples are given to demonstrate the wide applicability and high accuracy of the proposed algorithm.

Tbilisi Mathematical Journal, 2018

Iranian Journal of Science and Technology, Transactions A: Science, 2017

This paper concerns new numerical solutions for certain coupled system of fractional differential... more This paper concerns new numerical solutions for certain coupled system of fractional differential equations through the employment of the so-called generalized Fibonacci polynomials. These polynomials include two parameters and they generalize some important well-known polynomials such as Fibonacci, Pell, Fermat, second kind Chebyshev, and second kind Dickson polynomials. The proposed numerical algorithm is essentially built on applying the spectral tau method together with utilizing a Fejer quadrature formula. For the implementation of our algorithm, we introduce a new operational matrix of fractional-order differentiation of generalized Fibonacci polynomials. A careful investigation of convergence and error analysis of the proposed generalized Fibonacci expansion is performed. The robustness of the proposed algorithm is tested through presenting some numerical experiments.

Computational and Applied Mathematics, 2017

The principal aim of the current paper is to present and analyze two new spectral algorithms for ... more The principal aim of the current paper is to present and analyze two new spectral algorithms for solving some types of linear and nonlinear fractional-order differential equations. The proposed algorithms are obtained by utilizing a certain kind of shifted Chebyshev polynomials called the shifted fifth-kind Chebyshev polynomials as basis functions along with the application of a modified spectral tau method. The class of fifth-kind Chebyshev polynomials is a special class of a basic class of symmetric orthogonal polynomials which are constructed with the aid of the extended Sturm–Liouville theorem for symmetric functions. An investigation for the convergence and error analysis of the proposed Chebyshev expansion is performed. For this purpose, a new connection formulae between Chebyshev polynomials of the first and fifth kinds are derived. The obtained numerical results ascertain that our two proposed algorithms are applicable, efficient and accurate.

Contemporary Mathematics

In this study, we employ a rational Jacobi collocation technique to effectively address linear ti... more In this study, we employ a rational Jacobi collocation technique to effectively address linear time-fractional subdiffusion and reaction sub-diffusion equations. The semi-analytic approximation solution, in this case, represents the spatial and temporal variables as a series of rational Jacobi polynomials. Subsequently, we apply the operational collocation method to convert the target equations into a system of algebraic equations. A comprehensive investigation into the convergence properties of the dual series expansion employed in this approximation is conducted, demonstrating the robustness of the numerical method put forth. To illustrate the method's accuracy and practicality, we present several numerical examples. The advantages of this method are: high accuracy, efficiency, applicability, and high rate of convergence.

Fractal and Fractional

In this study, we present an innovative approach involving a spectral collocation algorithm to ef... more In this study, we present an innovative approach involving a spectral collocation algorithm to effectively obtain numerical solutions of the nonlinear time-fractional generalized Kawahara equation (NTFGKE). We introduce a new set of orthogonal polynomials (OPs) referred to as “Eighth-kind Chebyshev polynomials (CPs)”. These polynomials are special kinds of generalized Gegenbauer polynomials. To achieve the proposed numerical approximations, we first derive some new theoretical results for eighth-kind CPs, and after that, we employ the spectral collocation technique and incorporate the shifted eighth-kind CPs as fundamental functions. This method facilitates the transformation of the equation and its inherent conditions into a set of nonlinear algebraic equations. By harnessing Newton’s method, we obtain the necessary semi-analytical solutions. Rigorous analysis is dedicated to evaluating convergence and errors. The effectiveness and reliability of our approach are validated through ...

Mathematical Methods in the Applied Sciences

Fractal and Fractional

In this study, a spectral tau solution to the heat conduction equation is introduced. As basis fu... more In this study, a spectral tau solution to the heat conduction equation is introduced. As basis functions, the orthogonal polynomials, namely, the shifted fifth-kind Chebyshev polynomials (5CPs), are used. The proposed method’s derivation is based on solving the integral equation that corresponds to the original problem. The tau approach and some theoretical findings serve to transform the problem with its underlying conditions into a suitable system of equations that can be successfully solved by the Gaussian elimination method. For the applicability and precision of our suggested algorithm, some numerical examples are given.

Symmetry

This article proposes a numerical algorithm utilizing the spectral Tau method for numerically han... more This article proposes a numerical algorithm utilizing the spectral Tau method for numerically handling the Kawahara partial differential equation. The double basis of the fifth-kind Chebyshev polynomials and their shifted ones are used as basis functions. Some theoretical results of the fifth-kind Chebyshev polynomials and their shifted ones are used in deriving our proposed numerical algorithm. The nonlinear term in the equation is linearized using a new product formula of the fifth-kind Chebyshev polynomials with their first derivative polynomials. Some illustrative examples are presented to ensure the applicability and efficiency of the proposed algorithm. Furthermore, our proposed algorithm is compared with other methods in the literature. The presented numerical method results ensure the accuracy and applicability of the presented algorithm.

