Linear Fractional Partial Differential Equations Research Articles

An Approach for Approximating Analytical Solutions of the Navier-Stokes Time-Fractional Equation Using the Homotopy Perturbation Sumudu Transform’s Strategy

In this study, we utilize the properties of the Sumudu transform (SuT) and combine it with the homotopy perturbation method to address the time fractional Navier-Stokes equation. We introduce a new technique called the homotopy perturbation Sumudu transform Strategy (HPSuTS), which combines the SuT with the homotopy perturbation method using He’s polynomials. This approach proves to be powerful and practical for solving various linear and nonlinear fractional partial differential equations (FPDEs) in scientific and engineering fields. We demonstrate the efficiency and simplicity of this method through examples, showcasing its ability to approximate solutions for FPDEs. Additionally, we compare the numerical results obtained using this technique for different values of alpha, showing that as the value moves from a fractional order to an integer order, the solution becomes increasingly similar to the exact solution. Furthermore, we provide the tabular representations of the solution for each example.

Exploring the Remarkable Properties of the Double Sadik Transform and Its Applications to Fractional Caputo Partial Differential Equations

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.

Investigation to analytic solutions of modified conformable time–space fractional mixed partial differential equations

In this paper, the invariant subspace method (ISM) is developed to obtain the exact solution of linear and nonlinear time and space fractional mixed partial differential equations involving modified conformable fractional derivative (MCFD). Moreover, the method is successfully extended to the coupled system of modified conformable fractional differential equations. Variety of time-space fractional mixed PDEs are considered and solved to illustrate the established result of ISM. Finally, a comparison between exact solution of fractional PDE in the sense of MCFD and Caputo fractional derivative (Ca-FD) is presented through some examples of time–space fractional mixed partial differential equations.

Application of modified extended tanh method in solving fractional order coupled wave equations

In this paper, our purpose is to analyse the dynamical behaviour of the solitary waves of coupled wave equations, namely fractional order modified Volterra’s equations. Here, the local fractional derivative viz. beta derivative has been used and the solutions are analysed for different values of the parameter β. The fractional order modified Volterra’s equations consist of two non linear fractional partial differential equations. Fractional complex transformation has been applied to convert fractional order non linear partial differential equations into integer order non linear ordinary differential equations. The modified extended tanh method is used to find ample solitary wave solutions of fractional order modified Volterra’s equations. The solutions of modified Volterra’s equations have been given in the sense of Riccati equation. Proposed method has been utilized to pluck out periodic singular soliton and kink shape solution, which are illustrated by 2D and 3D plots.

Technique to Solve Linear Fractional Differential Equations Using B-Polynomials Bases

A multidimensional, modified, fractional-order B-polys technique was implemented for finding solutions of linear fractional-order partial differential equations. To calculate the results of the linear Fractional Partial Differential Equations (FPDE), the sum of the product of fractional B-polys and the coefficients was employed. Moreover, minimization of error in the coefficients was found by employing the Galerkin method. Before the Galerkin method was applied, the linear FPDE was transformed into an operational matrix equation that was inverted to provide the values of the unknown coefficients in the approximate solution. A valid multidimensional solution was determined when an appropriate number of basis sets and fractional-order of B-polys were chosen. In addition, initial conditions were applied to the operational matrix to seek proper solutions in multidimensions. The technique was applied to four examples of linear FPDEs and the agreements between exact and approximate solutions were found to be excellent. The current technique can be expanded to find multidimensional fractional partial differential equations in other areas, such as physics and engineering fields.

A Laplace substitution method of space‐time fractional order including jumbled partial derivatives

In this paper, we present new approximate solutions to some linear and nonlinear fractional partial differential equations (FPDEs), which include mixed partial derivatives using Laplace substitution method (LSM). This method is entirely based on the well‐known Laplace transform and Adomian polynomial. Looking at several studies, we find that the results obtained from this technique converge well with the exact solutions.

