Fractional Ordinary Differential Equations Research Articles

A novel numerical method to solve fractional ordinary differential equations with proportional Caputo derivatives

In this paper, we develop a novel numerical scheme, namely ‘NPCM-PCDE,’ to integrate fractional ordinary differential equations with proportional Caputo derivatives of the type pc D α u(t) = f 1(t, u(t)), t ≥ 0, 0 < α < 1 involving a non-linear operator f 1. A new method is developed using a natural discretization of the proportional Caputo derivative and the decomposition method to decompose the non-linear operator f 1. The error and stability analyses for the proposed method are provided. Some illustrated examples are given to compare the solution curves graphically with the exact solution and to prove the utility and efficiency of the method. The proposed NPCM-PCDE is found to be efficient, easy to implement, convergent, and stable.

On the inverse problem of time dependent coefficient in a time fractional diffusion problem by sinc wavelet collocation method

The object of this study is to establish the unknown function in a time fractional diffusion problem and the solution as well by utilizing Sinc wavelet collocation method (SWCM) and residual power series method (RPSM) together. SWCM enables us to convert time fractional diffusion problem into a system of fractional ordinary differential and algebraic equations. At this stage, the unknown function and the solution are constructed in the series form by employing RPSM. The novelty of this study is that the combination of SWCM and RPSM is utilized to establish the solution of inverse coefficient problem for the first time. Demonstrative examples are presented to articulate the implementation and importance of the proposed method.

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

[1] M. Duz, Solution of complex differential equations with variable coefficients by using reduced differential transform, Miskolc Mathematical Notes Vol. 21 (2020), No. 1, pp. 161–170 [2] M. Duz, “Application of Elzaki Transform to first order constant coefficients complex equa ions.” Bulletin of the international mathematical virtual institute., vol. 7, pp. 387–393, 2017, doi: 10.7251/BIMVI1702387D. [3] M. Duz, “On an application of Laplace transforms.” ¨ NTMSCI., vol. 5, no. 2, pp. 193–198, 2017, doi: 10.12732/ijam.v31i1.2. [4] M. Duz, “Solution of complex equations with Adomian Decomposition method.” ¨ TWMS J. App. Eng. Math.., vol. 7, no. 1, pp. 66–73, 2017. [5] Watugala GK (1993) Sumudu transform: a new integral transform to solve differential equations and control engineering problems. Int J Math Educ Sci Technol 24(1):35–43 [6] Anac, H, Merdan, M and Kesemen, Solving for the random component time-fractional partial differential equations with the new Sumudu transform iterative method, SN Applied Sciences (2020) 2:1112 | https://doi.org/10.1007/s42452-020-2625-3 [7] Weerakoon S (1994) Application of Sumudu transform to partial differential equations. Int J Math Educ Sci Technol 25(2):277–283 [8] Weerakoon S (1998) Complex inversion formula for Sumudu transform. Int J Math Edu Sci Technol 29(4):618–621 [9] Demiray ST, Bulut H, Belgacem FBM (2015) Sumudu transform method for analytical solutions of fractional type ordinary differential equations. Math Prob Eng. https ://doi. org/10.1155/2015/13169 0 [10] Kumar M, Daftardar-Gejji V (2018) Exact solutions of fractional partial differential equations by Sumudu transform iterative method. arXiv :1806.03057 v1 [11] Rahman NAA, Ahmad MZ (2017) Solving fuzzy fractional differential equations using fuzzy Sumudu transform. J Nonlinear Sci Appl 10(5):2620–2632 [12] Prakash A, Kumar M, Baleanu D (2018) A new iterative technique for a fractional model of nonlinear Zakharov–Kuznetsov equations via Sumudu transform. Appl Math Comput 334:30–40 [13] H. Eltayeb and A. Kılıçman, A Note on the Sumudu Transforms and Differential Equations, Applied Mathematical Sciences, Vol. 4, 2010, no. 22, 1089 – 1098 [14] Watugala, G.K. The Sumudu transform for functions of two variables, Mathematical Engineering in Industry, 8(4)(2002), pp. 293–302. [15] A. Kılıçman, H. Eltayeb and R. P. Agarwal, On Sumudu Transform and System of Differential Equations, Abstract and Applied Analysis Volume 2010, Article ID 598702, 11 pages [16] Merdan, M., Anac, H., Bekiryazici, Z. and Kesemen, T. 2019. Solving of Some Random Partial Differential Equations by Using Differential Transformation Method and Laplace-Padé Method, Journal of Gumushane University Institute of Science and Technology, 9(1), 108-118. [17] Merdan, M., Anac, H., Bekiryazici, Z. and Kesemen, T. 2019. Solving of Some Random Partial Differential Equations by Using Differential Transformation Method and Laplace-Padé Method, Journal of Gumushane University Institute of Science and Technology, 9(1), 108-118. [18] Merdan,M., Şişman,Ş., 2020. Analysıs Of Random Dıscrete Tıme Logıstıc Model, Sigma Journal of Engineering and Natural Sciences Sigma, 38(3),1269-1298. [19] Merdan,M., Altay, Ö. and Bekiryazici, Z.2020. Investigation of the Behaviour of Volterra Integral Equations with Random Effects, Journal of Gumushane University Institute of Science and Technology, 10(1), 205-216. [20] Merdan. M, 2020, Ordinary and partial complex differential equations with random effects. Gumushane University, Institute of Science, Master's Thesis. [21] Feller W., 1968. An Introduction to Probability Theory and Its Applications, volume 1, 3rd edition. New York: John Wiley &amp; Sons.

