# Computational Methods for Differential Equations Volume:12 Issue: 3, Summer 2024

• تاریخ انتشار: 1403/02/12
• تعداد عناوین: 15
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• Amirreza Azad, Ehsan Yaghoubi, Azadeh Jafari * Pages 425-437
In this paper, we transform the given nonlocal boundary condition problem into a manageable local equation. By introducing an additional transformation of the variables, we can simplify this equation into conformable Burgers’ equation. Thus, the Keller Box method is used as a numerical scheme to solve the equation. A comparison is made between numerical results and the analytic solution to validate the results of our proposed method.
Keywords: Nonlocal Boundary Condition, Burgers’ Equation, Keller-Box Scheme
• Abdelati El Allaoui *, Youssef Allaoui, Said Melliani, Hamid El Khalfi Pages 438-453
This paper delves into the identification of upper and lower solutions for a high-order fractional integro-differential equation featuring non-linear boundary conditions. By introducing an order relation, we define these upper and lower solutions. Through a rigorous approach, we demonstrate the existence of these solutions as the limits of sequences derived from carefully selected problems, supported by the application of Arzel\a-Ascoli's theorem. To illustrate the significance of our findings, we provide an illustrative example. This research contributes to a deeper understanding of solutions in the context of complex fractional integro-differential equations.
Keywords: Arzel, A-Ascoli's Theorem, Caputo Fractional Derivative, Integro-Differential Equation, Nonlinear Boundary Conditions
• Mohammad Samei *, Lotfollah Karimi, Mohammed K. A. Kaabar, Roya Raeisi, Jehad Alzabut, Francisco Martinez Gonzalez Pages 454-470
The objectives of this study are to develop the SEIR model for COVID-19 and evaluate its main parameters such as therapeutic vaccines, vaccination rate, and effectiveness of prophylactic. Global and local stability of the model and numerical simulation are examined. The local stability of equilibrium points was classified. A Lyapunov function is constructed to analyze the global stability of the disease-free equilibrium. The simulation part is based on two situations, including the USA and Iran. Our results provide a good contribution to the current research on this topic.
Keywords: Efficiency Of Vaccines, Numerical Simulation, Equilibrium Point, Covid-19
• Hanif Mirzaei *, Mahmood Emami, Kazem Ghanbari, Mohammad Shahriari Pages 471-483
In this paper, Computing the eigenvalues of the Conformable Sturm-Liouville Problem (CSLP) of order $2 \alpha$, $\frac{1}{2}<\alpha \leq 1$, and dirichlet boundary conditions is considered. For this aim, CSLP is discretized to obtain a matrix eigenvalue problem (MEP) using finite element method with fractional shape functions. Then by a method based on the asymptotic form of the eigenvalues, we correct the eigenvalues of MEP to obtain efficient approximations for the eigenvalues of CSLP. Finally, some numerical examples to show the efficiency of the proposed method are given. Numerical results show that for the $n$th eigenvalue, the correction technique reduces the error order from $O(n^4h^2)$ to $O(n^2h^2)$.
Keywords: Sturm-Liouville Problem, Conformable Derivative, Finite Elements Method, Correction Idea
• Minoo Kamrani * Pages 484-501
This study aims to investigate a stochastic Volterra integral equation driven by fractional Brownian motion with Hurst parameter $H\in (\frac 12, 1)$. We employ the Wong-Zakai approximation to simplify this intricate problem, transforming the stochastic integral equation into an ordinary integral equation. Moreover, we consider the convergence and the rate of convergence of the Wong-Zakai approximation for this kind of equation.
Keywords: Wong-Zakai Approximation, Fractional Brownian Motion, Volterra Integral Equation
• Nada Mahdi *, Ayad Khudair Pages 502-510
In this paper, we introduce a generalized delta q−Mittag-Leffler function. Also, we solve some Caputo delta q−fractional dynamic equations and these solutions are expressed by means of the newly introduced delta q−Mittag-Leffler function.
