International Journal of Analysis and Applications
The Double Sadik Transform (DST) represents a generalized double integral transform that has emer... more The Double Sadik Transform (DST) represents a generalized double integral transform that has emerged as a highly effective analytical technique for solving numerous scientific problems. This study aims to investigate the DST applied to elementary functions and explore its notable properties, including its duality with the Double Laplace Transform and its capability to transform shifting functions, periodic functions, and convolution functions. Furthermore, the DST methodology is employed to resolve prominent linear fractional Caputo partial differential equations with known solutions commonly encountered in diverse mathematical models. The obtained outcomes are expressed in exact closed form, with the most precise results articulated through the Mittag-Leffler function. These results serve to validate the effectiveness and efficiency of the DST approach, establishing it as a valuable tool for addressing scientific problems involving fractional calculus.
In this manuscript, the Laplace Adomian decomposition method (LADM) is presented to solve Newell-... more In this manuscript, the Laplace Adomian decomposition method (LADM) is presented to solve Newell-Whitehead-Segel equation. The method can be applied to linear and nonlinear problems. Some examples have been carried out in order to illustrate the efficiency and reliability of the method.
This article presents the concept of ı ⋆ -continuous multifunctions in ideal topological spaces. ... more This article presents the concept of ı ⋆ -continuous multifunctions in ideal topological spaces. Especially, several characterizations of ı ⋆ -continuous multifunctions are investigated. Moreover, the relationships between ı ⋆ -continuity and the other types of continuity for multifunctions are established.
The aim of this paper is to introduce and study upper and lower almost γ-M continuous multifuncti... more The aim of this paper is to introduce and study upper and lower almost γ-M continuous multifunctions from an m-space into a topological space.
In this work, an incorporated form of Sadik transform and Adomian decomposition method which is c... more In this work, an incorporated form of Sadik transform and Adomian decomposition method which is called the Sadik decomposition method is presented. The method is applied to solve a system of nonlinear fractional Volterra integro-differential equations in the convolution form. To avoid collecting the noise terms that lead the method to fail for seeking the solution, the proposed method is modified by selecting a suitable initial solution. The obtained results are expressed in the explicit form of a power series with easily computable terms. In addition, illustrative examples are shown to demonstrate the effectiveness of the method.
The linear delay ordinary differential equation y”(x) + a(x) y’(x) + b(x) y’(x −τ ) + c(x) y(x) +... more The linear delay ordinary differential equation y”(x) + a(x) y’(x) + b(x) y’(x −τ ) + c(x) y(x) + d(x) y(x −τ ) = g(x) is studied, where the coefficients a(x),b(x), c(x) and d(x) and function g(x) are arbitrary. In this manuscript, group analysis is applied to find equivalent symmetries of the equation.
In this manuscript, the fractional residual power series (FRPS) method is employed in solving a s... more In this manuscript, the fractional residual power series (FRPS) method is employed in solving a system of linear fractional Fredholm integro-differential equations. The significant role of this system in various fields has attracted the attention of researchers for a decade. The definition of fractional derivative here is described in the Caputo sense. The proposed method relies on the generalized Taylor series expansion as well as the fact that the fractional derivative of stationary is zero. The process starts by constructing a residual function by supposing the finite order of an approximate power series solution that prescribes the initial conditions. Then, utilizing some conditions, the residual functions are converted to a linear system for the power series coefficients. Solving the linear system reveals the coefficients of the fractional power series solution. Finally, by substituting these coefficients into the supposed form of a solution, the approximate fractional power series solutions are derived. This technique has the advantage of being able to be applied directly to the problem and spending less time on computation. It is not only an easy method for implementation of the problem, but also provides productive results after a few iterations. Some problems with known solutions emphasize the procedure's simplicity and reliability. Moreover, the obtained exact solution demonstrated the efficiency and accuracy of the presented method.
Communications in Nonlinear Science and Numerical Simulation, 2010
This research gives a complete Lie group classification of second-order delay ordinary differenti... more This research gives a complete Lie group classification of second-order delay ordinary differential equations of the form y 00 ¼ f x; y; y s ; y 0 ; y 0 s À Á ; where s > 0 is the delay, y s ¼ yðx À sÞ and y 0 s ¼ y 0 ðx À sÞ. All classes of second-order delay ordinary differential equations admitting a Lie algebra are presented in the manuscript. The method for finding the solution for this type of differential equation is developed.