Advancing differential equation solutions: A novel Laplace-reproducing kernel Hilbert space method for nonlinear fractional Riccati equations

Fractional ordinary differential equations are crucial for modeling various phenomena in engineering and physics. This study aims to enhance solution methodologies by combining the Laplace transform with the reproducing kernel Hilbert space method (RKHSM). This integration leads to a more effective approach compared to the classical RKHSM. We apply the Laplace-reproducing kernel Hilbert space method (L-RKHSM) to develop novel numerical solutions for nonlinear fractional Riccati differential equations. The L-RKHSM systematically produces both approximate and analytic solutions in series form. We present detailed results for four illustrative examples, showcasing the superior performance of the L-RKHSM over traditional methods. This innovative approach not only advances our understanding of nonlinear fractional ordinary differential equations but also demonstrates its effectiveness through significantly improved outcomes in various applications.

A Comparative Analysis of Conformable, Non-conformable, Riemann-Liouville, and Caputo Fractional Derivatives

This study undertakes a comparative analysis of the non conformable and conformable fractional derivatives alongside the Riemann-Liouville and Caputo fractional derivatives. It examines their efficacy in solving fractional ordinary differential equations and explores their applications in physics through numerical simulations. The findings suggest that the conformable fractional derivative emerges as a promising substitute for the non conformable, Riemann-Liouville and Caputo fractional derivatives within the range of order $\alpha $ where $1/2 &lt; \alpha &lt; 1$.

DETERMINATION OF A KERNEL IN A NONLOCAL PROBLEM FOR THE TIME-FRACTIONAL INTEGRO-DIFFERENTIAL EQUATION

In this work, we consider an inverse problem of determining the kernel of a fractional diffusion equation. The backward problem is the initial-boundary/value problem for this equation with non-local initial and homogeneous Dirichlet conditions. To determine the kernel, an overdetermination condition of the integral form is specified for the solution of the backward problem. Using the Fourier method and an ordinary fractional differential equation with a non-local boundary condition, the inverse problem is reduced to an equivalent problem. Further by using the fixed point argument in suitable Sobolev spaces, the global theorems of existence and uniqueness for the solution of the inverse problem are obtained.

Similarity reduction, group analysis, conservation laws, and explicit solutions for the time-fractional deformed KdV equation of fifth order

Through this paper, we consider the time-fractional deformed fifth-order Korteweg–de Vries (KdV) equation. First of all, we detect its symmetries by Lie group analysis with the help of Riemann–Liouville (R-L) fractional derivatives. These symmetries are employed to convert the considered equation into a fractional ordinary differential (FOD) equation in the sense of Erdélyi-Kober (E-K) fractional operator. Also, a set of new analytical solutions for the equation under study are obtained via the power series method. We test the accuracy and effectiveness of this method by providing a numerical simulation of the obtained solution and studying the effect of [Formula: see text] which is represented graphically in 2D and 3D plots. Added to that, we prove the convergence of the power series solutions. Finally, the computation of the conservation laws is introduced in detail.

Construction of Fractional Pseudospectral Differentiation Matrices with Applications

Differentiation matrices are an important tool in the implementation of the spectral collocation method to solve various types of problems involving differential operators. Fractional differentiation of Jacobi orthogonal polynomials can be expressed explicitly through Jacobi–Jacobi transformations between two indexes. In the current paper, an algorithm is presented to construct a fractional differentiation matrix with a matrix representation for Riemann–Liouville, Caputo and Riesz derivatives, which makes the computation stable and efficient. Applications of the fractional differentiation matrix with the spectral collocation method to various problems, including fractional eigenvalue problems and fractional ordinary and partial differential equations, are presented to show the effectiveness of the presented method.

