Elzaki Transform Research Articles

AN INNOVATIVE METHOD FOR SOLVING LINEAR AND NONLINEAR FRACTIONAL TELEGRAPH EQUATIONS

This work investigates and solves the time-fractional telegraph equations (TFTEs) occurring in electromagnetism, which serve as mathematical models in several practically significant applied research domains. Elzaki transform (ET) is employed in this process. Caputo sense describes fractional derivatives. Solutions of TFTEs were found in an easy-to-understand, step-by-step way using ET. In addition, instances are given to show how the phrase can be applied and how valid it is for the problem-solving form. The exact solutions and the analytical solutions accord well for the tested problems. This work also discusses the convergence of the ET technique to the exact solution of TFTEs. Several examples of linear and nonlinear TFTEs are used to demonstrate the suggested methodology. The novel technique’s results show that it is an effective way to solve TFTEs, and it makes the procedure easier.

Investigation of solution behavior of Differential Equations by Sumudu methods of random complex Partial Differential Equations

In this study, solutions of random complex partial differential equations were found using the two-dimensional Sumudu transformation method(STM). The initial conditions of a deterministic equation or the non-homogeneous part of the equation are transformed into random variables to obtain a random complex partial differential equation. With the help of the properties of two-dimensional Sumudu and inverse Sumudu transformation, an approximate analytical solution of a complex partial differential equation with random constant coefficients was obtained by selecting a random variable with an initial condition of Normal and Gamma distribution. The probability characteristics of the resulting solutions, such as expected value and variance, were obtained and graphically shown with the help of the Maple package program.

Novel approach for solving fractional partial differential equations using conformable Elzaki Transform

AbstractThis article discusses three different kinds of fractional partial differential equations (FPDEs) and how to solve them. First, a reformulation of the conformable fractional Elzaki Transform (ET) method is shown. The proposed methodology is then applied to the solution of three different types of FPDEs. The effectiveness and dependability of the proposed strategy are found to be validated by numerical simulations. We believe the provided strategy is very successful and straightforward to adapt of many types of FPDEs. This method is one of the completely new approaches for solving FPDEs because of its exceptional correctness and speed, which set it apart from other approaches.

Innovative Solutions to the Fractional Diffusion Equation Using the Elzaki Transform

This study explores the application of advanced mathematical techniques to solve fractional differential equations, focusing particularly on the fractional diffusion equation. The fractional diffusion equation, used to simulate a range of physical and engineering phenomena, poses considerable difficulties when applied to fractional orders. Thus, by utilizing the mighty powers of fractional calculus, we employ the variational iteration method (VIM) with the Elzaki transform to produce highly accurate approximations for these specific differential equations. The VIM provides an iterative framework for refining solutions progressively, while the Elzaki transform simplifies the complex integral transforms involved. By integrating these methodologies, we achieve accurate and efficient solutions to the fractional diffusion equation. Our findings demonstrate the robustness and effectiveness of combining the VIM and the Elzaki transform in handling fractional differential equations, offering explicit functional expressions that are beneficial for theoretical analysis and practical applications. This research contributes to the expanding field of fractional calculus, providing valuable insights and useful tools for solving complex, nonlinear fractional differential equations across various scientific and engineering disciplines.

Numerical Solution of Time-fractional Reaction Diffusion Equation via Elzaki Transform with Residual Power Series Method

Numerical Solution of Time-fractional Reaction Diffusion Equation via Elzaki Transform with Residual Power Series Method

Elzaki Transform Homotopy Analysis Techniques for Solving Fractional (2+1)-D and (3+1)-D Nonlinear Schrodinger Equations

Elzaki Transform Homotopy Analysis Techniques for Solving Fractional (2+1)-D and (3+1)-D Nonlinear Schrodinger Equations

Exploring the Elzaki Transform: Unveiling Solutions to Reaction-Diffusion Equations with Generalized Composite Fractional Derivatives

This article investigates the use of the Elzaki transform on a generalized composite fractional derivative. To establish the framework for this inquiry, numerous essential lemmas about the Elzaki transform are presented. We successfully extract the solution to the reaction-diffusion problem using both the Elzaki and Fourier transforms, which include a generalized composite fractional derivative. We also look at special examples of the generalized equation, which helps us understand its applications and consequences better. The results show that the Elzaki transform is successful in dealing with complicated fractional differential equations, introducing new analytical approaches and solutions to the subject of fractional calculus and its applications in reaction-diffusion systems.