Symmetry, 2023

This article proposes a numerical algorithm utilizing the spectral Tau method for numerically han... more This article proposes a numerical algorithm utilizing the spectral Tau method for numerically handling the Kawahara partial differential equation. The double basis of the fifth-kind Chebyshev polynomials and their shifted ones are used as basis functions. Some theoretical results of the fifth-kind Chebyshev polynomials and their shifted ones are used in deriving our proposed numerical algorithm. The nonlinear term in the equation is linearized using a new product formula of the fifth-kind Chebyshev polynomials with their first derivative polynomials. Some illustrative examples are presented to ensure the applicability and efficiency of the proposed algorithm. Furthermore, our proposed algorithm is compared with other methods in the literature. The presented numerical method results ensure the accuracy and applicability of the presented algorithm.

In this research article, a novel operational matrix of fractional-order differentiation of Lucas... more In this research article, a novel operational matrix of fractional-order differentiation of Lucas polynomials in the Caputo sense is established. Based on this matrix along with the application of tau and collocation spectral methods, two efficient numerical algorithms for solving multi-term fractional differential equations are proposed and analyzed. Some new formulae for Lucas polynomials are stated and proved for investigating the new algorithms. The convergence and error analysis of the suggested Lucas expansion are investigated carefully. Some new inequalities including the modified Bessel function of the first kind and the well-known golden ratio are stated and proved. Some numerical tests are carried out for some specific and important types of problems including the Bagley-Torvik, Ricatti, Lane-Emden and oscillator equations. The results obtained are compared with some existing ones in open literature and it is noticed that the two proposed algorithms are robust, accurate an...

Journal of Function Spaces

The main goal of this paper is to develop a new formula of the fractional derivatives of the shif... more The main goal of this paper is to develop a new formula of the fractional derivatives of the shifted Chebyshev polynomials of the third kind. This new formula expresses approximately the fractional derivatives of these polynomials in the Caputo sense in terms of their original ones. The linking coefficients are given in terms of a certain 4 F 3 1 terminating hypergeometric function. The integer derivatives of the shifted third-kind Chebyshev polynomials can be calculated using this formula after performing some reductions. To solve a nonlinear fractional pantograph differential equation with quadratic nonlinearity, the fractional derivative formula is used in conjunction with the tau technique. The role of the tau method is to convert the pantograph differential equation with its governing initial/boundary conditions into a nonlinear system of algebraic equations that can be treated with the aid of Newton’s iterative scheme. To test the method’s convergence, certain estimations ar...

Contemporary mathematics, Jun 13, 2024

International Journal of Applied and Computational Mathematics, 2021

Arabian Journal of Mathematics, 2019

This work reports a collocation algorithm for the numerical solution of a Volterra–Fredholm integ... more This work reports a collocation algorithm for the numerical solution of a Volterra–Fredholm integral equation (V-FIE), using shifted Chebyshev collocation (SCC) method. Some properties of the shifted Chebyshev polynomials are presented. These properties together with the shifted Gauss–Chebyshev nodes were then used to reduce the Volterra–Fredholm integral equation to the solution of a matrix equation. Nextly, the error analysis of the proposed method is presented. We compared the results of this algorithm with others and showed the accuracy and potential applicability of the given method.

International Journal of Applied and Computational Mathematics, 2019

International Journal of Modern Physics C

The time-fractional diffusion equation is applied to a wide range of practical applications. We s... more The time-fractional diffusion equation is applied to a wide range of practical applications. We suggest using a potent spectral approach to solve this equation. These techniques’ main objective is to efficiently solve the linear time-fractional problem by transforming it into a system of linear algebraic equations in the expansion coefficients, together with the problem’s initial and boundary conditions. The main advantage of our technique is that the resulting linear systems have special structures which facilitate their computational solution. The numerical methods are supported by a thorough convergence study for the suggested Chebyshev expansion. Some test problems are offered to demonstrate the suggested methods’ broad applicability and a high degree of accuracy.

Computational and Applied Mathematics, 2018

We developed a numerical scheme to solve the variable-order fractional linear subdiffusion and no... more We developed a numerical scheme to solve the variable-order fractional linear subdiffusion and nonlinear reaction-subdiffusion equations using the shifted Jacobi collocation method. Basically, a time-space collocation approximation for temporal and spatial discretizations is employed efficiently to tackle these equations. The convergence and stability analyses of the suggested basis functions are presented in-depth. The validity and efficiency of the proposed method are investigated and verified through numerical examples.

