Fractional Differential Equations Research Articles

Coupled system of sequential partial $\sigma \left( .,.\right)-$Hilfer fractional differential equations with weighted double phase operator: Existence, Hyers-Ulam stability and controllability

In this paper, we are concerned by a sequential partial Hilfer fractional di erential system with weighted double phase operator. First, we introduce the concept of Hyers-Ulam stability with respect to an operator L for an abstract equation of the form u = LFu in Banach lattice by using the xed point ar- guments and spectral theory. Then, we prove the controllability and apply the previous results obtained for abstract equation to prove existence and Hyers- Ulam stability of a coupled system of sequential fractional partial di erential equations involving a weighted double phase operator. Finally, example illustrat- ing the main results is constructed. This work contains several new ideas, and gives a uni ed approach applicable to many types of di erential equations.

Analysis of fractional Swift-Hohenberg models using highly accurate techniques within the Caputo operator framework

This investigation aims to analyze and solve the fractional SwiftHohenberg (FSH) equation using the Aboodh residual power series method (ARPSM) and Aboodh transform iterative method (ATIM) within the Caputo operator framework. This equation is widely used in modeling pattern formation phenomena in various physical systems. Thus, the current study focuses on understanding the mechanics and dynamics of wave propagation described by this equation. Additionally, it investigates the impact of the fractional parameter on the behavior of these waves. By employing both ARPSM and ATIM, we aim to obtain highly accurate and efficient approximations to this equation. The effectiveness of these methods is demonstrated through numerical simulations, where we compare the obtained results with existing analytical and numerical solutions. Our findings highlight the utility of the ARPSM and ATIM in studying complex nonlinear fractional differential equations, providing valuable insights into pattern formation dynamics governed by the Swift-Hohenberg equation.

Numerical and visual analysis of Maclaurin type inequalities in the setting of generalized fractional calculus and applications

Numerical and visual analysis of Maclaurin type inequalities in the setting of generalized fractional calculus and applications

Fractional Partial Differential Equation Modeling for Solar Cell Charge Dynamics

Fractional Partial Differential Equation Modeling for Solar Cell Charge Dynamics

Certain Subclass of Harmonic Univalent Functions Associated With Fractional Calculus Operator

In this paper, a class of complex-valued harmonic univalent functions f (z) in the open disk U is defined by fractional calculus operator. We obtained coefficient bounds, distortion inequalities, convex combination, extreme points and convolution conditions for this class.

On a Model for Solving Mixed Fractional Integro Differential Equation

In this work, the mixed fractional integro differential equation (MfrIo-DE) of the second kind, under certain condition is considered, in the space L2(−1, 1)× C [0, T]; T < 1 T is the time. The position kernel k (|x−y|) of IE has a singularity. After integrating and using the properties of fractional integral, we have a MIE in position and time, where the kernel of position takes the singular form k (|x−y|), and the kernel of time takes the singular Abel form (t −τ)α−1 , 0 < α < 1. Then, using separation of variable method, under certain substitution, we obtain FIE in position, with variable fractional coefficients in time. Using the Toeplitz matrix method (TMM), we have a nonlinear algebraic system (NAS). Moreover, numerical results are obtained and discussed, especially when 0 < α < 1. Also, the solutions of the mixed equation are considered when α = 0, α = 1. Finally, the error estimate, in each case, is computed.

Analysis and Modeling of Fractional Order LC Series Resonant Boost Converter Based on Fractional Calculus and Laplace Transform

ABSTRACTBased on the fractional‐order (FO) characteristics of inductors and capacitors, many basic PWM DC–DC converters are defined and modeled by FO calculus in previous studies, but the research of FO resonant dc converters is still in its early stage. Therefore, this paper adopts FO calculus and Laplace transform to model and analyze a ZCS boost converter with an LC resonant tank, which mainly focuses on the influence of the FO component on the soft switching characteristics, converter efficiency, and output voltage gain of this isolated dc converter. This paper extends the topology to the fractional order domain and conducts FO modeling. The order of the resonant inductor and capacitor will affect the amplitude/phase of the resonant current and then affect the converter ZCS characteristic. The analysis demonstrates that the reduction of FOI and FOC order is not conducive to the ZCS of switching devices and converter efficiency. Based on the voltage and current relationship of the FO LC resonant tank, a parameter design guideline is presented. Numerical tools are used to solve the FO output voltage gain and draw the gain curve. And the order of FOI and FOC provides new flexibility for the converter output gain and soft switching design. Finally, simulations and a 360 W hardware experimental prototype are conducted to verify the accuracy and effectiveness of theoretical analysis.