Numerical Study of Time-Fractional Schrödinger Model in One-Dimensional Space Arising in Mathematical Physics

This study provides an innovative and attractive analytical strategy to examine the numerical solution for the time-fractional Schrödinger equation (SE) in the sense of Caputo fractional operator. In this research, we present the Elzaki transform residual power series method (ET-RPSM), which combines the Elzaki transform (ET) with the residual power series method (RPSM). This strategy has the advantage of requiring only the premise of limiting at zero for determining the coefficients of the series, and it uses symbolic computation software to perform the least number of calculations. The results obtained through the considered method are in the form of a series solution and converge rapidly. These outcomes closely match the precise results and are discussed through graphical structures to express the physical representation of the considered equation. The results showed that the suggested strategy is a straightforward, suitable, and practical tool for solving and comprehending a wide range of nonlinear physical models.

Elzaki transform decomposition approach to solve Riccati matrix differential equations

Elzaki Transform Adomian decomposition technique (ETADM), which an elegant combine, has been employed in this work to solve non-linear Riccati matrix differential equations. Solutions are presented to demonstrate the relevance of the current approach. With the use of figures, the results of the proposed strategy are displayed and evaluated. It is demonstrated that the suggested approach is effective, dependable, and simple to apply to a range of related scientific and technical problems.

Analytical solutions via coupled Elzaki adomian decomposition method for some applications

An efficient combination of Adomian Decomposition iterative technique coupled Elzaki transformation (ETADM) for solving Telegraph equation and Riccati non-linear differential equation (RNDE) is introduced in a novel way to get an accurate analytical solution. An elegant combination of the Elzaki transform, the series expansion method, and the Adomian polynomial. The suggested method will convert differential equations into iterative algebraic equations, thus reducing processing and analytical work. The technique solves the problem of calculating the Adomian polynomials. The method’s efficiency was investigated using some numerical instances, and the findings demonstrate that it is easier to use than many other numerical procedures. It has also been established that (ETADM) is a trustworthy technique for solving differential equations. Using the Mathematica 13.3 programme, the graphs of the approximate solutions are presented.

Application of Elzaki Transform Method to Market Volatility Using the Black-Scholes Model

Application of Elzaki Transform Method to Market Volatility Using the Black-Scholes Model

A homotopy perturbation method with Elzaki transformation for solving the fractional Biswas–Milovic model

Abstract In this research, we use the homotopy perturbation method (HPM) combined with the Elzaki transform to investigate the fractional Biswas–Milovic equation (BME) within the framework of the Caputo operator. The fractional BME is a significant mathematical model with applications in various scientific and engineering fields, including physics, biology, and chemistry. However, its fractional nature introduces analytical complexities. By integrating the HPM with the Elzaki transform, we aim to provide an effective approach for obtaining accurate solutions to this equation. The combination of these mathematical techniques allows us to explore the behavior of the fractional BME in a comprehensive manner. The research outcomes are supported by numerical results and comparisons, demonstrating the reliability and efficiency of the proposed methodology. This study contributes to advancing the tools for solving fractional equations and enhances our understanding of the intricate dynamics described by the fractional BME.

New Method of Cryptography With Python Code Using Elzaki Transform and Linear Combination of Function

New Method of Cryptography With Python Code Using Elzaki Transform and Linear Combination of Function

Numerical Solution of Two-Dimensional Fractional Order Diffusion Equation by using Elzaki Transform with Residual Power Series Method

The Elzaki transform with RPS method is an efficient and reliable approach for solution of non-integer order linear as well as non-linear differential equations. The major endeavourof current work is to find the solution of two dimensional non-integer order diffusion equation by Elzaki transform with RPS method. Firstly, Elzaki is applied on this equation. Secondly, inverse Elzaki is taken on this equation for finding the expressionof solution. Thirdly, assumed approximate solution is substituted on this equation. Then unidentified coefficient functions are obtained by using residual function is equal to zero as well as combining its initial circumstances. At last, coefficient functions are substituted in power series form for finding finite approximate logical solutions. The judgment between exact solution and approximate analytic solutions with different number of terms of this equation are determined and compared for reliability. This method reduces the size of computational works of solution of non-integer order non-linear diffusion equation

Application of Elzaki Transform Method for Solving and Interpreting HIV Superinfection Model

In this work, a superinfection model of two HIV strains was proposed. The proposed model was solved and interpreted using the Elzaki Transform Method (ETM). The proposed model presented some non-linear terms which are difficult to resolve using the ETM. Hence, we employed the Adomian Decomposition Method (ADM) to resolve the nonlinear terms. We derived an iterative scheme that was used to predict the behavior of the model. Results of data simulation showed that the population of healthy CD4+ T cells declined with respect to time in the presence of HIV strains. The viral loads for both viral strains are observed to be on a steady increase. The study reveals that ETM can be used to solve Superinfection models of HIV. The method is easier, more efficient, and more effective, and it converges faster to the solution when compared to other transform methods. We recommend that ETM can be applied to superinfection and co-infection models of other infectious diseases.