Keywords: Time Scale Calculus, Q−Fractional Calculus, Q−Mittag-Leffler Function
• Moslem Valizadeh, Yaghoub Mahmoudi *, Farhad Dastmalchi Saei Pages 511-522
This paper presents the development of a series of fractional multi-step linear finite difference methods (FLMMs) designed to address fractional multi-delay pantograph differential equations of order $0 < \alpha \leq 1$. These $p$ FLMMs are constructed using fractional backward differentiation formulas of first and second orders, thereby facilitating the numerical solution of fractional differential equations. Notably, we employ accurate approximations for the delayed components of the equation, guaranteeing the retention of stability and convergence characteristics in the proposed $p$-FLMMs. To substantiate our theoretical findings, we offer numerical examples that corroborate the efficacy and reliability of our approach.
Keywords: Fractional Derivative, Fractional Integration, Fractional Linear Multi-Step Methods
• Vembu Ananthaswamy *, Ramaiya Subanya, Sathyamoorthy Sivasankari Pages 523-543
An analytical study of two different models of rectangular porous fins are investigated using a new approximate analytical method, the Ananthaswamy-Sivasankari method. The obtained results are compared with the numerical solution, which results in a very good agreement. The impacts of several physical parameters involved in the problem are interlined graphically. Fin efficiency and the heat transfer rate are also calculated and displayed. The result obtained by this method is in the most explicit and simple form. The convergence of the solution determined is more accurate as compared to various analytical and numerical methods.
Keywords: Longitudinal Fin, Darcy’S Model, Insulated Tip, Ananthaswamy-Sivasankari Method (ASM), Magneto Hydrodynamics (MHD)
• Melike Karta * Pages 544-560
A hybrid method utilizing the collocation technique with B-splines and Lie-Trotter splitting algorithm applied for 3 model problems which include a single solitary wave,  two solitary wave interaction, and a Maxwellian initial condition is designed for getting the approximate solutions for the generalized equal width (GEW) equation. Initially, the considered problem has been split into 2 sub-equations as linear $U_t=\hat{A}(U)$ and nonlinear $U_t=\hat{B}(U)$ in the  terms of time.  After, numerical schemes have been constructed for these sub-equations utilizing the finite element method (FEM) together with quintic B-splines. Lie-Trotter splitting technique $\hat{A}o\hat{B}$ has been  used to generate approximate solutions of the main equation. The stability analysis of acquired numerical schemes has been examined by the Von Neumann method. Also, the error norms $L_2$ and $L_\infty$ with mass, energy, and momentum conservation constants $I_1$, $I_2$, and $I_3$, respectively are calculated to illustrate how perfect solutions this new algorithm applied to the problem generates and the ones produced are compared with those in the literature. These new results exhibit that the algorithm presented in this paper is more accurate and successful, and easily applicable to other non-linear partial differential equations (PDEs) as the present equation.
Keywords: B-Splines, Lie-Trotter Splitting, Collocation Method, Generalized Equal Width Equation
• Ahmed S. Rashed, Mokhtar Mohamed, Mustafa Inc * Pages 561-570
An optimal system of Lie infinitesimals has been used in an investigation to find a solution to the (2+1)-dimensional Bogoyavlensky-Konopelchenko equation (BKE). This investigation was conducted to characterize certain fantastic characteristics of plasma-particle dispersion. A careful investigation into the Lie space with an unlimited number of dimensions was carried out to locate the relevant arbitrary functions. When developing accurate solutions for the BKE, it was necessary to establish an optimum system that could be employed in single, double, and triple combination forms.There were some fantastic wave solutions developed, and these were depicted visually. The Optimal Lie system demonstrates that it can obtain many accurate solutions to evolution equations.
Keywords: Evolution Equations, BKE, Lie Infinitesimals, Optimal Lie System
• Mohsen Khasteh, Amir Hosein Refahi Sheikhani *, Farhad Shariffar Pages 571-584
In this paper, we proposed a numerical method based on the shifted fractional order Jacobi and trapezoid methods to solve a type of distributed partial differential equations. The fractional derivatives are considered in the Caputo-Prabhakar type. By shifted fractional-order Jacobi polynomials our proposed method can provide highly accurate approximate solutions by reducing the problem under study to a set of algebraic equations which is technically simpler to handle. In order to demonstrate the error estimates, several lemmas are provided. Finally, numerical results are provided to demonstrate the validity of the theoretical analysis.