International Journal of Analysis and Applications
The Double Sadik Transform (DST) represents a generalized double integral transform that has emer... more The Double Sadik Transform (DST) represents a generalized double integral transform that has emerged as a highly effective analytical technique for solving numerous scientific problems. This study aims to investigate the DST applied to elementary functions and explore its notable properties, including its duality with the Double Laplace Transform and its capability to transform shifting functions, periodic functions, and convolution functions. Furthermore, the DST methodology is employed to resolve prominent linear fractional Caputo partial differential equations with known solutions commonly encountered in diverse mathematical models. The obtained outcomes are expressed in exact closed form, with the most precise results articulated through the Mittag-Leffler function. These results serve to validate the effectiveness and efficiency of the DST approach, establishing it as a valuable tool for addressing scientific problems involving fractional calculus.
In this manuscript, the Laplace Adomian decomposition method (LADM) is presented to solve Newell-... more In this manuscript, the Laplace Adomian decomposition method (LADM) is presented to solve Newell-Whitehead-Segel equation. The method can be applied to linear and nonlinear problems. Some examples have been carried out in order to illustrate the efficiency and reliability of the method.
This article presents the concept of ı ⋆ -continuous multifunctions in ideal topological spaces. ... more This article presents the concept of ı ⋆ -continuous multifunctions in ideal topological spaces. Especially, several characterizations of ı ⋆ -continuous multifunctions are investigated. Moreover, the relationships between ı ⋆ -continuity and the other types of continuity for multifunctions are established.
The aim of this paper is to introduce and study upper and lower almost γ-M continuous multifuncti... more The aim of this paper is to introduce and study upper and lower almost γ-M continuous multifunctions from an m-space into a topological space.
In this work, an incorporated form of Sadik transform and Adomian decomposition method which is c... more In this work, an incorporated form of Sadik transform and Adomian decomposition method which is called the Sadik decomposition method is presented. The method is applied to solve a system of nonlinear fractional Volterra integro-differential equations in the convolution form. To avoid collecting the noise terms that lead the method to fail for seeking the solution, the proposed method is modified by selecting a suitable initial solution. The obtained results are expressed in the explicit form of a power series with easily computable terms. In addition, illustrative examples are shown to demonstrate the effectiveness of the method.
The linear delay ordinary differential equation y”(x) + a(x) y’(x) + b(x) y’(x −τ ) + c(x) y(x) +... more The linear delay ordinary differential equation y”(x) + a(x) y’(x) + b(x) y’(x −τ ) + c(x) y(x) + d(x) y(x −τ ) = g(x) is studied, where the coefficients a(x),b(x), c(x) and d(x) and function g(x) are arbitrary. In this manuscript, group analysis is applied to find equivalent symmetries of the equation.
In this manuscript, the fractional residual power series (FRPS) method is employed in solving a s... more In this manuscript, the fractional residual power series (FRPS) method is employed in solving a system of linear fractional Fredholm integro-differential equations. The significant role of this system in various fields has attracted the attention of researchers for a decade. The definition of fractional derivative here is described in the Caputo sense. The proposed method relies on the generalized Taylor series expansion as well as the fact that the fractional derivative of stationary is zero. The process starts by constructing a residual function by supposing the finite order of an approximate power series solution that prescribes the initial conditions. Then, utilizing some conditions, the residual functions are converted to a linear system for the power series coefficients. Solving the linear system reveals the coefficients of the fractional power series solution. Finally, by substituting these coefficients into the supposed form of a solution, the approximate fractional power series solutions are derived. This technique has the advantage of being able to be applied directly to the problem and spending less time on computation. It is not only an easy method for implementation of the problem, but also provides productive results after a few iterations. Some problems with known solutions emphasize the procedure's simplicity and reliability. Moreover, the obtained exact solution demonstrated the efficiency and accuracy of the presented method.
Communications in Nonlinear Science and Numerical Simulation, 2010
This research gives a complete Lie group classification of second-order delay ordinary differenti... more This research gives a complete Lie group classification of second-order delay ordinary differential equations of the form y 00 ¼ f x; y; y s ; y 0 ; y 0 s À Á ; where s > 0 is the delay, y s ¼ yðx À sÞ and y 0 s ¼ y 0 ðx À sÞ. All classes of second-order delay ordinary differential equations admitting a Lie algebra are presented in the manuscript. The method for finding the solution for this type of differential equation is developed.
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