Search Results (16)

This work investigates the solutions of fractional integro-differential equations (FIDEs) using a unique kernel operator within the Caputo framework. The problem is addressed using both analytical and numerical techniques. First, the two-step Adomian decomposition method (TSADM) is applied to obtain an exact solution (if it exists). In the second part, numerical methods are used to generate approximate solutions, complementing the analytical approach based on the Adomian decomposition method (ADM), which is further extended using the Sumudu and Shehu transform techniques in cases where TSADM fails to yield an exact solution. Additionally, we establish the existence and uniqueness of the solution via fixed-point theorems. Furthermore, the Ulam–Hyers stability of the solution is analyzed. A detailed error analysis is performed to assess the precision and performance of the developed approaches. The results are demonstrated through validated examples, supported by comparative graphs and detailed error norm tables ($L_{\infty }$, $L_2$, and $L_1$). The graphical and tabular comparisons indicate that the Sumudu-Adomian decomposition method (Sumudu-ADM) and the Shehu-Adomian decomposition method (Shehu-ADM) approaches provide highly accurate approximations, with Shehu-ADM often delivering enhanced performance due to its weighted formulation. The suggested approach is simple and effective, often producing accurate estimates in a few iterations. Compared to conventional numerical and analytical techniques, the presented methods are computationally less intensive and more adaptable to a broad class of fractional-order differential equations encountered in scientific applications. The adopted methods offer high accuracy, low computational cost, and strong adaptability, with potential for extension to variable-order fractional models. They are suitable for a wide range of complex systems exhibiting evolving memory behavior. Full article

The study of vibration equations of large membranes is crucial in various scientific and engineering fields. Analyzing the vibration equations of bridges, roofs, and spacecraft structures helps in designing structures that resist excessive oscillations and potential failures. Aircraft wings, parachutes, and satellite components often behave like large membranes. Understanding their vibration characteristics is essential for stability, efficiency, and durability. Studying large membrane vibration involves solving partial differential equations and eigenvalue problems, contributing to advancements in numerical methods and computational physics. In this paper, the Elzaki transformation decomposition method and the Shehu transformation decomposition method, along with inverse Elzaki and inverse Shehu transformations, are used to investigate the fractional vibration equation of a large membrane. The solutions are obtained in terms of Mittag–Leffler functions. Full article

The development of numerical or analytical solutions for fractional mathematical models describing specific phenomena is an important subject in physics, mathematics, and engineering. This paper’s main objective is to investigate the approximation of the fractional order Caudrey–Dodd–Gibbon ($CDG$) nonlinear equation, which appears in the fields of laser optics and plasma physics. The physical issue is modeled using the Caputo derivative. Adomian and homotopy polynomials facilitate the handling of the nonlinear term. The main innovation in this paper is how the recurrence relation, which generates the series solutions after just a few iterations, is handled. We examined the assumed model in fractional form in order to demonstrate and verify the efficacy of the new methods. Moreover, the numerical simulation is used to show how the physical behavior of the suggested method’s solution has been represented in plots and tables for various fractional orders. We provide three problems of each equation to check the validity of the offered schemes. It is discovered that the outcomes derived are close to the accurate result of the problems illustrated. Additionally, we compare our results with the Laplace residual power series method (LRPSM), the natural transform decomposition method (NTDM), and the homotopy analysis shehu transform method (HASTM). From the comparison, our methods have been demonstrated to be more accurate than alternative approaches. The results demonstrate the significant benefit of the established methodologies in achieving both approximate and accurate solutions to the problems. The results show that the technique is extremely methodical, accurate, and very effective for examining the nature of nonlinear differential equations of arbitrary order that have arisen in related scientific fields. Full article

The Kawahara equation exhibits signal dispersion across lines of transmission and the production of unstable waves from the water in the broad wavelength area. This article explores the computational analysis for the approximate series of time fractional Kawahara (TFK) and modified Kawahara (TFMK) problems. We utilize the Shehu homotopy transform method (SHTM), which combines the Shehu transform (ST) with the homotopy perturbation method (HPM). He’s polynomials using HPM effectively handle the nonlinear terms. The derivatives of fractional order are examined in the Caputo sense. The suggested methodology remains unaffected by any assumptions, restrictions, or hypotheses on variables that could potentially pervert the fractional problem. We present numerical findings via visual representations to indicate the usability and performance of fractional order derivatives for depicting water waves in long-wavelength regions. The significance of our proposed scheme is demonstrated by the consistency of analytical results that align with the exact solutions. These derived results demonstrate that SHTM is an effective and powerful scheme for examining the results in the representation of series for time-fractional problems. Full article