Keywords: Distributed Order, Caputo-Prabhakar Fractional Derivative, Shifted Jacobi Polynomials, Trapezoid, Numerical Method
• Prasad CN *, R Suchithra Pages 585-598
This paper proposes a reversible data embedding algorithm in encrypted images of cloud storage where the embedding was performed by detecting a predictor that provides a maximum embedding rate. Initially, the scheme generates trail data which are embedded using the prediction error expansion in the encrypted training images to obtain the embedding rate of a predictor. The process is repeated for different predictors from which the predictor that offers the maximum embedding rate is estimated. Using the estimated predictor as the label the Convolutional neural network (CNN) model is trained with the encrypted images. The trained CNN model is used to estimate the best predictor that provides the maximum embedding rate. The estimation of the best predictor from the test image does not use the trail data embedding process. The evaluation of proposed reversible data hiding uses the datasets namely BossBase and BOWS-2 with the metrics such as embedding rate, SSIM, and PSNR. The proposed predictor classification was evaluated with the metrics such as classification accuracy, recall, and precision. The predictor classification provides an accuracy, recall, and precision of 92.63\%, 91.73\%, and 90.13\% respectively. The reversible data hiding using the proposed predictor selection approach provides an embedding rate of 1.955 bpp with a PSNR and SSIM of 55.58dB and 0.9913 respectively.
Keywords: Reversible Data Hiding, Image Encryption, Prediction Error Expansion, Convolutional Neural Network, Embedding Capacity
• Vali Torkashvand *, Manochehr Kazemi, Masoud Azimi Pages 599-609
In this work, we have proposed a general manner to extend some two-parametric with-memory methods to obtain simple roots of nonlinear equations. Novel improved methods are two-step without memory and have two self-accelerator parameters that do not have additional evaluation. The methods have been compared with the nearest competitions in various numerical examples. Anyway, the theoretical order of convergence is verified. The basins of attraction of the suggested methods are presented and corresponded to explain their interpretation.
Keywords: With-Memory Method, Basin Of Attraction, Accelerator Parameter, $R$-Order Convergence, Nonlinear Equations
• Shahrbanoo Akbarpoor, Abdol Dabbaghian * Pages 610-623
Our research is about the Sturm-Liouville equation which contains conformable fractional derivatives of order $\alpha \in (0,1]$ in lieu of the ordinary derivatives. First, we present the eigenvalues, eigenfunctions, and nodal points, and the properties of nodal points are used for the reconstruction of an integral equation. Then, the Bernstein technique was utilized to solve the inverse problem, and the approximation of solving this problem was calculated. Finally, the numerical examples were introduced to explain the results. Moreover,  the analogy of this technique is shown in a numerical example with the Chebyshev interpolation technique .
Keywords: Inverse Problem, Conformable Fractional, Nodal Points, Bernstein Technique, Numerical Solution
• Eman Mansour, Emad Kuffi * Pages 624-637
Differential equations are used to represent different scientific problems are handled efficiently by integral transformations, where integral transforms represent an easy and effective tool for solving many problems in the mentioned fields. This work utilizes the integral transform of the Complex SEE integral transformation to provide an efficient solution method for the difference and differential-difference equations by benefiting from the properties of this complex transform to solve some problems related to difference and differential-difference equations. The 3D, contour and 2D surfaces, as well as the related density plot surfaces of some acquired data, are used to draw the physical aspect of the obtained findings. The proposed approach offers an efficient and rapid solution for addressing the inherent complexity of differential-difference problems with initial conditions.
Keywords: The Complex SEE Transform, Inverse Of Complex SEE Transform, Differential Equations, Difference Equations, Differential-Difference Equations