Cmes-computer Modeling in Engineering & Sciences, 2015

In this paper, a new spectral algorithm for solving linear and nonlinear fractional-order initial... more In this paper, a new spectral algorithm for solving linear and nonlinear fractional-order initial value problems is established. The key idea for obtaining the suggested spectral numerical solutions for these equations is actually based on utilizing the ultraspherical wavelets along with applying the collocation method to reduce the fractional differential equation with its initial conditions into a system of linear or nonlinear algebraic equations in the unknown expansion coefficients. The convergence and error analysis of the suggested ultraspherical wavelets expansion are carefully discussed. For the sake of testing the proposed algorithm, some numerical examples are considered. The numerical results indicate that the resulting approximate solutions are close to the analytical solutions and they are more accurate than those obtained by some other existing techniques in literature.

International Journal of Applied and Computational Mathematics, 2021

This paper deals with the implementation and presentation of numerical solutions of fractional pa... more This paper deals with the implementation and presentation of numerical solutions of fractional pantograph differential equations (FPDEs) using generalized Lucas polynomials (GLPs). The derivation of our proposed algorithms is built on introducing an operational matrix of derivatives (OMDs) of the GLPs and after that employing it to convert the problem into an algebraic system of equations whose solution can be found through some suitable algorithms such as Gauss elimination and Newton–Raphson methods. Finally, by providing various illustrative examples, including comparisons with the results obtained by some other existing literature methods, the efficiency and applicability of our proposed algorithms are demonstrated.

International Journal of Computational Methods, 2018

In this paper, we analyze and implement a new efficient spectral Galerkin algorithm for handling ... more In this paper, we analyze and implement a new efficient spectral Galerkin algorithm for handling linear one-dimensional telegraph type equation. The principle idea behind this algorithm is to choose appropriate basis functions satisfying the underlying boundary conditions. This choice leads to systems with specially structured matrices which can be efficiently inverted. The proposed numerical algorithm is supported by a careful investigation for the convergence and error analysis of the suggested approximate double expansion. Some illustrative examples are given to demonstrate the wide applicability and high accuracy of the proposed algorithm.

Tbilisi Mathematical Journal, 2018

Iranian Journal of Science and Technology, Transactions A: Science, 2017

This paper concerns new numerical solutions for certain coupled system of fractional differential... more This paper concerns new numerical solutions for certain coupled system of fractional differential equations through the employment of the so-called generalized Fibonacci polynomials. These polynomials include two parameters and they generalize some important well-known polynomials such as Fibonacci, Pell, Fermat, second kind Chebyshev, and second kind Dickson polynomials. The proposed numerical algorithm is essentially built on applying the spectral tau method together with utilizing a Fejer quadrature formula. For the implementation of our algorithm, we introduce a new operational matrix of fractional-order differentiation of generalized Fibonacci polynomials. A careful investigation of convergence and error analysis of the proposed generalized Fibonacci expansion is performed. The robustness of the proposed algorithm is tested through presenting some numerical experiments.

Computational and Applied Mathematics, 2017

The principal aim of the current paper is to present and analyze two new spectral algorithms for ... more The principal aim of the current paper is to present and analyze two new spectral algorithms for solving some types of linear and nonlinear fractional-order differential equations. The proposed algorithms are obtained by utilizing a certain kind of shifted Chebyshev polynomials called the shifted fifth-kind Chebyshev polynomials as basis functions along with the application of a modified spectral tau method. The class of fifth-kind Chebyshev polynomials is a special class of a basic class of symmetric orthogonal polynomials which are constructed with the aid of the extended Sturm–Liouville theorem for symmetric functions. An investigation for the convergence and error analysis of the proposed Chebyshev expansion is performed. For this purpose, a new connection formulae between Chebyshev polynomials of the first and fifth kinds are derived. The obtained numerical results ascertain that our two proposed algorithms are applicable, efficient and accurate.

Contemporary Mathematics

In this study, we employ a rational Jacobi collocation technique to effectively address linear ti... more In this study, we employ a rational Jacobi collocation technique to effectively address linear time-fractional subdiffusion and reaction sub-diffusion equations. The semi-analytic approximation solution, in this case, represents the spatial and temporal variables as a series of rational Jacobi polynomials. Subsequently, we apply the operational collocation method to convert the target equations into a system of algebraic equations. A comprehensive investigation into the convergence properties of the dual series expansion employed in this approximation is conducted, demonstrating the robustness of the numerical method put forth. To illustrate the method's accuracy and practicality, we present several numerical examples. The advantages of this method are: high accuracy, efficiency, applicability, and high rate of convergence.