Integral transform methods are a common tool employed to study the Hyers–Ulam stability of differential equations, including Laplace, Kamal, Tarig, Aboodh, Mahgoub, Sawi, Fourier, Shehu, and Elzaki integral transforms. This work provides improved techniques for integral transforms in relation to establishing the Hyers–Ulam stability of differential equations with constant coefficients, utilizing the Kamal transform, where we focus on first- and second-order linear equations. In particular, in this work, we employ the Kamal transform to determine the Hyers–Ulam stability and Hyers–Ulam stability constants for first-order complex constant coefficient differential equations and, for second-order real constant coefficient differential equations, improving previous results obtained by using the Kamal transform. In a section of examples, we compare and contrast our results favorably with those established in the literature using means other than the Kamal transform. Full article

In this paper, we suggest a modification for the residual power series method that is used to solve fractional-order Helmholtz equations, which is called the Shehu-transform residual power series method ($\mathbb{S}$T-RPSM). This scheme uses a combination of the Shehu transform ($\mathbb{S}$T) and the residual power series method (RPSM). The fractional derivatives are taken with respect to Caputo order. The novelty of this approach is that it does not restrict the fractional order and reduces the need for heavy computational work. The results were obtained using an iterative series that led to an exact solution. The 3D graphical plots for different values of fractional orders are shown to compare $\mathbb{S}$T-RPSM results with exact solutions. Full article

The entirety of the information regarding a subatomic particle is encoded in a wave function. Solving quantum mechanical models (QMMs) means finding the quantum mechanical wave function. Therefore, great attention has been paid to finding solutions for QMMs. In this study, a novel algorithm that combines the conformable Shehu transform and the Adomian decomposition method is presented that establishes approximate and exact solutions to QMMs in the sense of conformable derivatives with zero and nonzero trapping potentials. This solution algorithm is known as the conformable Shehu transform decomposition method (CSTDM). To evaluate the efficiency of this algorithm, the numerical results in terms of absolute and relative errors were compared with the reduced differential transform and the two-dimensional differential transform methods. The comparison showed excellent agreement with these methods, which means that the CSTDM is a suitable alternative tool to the methods based on the Caputo derivative for the solutions of time-fractional QMMs. The advantage of employing this approach is that, due to the use of the conformable Shehu transform, the pattern between the coefficients of the series solutions makes it simple to obtain the exact solution of both linear and nonlinear problems. Consequently, our approach is quick, accurate, and easy to implement. The convergence, uniqueness, and error analysis of the solution were examined using Banach’s fixed point theory. Full article

This work aims at a new semi-analytical method called the variational iteration transformation method for solving nonlinear homogeneous and nonhomogeneous fractional-order gas dynamics equations. The Shehu transformation and the iterative technique are applied to solve the suggested problems. The proposed method has an advantage over existing approaches because it does not require additional materials or computations. Four problems are used to test the authenticity of the proposed method. Using the suggested method, the solution proves to be more accurate. The proposed method can be implemented to solve many nonlinear fractional order problems because it has a straightforward implementation. Full article

In the present research paper, an iterative approach named the iterative Shehu transform method is implemented to solve time-fractional hyperbolic telegraph equations in one, two, and three dimensions, respectively. These equations are the prominent ones in the field of physics and in some other significant problems. The efficacy and authenticity of the proposed method are tested using a comparison of approximated and exact results in graphical form. Both 2D and 3D plots are provided to affirm the compatibility of approximated-exact results. The iterative Shehu transform method is a reliable and efficient tool to provide approximated and exact results to a vast class of ODEs, PDEs, and fractional PDEs in a simplified way, without any discretization or linearization, and is free of errors. A convergence analysis is also provided in this research. Full article

In this paper, we introduce a new integral transform called the Formable integral transform, which is a new efficient technique for solving ordinary and partial differential equations. We introduce the definition of the new transform and give the sufficient conditions for its existence. Some essential properties and examples are introduced to show the efficiency and applicability of the new transform, and we prove the duality between the new transform and other transforms such as the Laplace transform, Sumudu transform, Elzaki transform, ARA transform, Natural transform and Shehu transform. Finally, we use the Formable transform to solve some ordinary and partial differential equations by presenting five applications, and we evaluate the Formable transform for some functions and present them in a table. A comparison between the new transform and some well-known transforms is made and illustrated in a table. Full article

