Fractional Differential Equations

Scattered Data Approximation

Most engineering and scientific phenomena, such as the surface of a landscape or the continuously changing temperature at a location are inherently infinite in space or time or both. We cannot measure all the data. Generally it is possible to record surface elevation values or the temperature only at some specific locations and times.

Interpolation and Approximation

Strong Formulation Finite Element Method Based on Differential Quadrature: A Survey

A method based on the Jacobi tau approximation for solving multi-term time-space fractional partial differential equations

Optimal solution of the fractional-order smoking model and its public health implications

Fractional differential equations (FDEs) arising in engineering and other sciences describe nature sufficiently in terms of symmetry properties. This paper proposes a numerical technique to approximate ordinary fractional initial value problems by applying fractional radial basis function neural network. The fractional derivative used in the method is considered Riemann-Liouville type. This method is simple to implement and approximates the solution of any arbitrary point inside or outside the domain after training the ANN model. Finally, three examples are presented to show the validity and applicability of the method.

https://www.mdpi.com/2073-8994/15/6/1275/pdf?version=1687161782

Solving Fractional Order Differential Equations by Using Fractional Radial Basis Function Neural Network

The paper’s main purpose is to find the unknown source function for the conformable heat equation. In the case of (Φ,g)∈L2(0,T)×L2(Ω), we give a modified Fractional Landweber solution and explore the error between the approximate solution and the desired solution under a priori and a posteriori parameter choice rules. The error between the regularized and exact solution is then calculated in Lq(D), with q≠2 under some reasonable Cauchy data assumptions.

/pdf/reconstructing-the-unknown-source-function-of-a-fractional-1unemvwl.pdf

Reconstructing the Unknown Source Function of a Fractional Parabolic Equation from the Final Data with the Conformable Derivative

In this study, the Caputo-Fabrizio fractional derivative is applied to generate a new category of variable-order fractional 2D optimization problems in an unbounded domain. To approach this problem, a novel hybrid method is devised which simultaneously exploits two sets of basis functions: a new class of basis functions namely the modified second kind Chebyshev functions and the shifted second kind Chebyshev cardinal functions. With the aid of the Lagrange multipliers method, the presented method converts the optimization problem under study into a system of algebraic equations, which are uncomplicated to solve. To confirm the precision of this new hybrid method, a convincing number of test problems are examined.

A hybrid method for variable-order fractional 2D optimal control problems on an unbounded domain

This paper introduces the blossomed and grafted blossomed trees (BT and GBT, respectively) which are two new types of rooted trees. The trees consist of a finite number of solid and hollow vertices that represent buds and blossoms, respectively. Then the relation between them and derivative operators in a differential equation is analyzed. These concepts not only demonstrate how natural phenomena can be inspiring in mathematics, but also we can devise a method based on the BT and GBT for finding s-stage Runge–Kutta coefficients, with the appropriate degree of accuracy. However, solving higher-order differential equations in the general form entails dealing with numerous complex expressions, while the new algorithm based on the BT and GBT provides simplicity and practicability. One of the advantages of using BT and GBT is easier ordering and standardizing the relations derived from derivative operators in differential equations and synchronizing them with numerical methods such as the Runge–Kutta algorithms. In brief, this approach helps avoiding mistakes in spite of a high volume of processing operations. Moreover, the kth-order derivative for monotonically labeled GBT having n buds and blossoms using these types of trees with $$k+n$$
 buds and blossoms is studied. This strategy is also adopted for GBT without labeling.

Zakieh Avazzadeh

Papers

Optimal solution of the fractional-order smoking model and its public health implications

Solving Fractional Order Differential Equations by Using Fractional Radial Basis Function Neural Network

Reconstructing the Unknown Source Function of a Fractional Parabolic Equation from the Final Data with the Conformable Derivative

A hybrid method for variable-order fractional 2D optimal control problems on an unbounded domain

Relation Between New Rooted Trees and Derivatives of Differential Equations