Fractal and Fractional

In this study, we present an innovative approach involving a spectral collocation algorithm to ef... more In this study, we present an innovative approach involving a spectral collocation algorithm to effectively obtain numerical solutions of the nonlinear time-fractional generalized Kawahara equation (NTFGKE). We introduce a new set of orthogonal polynomials (OPs) referred to as “Eighth-kind Chebyshev polynomials (CPs)”. These polynomials are special kinds of generalized Gegenbauer polynomials. To achieve the proposed numerical approximations, we first derive some new theoretical results for eighth-kind CPs, and after that, we employ the spectral collocation technique and incorporate the shifted eighth-kind CPs as fundamental functions. This method facilitates the transformation of the equation and its inherent conditions into a set of nonlinear algebraic equations. By harnessing Newton’s method, we obtain the necessary semi-analytical solutions. Rigorous analysis is dedicated to evaluating convergence and errors. The effectiveness and reliability of our approach are validated through ...

Mathematical Methods in the Applied Sciences

Fractal and Fractional

In this study, a spectral tau solution to the heat conduction equation is introduced. As basis fu... more In this study, a spectral tau solution to the heat conduction equation is introduced. As basis functions, the orthogonal polynomials, namely, the shifted fifth-kind Chebyshev polynomials (5CPs), are used. The proposed method’s derivation is based on solving the integral equation that corresponds to the original problem. The tau approach and some theoretical findings serve to transform the problem with its underlying conditions into a suitable system of equations that can be successfully solved by the Gaussian elimination method. For the applicability and precision of our suggested algorithm, some numerical examples are given.

Symmetry

This article proposes a numerical algorithm utilizing the spectral Tau method for numerically han... more This article proposes a numerical algorithm utilizing the spectral Tau method for numerically handling the Kawahara partial differential equation. The double basis of the fifth-kind Chebyshev polynomials and their shifted ones are used as basis functions. Some theoretical results of the fifth-kind Chebyshev polynomials and their shifted ones are used in deriving our proposed numerical algorithm. The nonlinear term in the equation is linearized using a new product formula of the fifth-kind Chebyshev polynomials with their first derivative polynomials. Some illustrative examples are presented to ensure the applicability and efficiency of the proposed algorithm. Furthermore, our proposed algorithm is compared with other methods in the literature. The presented numerical method results ensure the accuracy and applicability of the presented algorithm.

Symmetry, 2023

This article proposes a numerical algorithm utilizing the spectral Tau method for numerically han... more This article proposes a numerical algorithm utilizing the spectral Tau method for numerically handling the Kawahara partial differential equation. The double basis of the fifth-kind Chebyshev polynomials and their shifted ones are used as basis functions. Some theoretical results of the fifth-kind Chebyshev polynomials and their shifted ones are used in deriving our proposed numerical algorithm. The nonlinear term in the equation is linearized using a new product formula of the fifth-kind Chebyshev polynomials with their first derivative polynomials. Some illustrative examples are presented to ensure the applicability and efficiency of the proposed algorithm. Furthermore, our proposed algorithm is compared with other methods in the literature. The presented numerical method results ensure the accuracy and applicability of the presented algorithm.

In this research article, a novel operational matrix of fractional-order differentiation of Lucas... more In this research article, a novel operational matrix of fractional-order differentiation of Lucas polynomials in the Caputo sense is established. Based on this matrix along with the application of tau and collocation spectral methods, two efficient numerical algorithms for solving multi-term fractional differential equations are proposed and analyzed. Some new formulae for Lucas polynomials are stated and proved for investigating the new algorithms. The convergence and error analysis of the suggested Lucas expansion are investigated carefully. Some new inequalities including the modified Bessel function of the first kind and the well-known golden ratio are stated and proved. Some numerical tests are carried out for some specific and important types of problems including the Bagley-Torvik, Ricatti, Lane-Emden and oscillator equations. The results obtained are compared with some existing ones in open literature and it is noticed that the two proposed algorithms are robust, accurate an...

Journal of Function Spaces

The main goal of this paper is to develop a new formula of the fractional derivatives of the shif... more The main goal of this paper is to develop a new formula of the fractional derivatives of the shifted Chebyshev polynomials of the third kind. This new formula expresses approximately the fractional derivatives of these polynomials in the Caputo sense in terms of their original ones. The linking coefficients are given in terms of a certain 4 F 3 1 terminating hypergeometric function. The integer derivatives of the shifted third-kind Chebyshev polynomials can be calculated using this formula after performing some reductions. To solve a nonlinear fractional pantograph differential equation with quadratic nonlinearity, the fractional derivative formula is used in conjunction with the tau technique. The role of the tau method is to convert the pantograph differential equation with its governing initial/boundary conditions into a nonlinear system of algebraic equations that can be treated with the aid of Newton’s iterative scheme. To test the method’s convergence, certain estimations ar...