An Attractive Approach Associated with Transform Functions for Solving Certain Fractional Swift-Hohenberg Equation

Many phenomena in physics and engineering can be built by linear and nonlinear fractional partial differential equations which are considered an accurate instrument to interpret these phenomena. In the current manuscript, the approximate analytical solutions for linear and nonlinear time-fractional Swift-Hohenberg equations are created and studied by means of a recent superb technique, named the Laplace residual power series (LRPS) technique under the time-Caputo fractional derivatives. The proposed technique is a combination of the generalized Taylor’s formula and the Laplace transform operator, which depends mainly on the concept of limit at infinity to find the unknown functions for the fractional series expansions in the Laplace space with fewer computations and more accuracy comparing with the classical RPS that depends on the Caputo fractional derivative for each step in obtaining the coefficient expansion. To test the simplicity, performance, and applicability of the present method, three numerical problems of the time-fractional Swift-Hohenberg initial value problems are considered. The impact of the fractional order β on the behavior of the approximate solutions at fixed bifurcation parameter is shown graphically and numerically. Obtained results emphasized that the LRPS technique is an easy, efficient, and speed approach for the exact description of the linear and nonlinear time-fractional models that arise in natural sciences.

Solution of linear and non-linear partial differential equation of fractional order

We know that the solution of partial differential equations by analytical method is better than the solution by approximate or series solution method. In this paper, we discuss the solution of linear and non-linear fractional partial differential equations involving derivatives with respect to time or space variables by converting them into the partial differential equations of integer order. Also we develop an analytical formulation to solve such fractional partial differential equations. Moreover, we discuss the method to solve the fractional partial differential equations in space as well as time variables simultaneously with the help of some examples.

New Iterative Method of Solving Nonlinear Equations in Fluid Mechanics

This paper presents the results of applying a new iterative method to linear and nonlinear fractional partial differential equations in fluid mechanics. A numerical analysis was performed to find an exact solution of the fractional wave equation and fractional Burgers’ equation, as well as an approximate solution of fractional KdV equation and fractional Boussinesq equation. Fractional derivatives of the order α are described using Caputo's definition with <i>0</i> < α ≤ <i>1</i> or <i>1</i> < α ≤ <i>2</i>. A comparative analysis of the results obtained using a new iterative method with those obtained by the Adomian decomposition method showed the first method to be more efficient and simple, providing accurate results in fewer computational operations. Given its flexibility and ability to solve nonlinear equations, the iterative method can be used to solve more complex linear and nonlinear fractional partial differential equations.

Approximate Solution of 2-Dimensional VO Linear Fractional Partial Differential Equation

The non-polynomial spline method has been used to solving 2-dimensional variable-order(VO) fractional partial differential equations (FPDE). For VO fractional derivative, described in the sense of the Caputo. The main objective of this study and advantage of the proposed method is to investigate a public approximation for the frequency of the trigonometric functions of the non-polynomial part of the spline function. The powerful algorithm of the proposed method gives high accuracy results.

A space-time finite element method for solving linear Riesz space fractional partial differential equations

In this paper, numerical solutions for linear Riesz space fractional partial differential equations with a second-order time derivative are considered. A space-time finite element method is proposed to solve these equations numerically. In the time direction, the C0-continuous Galerkin method is used to approximate the second-order time derivative. In the space direction, the usual linear finite element method is developed to approximate the space fractional derivative. The matrix equivalent form of this numerical method is deduced. The stability of the discrete solution is established and the optimal error estimates are investigated. Some numerical tests are given to validate the theoretical results.

Solution to linear KdV and nonLinear space fractional PDEs

In this work, the author will briefly discuss applications of the Fourier and Laplace transforms in the solution of certain singular integral equations and evaluation of integrals. By combining integral transforms and operational methods we get more powerful analytical tool for solving a wide class of linear or even non- linear fractional differential or fractional partial differential equation. Numerous examples and exercises occur throughout the paper.

Exact analytical solutions of fractional order telegraph equations via triple Laplace transform

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Fractional shifted legendre tau method to solve linear and nonlinear variable-order fractional partial differential equations

Here, we shed light on the fractional linear and nonlinear Klein–Gorden partial differential equations via Fractional Shifted Legendre Tau Method. With this objective, the operational matrices of fractional-order shifted Legendre functions (FSLFs) are derived and combined with the Tau method to convert the fractional-order differential equations to a system of solvable algebraic equations. The validity and the efficiency of the operational matrices are tested. Our findings yield an affirmative consequence, indicating applicability of the proposed method for nonlinear equations appearing in science and engineering.