The present research correlates with a fuzzy hybrid approach merged with a homotopy perturbation transform method known as the fuzzy Shehu homotopy perturbation transform method (SHPTM). With the aid of Caputo and Atangana–Baleanu under generalized Hukuhara differentiability, we illustrate the reliability of this scheme by obtaining fuzzy fractional Cauchy reaction–diffusion equations (CRDEs) with fuzzy initial conditions (ICs). Fractional CRDEs play a vital role in diffusion and instabilities may develop spatial phenomena such as pattern formation. By considering the fuzzy set theory, the proposed method enables the solution of the fuzzy linear CRDEs to be evaluated as a series of expressions in which the components can be efficiently identified and generating a pair of approximate solutions with the uncertainty parameter $\lambda \in [0,1].$ To demonstrate the usefulness and capabilities of the suggested methodology, several numerical examples are examined to validate convergence outcomes for the supplied problem. The simulation results reveal that the fuzzy SHPTM is a viable strategy for precisely and accurately analyzing the behavior of a proposed model. Full article

We put into practice relatively new analytical techniques, the Shehu decomposition method and the Shehu iterative transform method, for solving the nonlinear fractional coupled Korteweg-de Vries (KdV) equation. The KdV equation has been developed to represent a broad spectrum of physics behaviors of the evolution and association of nonlinear waves. Approximate-analytical solutions are presented in the form of a series with simple and straightforward components, and some aspects show an appropriate dependence on the values of the fractional-order derivatives that are, in a certain sense, symmetric. The fractional derivative is proposed in the Caputo sense. The uniqueness and convergence analysis is carried out. To comprehend the analytical procedure of both methods, three test examples are provided for the analytical results of the time-fractional KdV equation. Additionally, the efficiency of the mentioned procedures and the reduction in calculations provide broader applicability. It is also illustrated that the findings of the current methodology are in close harmony with the exact solutions. It is worth mentioning that the proposed methods are powerful and are some of the best procedures to tackle nonlinear fractional PDEs. Full article

In this article, modified techniques, namely the variational iteration transform and Shehu decomposition method, are implemented to achieve an approximate analytical solution for the time-fractional Fornberg–Whitham equation. A comparison is made between the results of the variational iteration transform method and the Shehu decomposition method. The solution procedure reveals that the variational iteration transform method and Shehu decomposition method is effective, reliable and straightforward. The variational iteration transform methods solve non-linear problems without using Adomian’s polynomials and He’s polynomials, which is a clear advantage over the decomposition technique. The solutions achieved are compared with the corresponding exact result to show the efficiency and accuracy of the existing methods in solving a wide variety of linear and non-linear problems arising in various science areas. Full article

In this paper, we introduce a new type of integral transforms, called the ARA integral transform that is defined as: ${\mathcal{G}}_n\left[g\left(t\right)\right]\left(s\right)=G\left(n,s\right)=s\underset{0}{\overset{\infty }{{\displaystyle \int }}}{t}^{n-1}{e}^{-st}g\left(t\right)dt,\hspace{1em}s>0$ . We prove some properties of ARA transform and give some examples. Also, some applications of the ARA transform are given. Full article

Mostly, it is very difficult to obtained the exact solution of fractional-order partial differential equations. However, semi-analytical or numerical methods are considered to be an alternative to handle the solutions of such complicated problems. To extend this idea, we used semi-analytical procedures which are mixtures of Laplace transform, Shehu transform and Homotopy perturbation techniques to solve certain systems with Caputo derivative differential equations. The effectiveness of the present technique is justified by taking some examples. The graphical representation of the obtained results have confirmed the significant association between the actual and derived solutions. It is also shown that the suggested method provides a higher rate of convergence with a very small number of calculations. The problems with derivatives of fractional-order are also solved by using the present method. The convergence behavior of the fractional-order solutions to an integer-order solution is observed. The convergence phenomena described a very broad concept of the physical problems. Due to simple and useful implementation, the current methods can be used to solve problems containing the derivative of a fractional-order. Full article