Fractional time differential equations as a singular limit of the Kobayashi–Warren–Carter system

This paper is concerned with a singular limit of the Kobayashi–Warren–Carter system, a phase field system modelling the evolutions of structures of grains. Under a suitable scaling, the limit system is formally derived when the interface thickness parameter tends to zero. Different from many other problems, it turns out that the limit system is a system involving fractional time derivatives, although the original system is a simple gradient flow. A rigorous derivation is given when the problem is reduced to a gradient flow of a single-well Modica–Mortola functional in a one-dimensional setting.

Study on Controllability for Ψ-Hilfer Fractional Stochastic Differential Equations

The goal of this paper is to study the existence of a mild solution and controllability for a class of neutral stochastic differential equations (SDEs) involving the Ψ-Hilfer fractional derivatives, a generalization of the well-known Riemann–Liouville fractional derivative using almost sectorial operators. Sufficient conditions for controllability are established using the notion of measure of noncompactness (MNC) and the Mönch fixed-point theorem. An example is given to illustrate the abstract findings.

Lie Symmetries With Exact Solution for Time‐Fractional Foam Drainage Equation

ABSTRACTIn this article, we consider a nonlinear time‐fractional foam drainage equation and investigate the behavior of fluid flow through the foam. First, we apply the symmetry analysis and obtain the infinitesimal group of transformations involving some arbitrary parameters. Further, we classify the one‐dimensional optimal algebra, and using that, we obtain the similarity variables, which consequently reduces the time fractional partial differential equation (FPDE) to fractional‐order ordinary differential equation (FODE). Next, under the invariance condition, we construct exact solution for the given model and presented graphically. Furthermore, we derive the two‐dimensional optimal systems and nonclassical symmetry for the governing equation. Finally, we study the nature of fluid flow through the foam by numerical simulation.

Value of the Stretched Exponential and Fractional-order Model in Differentiating Hepatocellular from Intrahepatic Cholangiocarcinoma.

It remains unknown whether the parameters obtained using the Stretched Exponential Model (SEM) and Fractional Order Calculus (FROC) models can help distinguish Hepatocellular Carcinoma (HCC) from Intrahepatic Cholangiocarcinoma (ICC). This study aimed to evaluate the application value of the parameters of the 3.0T Magnetic Resonance Imaging (MRI) high-order SEM and FROC diffusion model in differentiating HCC and ICC. Patients with pathologically confirmed HCC and ICC were prospectively enrolled. Diffusion-weighted imaging scans with multiple b-values were acquired 2 weeks before the surgery. The original MRI images were fitted using the mono-exponential model, SEM, and FROC, and several parameters were obtained for the analysis. In total, 74 patients with HCC and 21 with ICC were included in the study. Significant differences between the HCC and ICC groups were noted in the Apparent Diffusion Coefficient (ADC: p = 0.007), Distributed Diffusion Coefficient (DDC: p < 0.001), and Diffusion coefficient (D: p < 0.001), as each value was significantly lower in the HCC than in the ICC group. The area under the receiver operating characteristic curve of ADC, DDC, and D was 0.694, 0.812, and 0.825, respectively, and the most effective corresponding cut-off values were 1.135 μm2/ms, 1.477 μm2/ms, and 1.104 μm2/ms, respectively. The diffusion parameters DDC from the SEM and D from the FROC model have been found to be more effective in discriminating HCC and ICC than the ADC from the mono-exponential model. Combining these quantitative parameters can improve the MRI's diagnostic accuracy, providing useful information for the preoperative differential diagnosis between HCC and ICC.