A Comparative Study of the Fractional Partial Differential Equations via Novel Transform

In comparison to fractional-order differential equations, integer-order differential equations generally fail to properly explain a variety of phenomena in numerous branches of science and engineering. This article implements efficient analytical techniques within the Caputo operator to investigate the solutions of some fractional partial differential equations. The Adomian decomposition method, homotopy perturbation method, and Elzaki transformation are used to calculate the results for the specified issues. In the current procedures, we first used the Elzaki transform to simplify the problems and then applied the decomposition and perturbation methods to obtain comprehensive results for the problems. For each targeted problem, the generalized schemes of the suggested methods are derived under the influence of each fractional derivative operator. The current approaches give a series-form solution with easily computable components and a higher rate of convergence to the precise solution of the targeted problems. It is observed that the derived solutions have a strong connection to the actual solutions of each problem as the number of terms in the series solution of the problems increases. Graphs in two and three dimensions are used to plot the solution of the proposed fractional models. The methods used currently are simple and efficient for dealing with fractional-order problems. The primary benefit of the suggested methods is less computational time. The results of the current study will be regarded as a helpful tool for dealing with the solution of fractional partial differential equations.

Pathway Fractional Integral Formula Involving Extended Mittag-leffler Function in the Kernel of Generalized Elzaki Transform

Pathway Fractional Integral Formula Involving Extended Mittag-leffler Function in the Kernel of Generalized Elzaki Transform

An efficient analytical approach with novel integral transform to study the two-dimensional solute transport problem

The q-homotopy analysis method (q-HAM) in combine with the novel integral transform known as Elzaki transform (ET) leads to an efficient analytical technique called, the q-homotopy analysis Elzaki transform method (q-HAETM). In the present study, the two- dimensional advection–dispersion (AD) problem is investigated using an analytical technique q-HAETM. These equations are mainly used to describe the fate of pollutants in aquifers. The analytical solutions to the AD equations are more interesting since they serve as benchmarks against which numerical solutions can be compared. The novelty of the work is to discuss the two-dimensional (2D) solute transport problem in the fractional sense. The reliability and the efficiency of the considered algorithm are demonstrated by employing the 2D fractional solute transport problem. The solute concentration profile is shown in terms of surface plots. The comparison of the exact solution and the approximate solution is done by the 2D plots. The numerical approximate error solutions are presented for different fractional orders. q-HAETM offers us to modulate the range of convergence of the series solution using ℏ, called auxiliary parameter or convergence control parameter. By performing appropriate numerical simulations in comparison with other existing techniques, the effectiveness and reliability of the considered technique are validated. The obtained findings show that the proposed method is very gratifying and examines the complex challenges that arise in science and innovation.

Numerical Analysis of Fractional-Order Camassa–Holm and Degasperis–Procesi Models

This study proposes innovative methods for the time-fractional modified Degasperis–Procesi (mDP) and Camassa–Holm (mCH) models of solitary wave solutions. To formulate the concepts of the homotopy perturbation transform method (HPTM) and Elzaki transform decomposition method (ETDM), we mix the Elzaki transform (ET), homotopy perturbation method (HPM), and Adomian decomposition method (ADM). The Caputo sense is applied to this work. The solutions to a few numerical examples of the modified Degasperis–Procesi (mDP) and Camassa–Holm (mCH) are shown for integer and fractional orders of the issues. The derived and precise solutions are compared using two-dimensional and three-dimensional plots of the solutions, confirming the suggested method’s improved accuracy. Tables are created for each problem to display the suggested approach’s results, precise solutions, and absolute error. These methods provide the iterations as a series of solutions. To show the proposed techniques’ efficiency, we compute the absolute error. It is evident from the estimated values that the approaches are precise and simple and that they can therefore be further extended to linear and nonlinear issues.

Numerical analysis of multi-dimensional time-fractional diffusion problems under the Atangana-Baleanu Caputo derivative.

This paper presents the Elzaki homotopy perturbation transform scheme (EHPTS) to analyze the approximate solution of the multi-dimensional fractional diffusion equation. The Atangana-Baleanu derivative is considered in the Caputo sense. First, we apply Elzaki transform (ET) to obtain a recurrence relation without any assumption or restrictive variable. Then, this relation becomes very easy to handle for the implementation of the homotopy perturbation scheme (HPS). We observe that HPS produces the iterations in the form of convergence series that approaches the precise solution. We provide the graphical representation in 2D plot distribution and 3D surface solution. The error analysis shows that the solution derived by EHPTS is very close to the exact solution. The obtained series shows that EHPTS is a very simple, straightforward, and efficient tool for other problems of fractional derivatives.