Lie Symmetry Analysis and Conservation Laws of Fractional Benjamin–Ono Equation

In this paper, the fractional Benjamin–Ono differential equation with a Riemann–Liouville fractional derivative is considered using the Lie symmetry analysis method. Two symmetries admitted by the equation are obtained. Then, the equation is reduced to a fractional ordinary differential equation with an Erdélyi–Kober fractional derivative by one of the symmetries. Finally, conservation laws for the equations are constructed using the new conservation theorem.

On the generalized time fractional reaction–diffusion equation: Lie symmetries, exact solutions and conservation laws

In this paper, Lie symmetry analysis method is applied to the generalized time fractional reaction–diffusion equation. We obtain a conditional symmetric group and some conservation laws of the governing equation. The obtained Lie symmetries are used to reduce the studied fractional partial differential equation to some fractional ordinary differential equations with Riemann–Liouville fractional derivative or Erdélyi-Kober fractional derivative. Furthermore, we obtained asymptotic stable solutions and convergent power series solutions for the reduced equations. The dynamic behavior of these exact solutions is presented graphically.

Exact Solutions for a Class of Variable Coefficients Fractional Differential Equations Using Mellin Transform and the Invariant Subspace Method

AbstractIn this paper, we propose a class of variable coefficients fractional ordinary differential equations (FODEs). Using Mellin transform (MT), we have transformed this class into a functional equation which can’t be solved in general. So, we have selected many special cases of this functional equation that can be solved exactly. After solving these special cases of the functional equation and using the inverse MT, we obtained some exact solutions for the proposed class. The obtained solutions are given in the form of $$H$$ H -function and the Wright function. The results, as special cases, contain some special forms given in the literature. Also, the invariant subspace method (ISM) is utilized for solving a class of nonlinear fractional diffusion equations with variable coefficients. The solutions of this class of nonlinear fractional diffusion equations depend upon the solutions of the proposed class of FODEs.

Numerical analysis of COVID-19 model with Caputo fractional order derivative

This paper focuses on the numerical solutions of a six-compartment fractional model with Caputo derivative. In this model, we obtain non-negative and bounded solutions, equilibrium points, and the basic reproduction number and analyze the stability of disease free equilibrium point. The existence and uniqueness of the solution are proven by employing the Picard–Lindelof approach and fixed point theory. The product–integral trapezoidal rule is employed to simulate the system of FODEs (fractional ordinary differential equations). The numerical results are presented in the form of graphs for each compartment. Finally, the sensitivity of the most important parameter (β) and its impact on COVID-19 dynamics and the basic reproduction number are reported.

A new graph theoretic analytical method for nonlinear distributed order fractional ordinary differential equations by clique polynomial of cocktail party graph

In this paper, we presented a new analytical method for one of the rapidly emerging branches of fractional calculus, the distributed order fractional differential equations (DFDE). Due to its significant applications in modeling complex physical systems, researchers have shown profound interest in developing various analytical and numerical methods to study DFDEs. With this motivation, we proposed an easy computational technique with the help of graph theoretic polynomials from algebraic graph theory for nonlinear distributed order fractional ordinary differential equations (NDFODE). In the method, we used clique polynomials of the cocktail party graph as an approximation solution. With operational integration and fractional differentiation in the Caputo sense, the NDFODEs transformed into a system of algebraic equations and then solved by Newton–Raphson's method to determine the unknowns in the Clique polynomial approximation. The proficiency of the proposed Clique polynomial collocation method (CCM) is illustrated with four numerical examples. The convergence and error analysis are discussed in tabular and graphical depictions by comparing the CCM results with the results of existing numerical methods.

Generalized RKM Method for Solving Sixth-Order Fractional Ordinary Differential Equations

Generalized RKM Method for Solving Sixth-Order Fractional Ordinary Differential Equations

Fractional Boundary Layer Flow: Lie Symmetry Analysis and Numerical Solution

In this paper, we present a fractional version of the Sakiadis flow described by a nonlinear two-point fractional boundary value problem on a semi-infinite interval, in terms of the Caputo derivative. We derive the fractional Sakiadis model by substituting, in the classical Prandtl boundary layer equations, the second derivative with a fractional-order derivative by the Caputo operator. By using the Lie symmetry analysis, we reduce the fractional partial differential equations to a fractional ordinary differential equation, and, then, a finite difference method on quasi-uniform grids, with a suitable variation of the classical L1 approximation formula for the Caputo fractional derivative, is proposed. Finally, highly accurate numerical solutions are reported.

