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1

Novel integral transform treating some Ψ-fractional derivative equations

Chamekh Mourad, Latrache Mohamed Ali, Elzaki Tarig M.

Acta Mechanica et Automatica

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2024

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Vol. 18, no. 3

571--578

EN

The paper deals with a new integral transformation method called Ψ-Elzaki transform (PETM) in order to analyze some Ψ-fractional differential equations. The proposed method uses a modification of the Elzaki transform that is well adapted to deal with Ψ-fractional operators. To solve the nonlinear Ψ-fractional differential equations, we combine the PETM by an iterative method to overcome this nonlinearity. To validate the accuracy and efficiency of this approach, we compare the results of the discussed numerical examples with the exact solutions.

2

Modified integral transform for solving benney-luke and singular pseudo-hyperbolic equations

Elzaki Tarig M., Chamekh Mourad, Ahmed Shams A.

Acta Mechanica et Automatica

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2024

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Vol. 18, no. 1

139--143

EN

In this article, we propose a technique based on modified double integral transforms used to solve certain equations of materials science, namely Benney–Luke (BL) and singular pseudo-hyperbolic (SP-H) equations. We have established some analytical results. This method can provide accurate one-step solutions, although the equations used may exhibit a singularity in the initial conditions. Some numerical examples have been discussed for illustration and to show the effectiveness of the technique for certain types of equations. We have developed an exact solution in just one step, whereas other approaches require several stages to succeed in a particular solution, making the proposed strategy particularly successful and straightforward to apply to various varieties of the B–L and SP-H equations.

3

A novel analytical method for the exact solution of the fractional-order biological population model

Elzaki Tarig M., Mohamed Mohamed Z.

Acta Mechanica et Automatica

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2024

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Vol. 18, no. 3

564--570

EN

In this research, we develop a new analytical technique based on the Elzaki transform (ET) to solve the fractional-order biological population model (FBPM) with initial and boundary conditions (ICs and BCs). This approach can be used to locate both the closed approximate solution and the exact solution of a differential equation. The usefulness and validity of this strategy for managing the solution of FBPM are demonstrated using a few real-world scenarios. The dependability of the suggested strategy is also shown using a table and a few graphs. The approximate solutions that were achieved and the convergence analysis are shown in numerical simulations in a range of fractional orders. From the numerical simulations, it can be seen that the population density increases with increasing fractional order, whereas the population density drops with decreasing fractional order.