Analytical solutions of linear fractional partial differential equations using fractional Fourier transform

This paper discusses the analytical solutions of fractional partial differential equations using Integral Transform method. The fractional derivatives are considered with reference to modified Riemann–Liouville derivatives. Fractional Fourier transform (FrFT) is applied to solve fractional heat diffusion, fractional wave, fractional telegraph and fractional kinetic equations. The method proposed here is effective enough to work on these equations efficiently.

Evaluation of one dimensional fuzzy fractional partial differential equations

This manuscript is related to investigate analytical solutions to some linear fractional partial fuzzy differential equations under certain conditions. For the concerned investigation, we utilize Laplace transform along with some decomposition method to compute series type solution under fuzzy concept. Some examples are solved to demonstrate the proposed method.

Kesirli Mertebe Kısmi Diferensiyel Denklemlerin Ayrık Homotopi Perturbasyon Metodu ile Çözümü

This work is developed the discrete homotopy perturbation method with a space discrete version to solve the linear and nonlinear time derivative fractional partial differential equations. The fractional derivatives are considered in the sense of Caputo. The success and applicability of this method has been demonstrated by some sample problems. When fractional order is unit, obtained results are good agreement with the exact solutions. The method demonstrated in this study is expected to solve similar problems in fractional calculus.

A new analytical solution of Klein–Gordon equation with local fractional derivative

The work presented in this paper is to combine the Sumudu transform method with a variational iteration method for solving linear and nonlinear partial differential equations with local fractional derivative. We apply the proposed method to obtain approximate analytical solutions of Klein–Gordon equations with local fractional derivative. The results obtained for the local fractional Klein–Gordon equation in the three cases presented, are in the form of non-differentiable functions. Through this work it can be said that this method is an alternative analytical method for linear and nonlinear local fractional partial differential equations.

Fractional mesh-free linear diffusion method for image enhancement and segmentation for automatic tumor classification

Computer aided diagnostic (CAD) models have shown outstanding performance in identifying many kind of diseases. Tumor identification is one of the most useful application of CAD model. Benign and malignant are two categories of tumor cells. Both categories share some textural features due to which tumor classification becomes complex and difficult task. In the present manuscript, a novel CAD model is being presented for classifying tumor cells automatically. The proposed model has been divided into four modules (image enhancement, segmentation, feature extraction and classification). All the modules are equally important in tumor classification. The manuscript focuses on effect of first two modules namely image enhancement and segmentation on tumor classification. In the present work, a novel linear fractional mesh-free partial differential equation (FPDE) based image enhancement method has been proposed to improve quality of the images. The proposed enhancement model is able to preserve fine details in smooth areas also non-linearly increases high frequency information. A novel fractional mesh-free segmentation method has been proposed to extract tumor region. It has been found that the method is able segment tumor region more accurately. Thirteen textural features have been used for training and testing. Support vector machine (SVM) classifier has been used to classify extracted tumor region. Quantitative analysis of proposed image enhancement, segmentation and CAD model have been done with other popular models. Higher performance has been observed using proposed models.

Numerical solution of the linear time fractional Klein-Gordon equation using transform based localized RBF method and quadrature

This work aims to approximate the solution of the linear time-fractional Klein-Gordon equations in Caputo's sense. The Laplace transform is applied to linear time fractional Klein-Gordon equation to eliminate the time variable and avoid the time stepping procedure. Application of the Laplace transform avoids the time instability issues which commonly occurs in time stepping methods and reduces the computational cost. The transform problem is then solved using local RBFs and finally the solution is obtained by the inverse Laplace transform. The solution is represented as an integral along a smooth curve in the complex plane which is then approximated by quadrature rule. The proposed method is capable of solving linear time fractional partial differential equations. The stability and convergence of the method are discussed. The efficiency of the method is demonstrated with the help of numerical experiments.