Error analysis of a highly efficient and accurate temporal multiscale method for a fractional differential system

This work is concerned with a nonlinear coupled system of fractional ordinary differential equations (FODEs) with multiple scales in time. The micro system is a nonlinear differential equation with a periodic applied force and the macro system is a fractional differential equation. We are interested in the effective long-term computation of the solution. The main challenge is how to obtain the solution of the coupled fractional problem at a lower computational cost, which calls for us to develop an effective analysis and computation framework. We derive a periodic auxiliary system to approximate the original system and give error estimates. Then we propose a numerical scheme based on the auxiliary problem and analyze the discrete error. An efficient multiscale algorithm that is also applicable to 2-D and 3-D problems is designed. The results of numerical experiments demonstrate the accuracy and computational efficiency of the proposed multiscale method. It is observed that, the computational time is significantly reduced and the multiscale method performs very well in comparison to the fully resolved simulation. Furthermore, as a particular example of applications, we consider a simplified model of the atherosclerotic plaque growth problem, and simply evaluate the effect of fractional parameter and damping coefficient on plaque growth and discuss the physical implication.

Symmetry Analysis and Wave Solutions of Time Fractional Kupershmidt Equation

This study employs the Lie symmetry technique to explore the symmetry features of the time fractional Kupershmidt equation. Specifically, we use the Lie symmetry technique to derive the symmetry generators for this equation, which incorporates a conformal fractional derivative. We use the symmetry generators to transform the fractional partial differential equation into a fractional ordinary differential equation, thereby simplifying the analysis. The obtained reduced equation is of fourth order nonlinear ordinary differential equation. To find the wave solutions, F/G‐expansion process has been used to obatin different types of solutions of the time‐fractional Kuperschmidt equation. The obtained wave solutions are hyperbolic and trigonometric in nature. We then use Maple software to visually depict these wave solutions for specific parameter values, providing insights into the behaviour of the system under investigation. Peak and kink wave solutions are achieved for the given problem.

A MODERN TRAVELING WAVE SOLUTION FOR CAPUTO-FRACTIONAL KLEIN–GORDON EQUATIONS

This research paper introduces a novel approach to deriving traveling wave solutions (TWSs) for the Caputo-fractional Klein–Gordon equations. This research presents a distinct methodological advancement by introducing TWSs of a particular time-fractional partial differential equation, utilizing a non-local fractional operator, specifically the Caputo derivative. To achieve our goal, a novel transformation is considered, that converts a time-fractional partial differential equation into fractional ordinary differential equations, enabling analytical solutions through various analytical methods. This paper employs the homotopy analysis method to achieve the target objectives. To demonstrate the efficiency and applicability of the proposed transform and method, two examples are discussed and analyzed in figures.

New results for fractional ordinary differential equations in fuzzy metric space

&lt;abstract&gt;&lt;p&gt;In this paper, we primarily focused on the existence and uniqueness of the initial value problem for fractional order fuzzy ordinary differential equations in a fuzzy metric space. First, definitions and relevant properties of the Gamma function and Beta function within a fuzzy metric space were provided. Second, by employing the principle of fuzzy compression mapping and Choquet integral of fuzzy numerical functions, we established the existence and uniqueness of solutions to initial value problems for fuzzy ordinary differential equations. Finally, several examples were presented to demonstrate the validity of our obtained results.&lt;/p&gt;&lt;/abstract&gt;

A third-order numerical method for solving fractional ordinary differential equations

In this paper, we developed a novel numerical method for solving general nonlinear fractional ordinary differential equations (FODEs). First, we transformed the nonlinear FODEs into the equivalent Volterra integral equations. We then developed a time-stepping algorithm for the numerical solution of the Volterra integral equations based on the third-order Taylor expansion for approximating the integrands in the Volterra integral equations on a chosen mesh with the mesh parameter $ h $. This approximation led to implicit nonlinear algebraic equations in the unknowns at each given mesh point, and an iterative algorithm based on Newton's method was developed to solve the resulting implicit equations. A convergence analysis of this numerical scheme showed that the error between the exact solution and numerical solution at each mesh point is $ \mathcal{O}(h^{3}) $, independent of the fractional order. Finally, four numerical examples were solved to verify the theoretical results and demonstrate the effectiveness of the proposed method.