Existence and uniqueness of generalized normal solutions to first order fractional differential equations and applications

We study the existence and uniqueness of continuous generalized normal solutions to initial value problems of first order fractional differential equations. We use the Banach contraction principle and the Weissinger fixed point theorem to obtain our results. We assume that the absolute values of the nonlinearities have upper bound functions in a subspace of continuous functions. As an example, the results are applied to equations with nonlinearities arising in logistic type population models with heterogeneous environments, and to population models of Ricker type. For more information see https://ejde.math.txstate.edu/Volumes/2024/81/abstr.html

On a mixed partial Caputo derivative and its applications to a hyperbolic partial fractional differential equation

AbstractWe propose an alternative definition of a mixed partial derivative in the Caputo sense for functions of two variables defined on the rectangle $$P=[0,a]\times [0,b]$$ P = [ 0 , a ] × [ 0 , b ] ($$a&gt;0, b&gt;0$$ a &gt; 0 , b &gt; 0 ). We give an integral representation of functions possessing such a derivative. Moreover, we study the existence and uniqueness of a solution, as well as the Ulam–Hyers type stability of a fractional counterpart of a nonlinear continuous Goursat-Darboux system described by the introduced Caputo derivative. This paper is a continuation of our paper [R. Kamocki, C. Obczyński, On the single partial Caputo derivatives for functions of two variables, Periodica Mathematica Hungarica 87(2), (2023), 324–339].

Solving nonlinear fractional differential equations by common fixed point results for a pair of (α, Θ)-type contractions in metric spaces

Abstract The problem of common solutions for nonlinear equations has significant theoretical and practical value. In this article, we first introduce a new concept of a pair of ( α , Θ ) \left(\alpha ,\Theta ) -type contractions, and then, we present some common fixed point results for the contractions in complete metric spaces. Finally, our results are applied to consider the existence, uniqueness and approximation of common solutions for two classes of nonlinear fractional differential equations.

New Studies for Dynamic Programming and Fractional Differential Equations in Partial Modular b-Metric Spaces

This study explores innovative insights into the realms of dynamic programming and fractional differential equations, situated explicitly within the framework of partial modular b-metric spaces enriched with a binary relation R, proposing a novel definition for a generalized ℷC-type Suzuki R-contraction specific to these spaces. By doing so, we pave the way for a range of relation-theoretical common fixed-point theorems, highlighting the versatility of our approach. To illustrate the practical relevance of our findings, we present a compelling example. Ultimately, this work aims to enrich the existing academic discourse and stimulate further research and practical applications within the field.

Mittag-Leffler Functions and the Sawi Transform: A New Approach to Fractional Calculus

This study uses the Sawi transform and Mittag-Leffler function to introduce a novel method of fractional calculus. The basic characteristics of the Sawi transform and its connection to other transforms such as the Laplace transform are explored. We demonstrate the application of the Sawi transform in deriving novel solutions to fractional differential equations and elucidate the importance of the Mittag-Leffler function in these scenarios. We show the efficiency of the presented method by presenting some nonhomogeneous equations and initial value problems. This study proves the efficacy of the Sawi transform by handling some problems in the sense of Caputo fractional calculus.

Multiple and Nonexistence of Positive Solutions for a Class of Fractional Differential Equations with p-Laplacian Operator

Research about multiple positive solutions for fractional differential equations is very important. Based on some outstanding results reported in this field, this paper continue the focus on this topic. By using the properties of the Green function and generalized Avery–Henderson fixed point theorem, we derive three positive solutions of a class of fractional differential equations with a p-Laplacian operator. We also study the nonexistence of positive solutions to the eigenvalue problem of the equation. Three examples are given to illustrate our main result.

Existence and Uniqueness Results for a Coupled System of Nonlinear Hadamard Fractional Differential Equations

In this paper, we examine the existence and uniqueness of solutions to a system of four-point non-separated Hadamard fractional differential equations. Applying Leray-Schauder’s alternative yields the existence of solutions; Banach’s contraction principle establishes the uniqueness of the solution. Lastly, we provide two examples to demonstrate our results.

Remotely multi-term and transparently single-term fractional differential equations

Remotely multi-term and transparently single-term fractional differential equations

Some best proximity point results on b-metric spaces with an application

In this paper, we introduce the concept of \vartheta -p-contraction mapping on b-metric spaces. Then, we obtain some best proximity results for these mappings. Also, an example to support the validity and superiority of our result has been given. Lastly, for the existence of solutions of nonlinear fractional differential equations of Caputo type we provide an application.