Search Results (151)

In this study, we apply the Laplace Transform Homotopy Analysis Method (LTHAM) to numerically solve a fractional-order telegraph equation with a Bessel operator. The iterative scheme developed is tested on multiple examples to evaluate its efficiency. Our observations indicate that the method generates an approximate solution in series form, which converges rapidly to the analytic solution in each instance. The convergence of these series solutions is assessed both geometrically and numerically. Our results demonstrate that LTHAM is a reliable, powerful, and straightforward approach to solving fractional telegraph equations, and it can be effectively extended to solve similar types of equations. Full article

Fractional-order differential equations are prevalent in many scientific fields; hence, their study has seen a renaissance in recent years. The fascinating realm of fractional calculus is explored in this research study, with particular emphasis on the Harry Dym equation. To solve this problem, we use the Laplace Residual Power Series Method (LRPSM) and introduce the New Iterative Method (NIM). Both the mathematical complexity of the Harry Dym problem and the viability of the Caputo operator in this setting are investigated in our work. We go beyond the limitations of traditional mathematical methods to provide novel insights into the results of fractional-order differential equations via careful analysis and cutting-edge procedures. In this paper, we combine theory and practice to provide a novel perspective to the results of high-order fractional differential equations. Our efforts pay off by expanding our knowledge of mathematics and revealing the latent potential of the Harry Dym equation. This study expands researchers’ and mathematicians’ perspectives, bringing in a new and exciting period of progress in the field of fractional calculus. Full article

In this work, we first develop the modified time Caputo fractional Kawahara Equations (MTCFKEs) in the usual Hilbert spaces and extend them to analogous structures within the theory of Hilbert algebras. Next, we employ the residual power series method, combined with the Laplace transform, to introduce a new effective technique called the Laplace Residual Power Series Method (LRPSM). This method is applied to derive the coefficients of the series solution for MTCFKEs in the context of Hilbert algebras. In real Hilbert algebras, we obtain approximate solutions for MTCFKEs under both exact and approximate initial conditions. We present both graphical and numerical results of the approximate analytical solutions to demonstrate the capability, efficiency, and reliability of the LRPSM. Furthermore, we compare our results with solutions obtained using the homotopy analysis method and the natural transform decomposition method. Full article

In this paper, we develop an analytical approximate solution for the nonlinear time-fractional Fisher’s equation using a right starting space function and a unique analytic-numeric technique referred to as the Laplace residual power series approach. The generalized Taylor’s formula and the Laplace transform operator are coupled in the aforementioned method, where the coefficients, obtained through fractional expansion in the Laplace space, are determined by applying the limit concept. In order to validate and illustrate the theoretical methodology of the LRPS technique, as well as to show its effectiveness, adaptability, and superiority in solving various types of nonlinear time and space fractional differential equations, numerical experiments are generated. The obtained analytical solutions are compatible with the precise solutions and concur with those proposed by the other approaches. The outcomes show that the Laplace residual power series strategy is incredibly successful, straightforward to implement, and well suited for handling the complexity of nonlinear problems. Full article

This paper presents a circuit design methodology for the analog realization of time-varying fractional-order chaotic systems. While most existing studies implement such systems by switching between two or more constant fractional orders, these approaches become impractical when the fractional order changes smoothly over time. To overcome this limitation, the proposed method introduces a transfer function approximation specifically designed for variable fractional-order integrators. The formulation relies on a linear and time-invariant definition of the fractional-order operator, ensuring compatibility with Laplace-domain analysis. Under the condition that the fractional-order function is Laplace-transformable and its Bode plot slope lies between $-20$ dB/decade and 0 dB/decade, the system is realized using op-amps and standard RC components. The Grünwald–Letnikov method is employed for numerical calculation of phase portraits, which are then compared with simulation and experimental results. The strong agreement among these results confirms the effectiveness of the proposed method. Full article

This study introduces a novel one-dimensional Fractional Time–Space Stochastic Advection–Diffusion Equation that revolutionizes the modeling of moisture transport within atmospheric boundary layers adjacent to oceanic surfaces. By synthesizing fractional calculus, advective transport mechanisms, and pink noise stochasticity, the proposed model captures the intricate interplay between temporal memory effects, non-local turbulent diffusion, and the correlated-fluctuations characteristic of complex ocean–atmosphere interactions. The framework employs the Caputo fractional derivative to represent temporal persistence and the fractional Laplacian to model non-local turbulent diffusion, and incorporates a stochastic term with a $1/f$ power spectral density to simulate environmental variability. An efficient numerical solution methodology is derived utilizing complementary Fourier and Laplace transforms, which elegantly converts spatial fractional operators into algebraic expressions and yields closed-form solutions via Mittag–Leffler functions. This method’s application to a benchmark coastal domain demonstrates that stronger advection significantly increases the spatial extent of conditions exceeding fog formation thresholds, revealing advection’s critical role in moisture transport dynamics. Numerical simulations demonstrate the model’s capacity to reproduce both anomalous diffusion phenomena and realistic stochastic variability, while convergence analysis confirms the numerical scheme’s robustness against varying noise intensities. This integrated fractional stochastic framework substantially advances atmospheric moisture modeling capabilities, with direct applications to meteorological forecasting, coastal climate assessment, and environmental engineering. Full article

Mathematics and physics are deeply interconnected. In fact, physics relies on mathematical tools like calculus and differential equations. The aim of this article is to introduce tempered Riemann–Liouville (RL) fractional operators and their properties with applications in mathematical physics. The tempered RL substantial fractional derivative is also defined, and its properties are discussed. The Laplace transforms (LTs) of the introduced fractional operators are evaluated. The Hyers–Ulam stability and the existence of a novel tempered fractional differential equation are examined. Moreover, a fractional integro-differential kinetic equation is formulated, and the LT is used to find its solution. A growth model and its graphical representation are established, highlighting the role of novel fractional operators in modeling complex dynamical systems. The developed mathematical framework offers valuable insights into solving a range of scenarios in mathematical physics. Full article

This study examines the asymptotic stability of a continuous stirred tank reactor (CSTR) used for poly(methyl methacrylate) (PMMA) polymerisation, utilizing nonlinear fractional-order mathematical models. By applying Taylor series and Laplace transform techniques analytically and incorporating real plant data, we focus exclusively on the chemical reaction effects in the kinetic constants, disregarding mass transport phenomena. Our results confirm that fractional derivatives significantly enhance the stability and performance of dynamic models compared to traditional integer-order approaches. Specifically, we analyze the stability of a linearized fractional-order system at steady state, demonstrating that the system maintains asymptotic stability within feasible operational limits. Variations in the fractional order reveal distinct impacts on stability regions and system performance, with optimal values leading to improved monomer conversion, polymer concentration, and weight-average molecular weight. Comparative analyses between fractional- and integer-order models show that fractional-order operators broaden stability regions and enable precise tuning of process variables. These findings underscore the efficiency gains achievable through fractional differential equations in polymerisation reactors, positioning fractional calculus as a powerful tool for optimizing CSTR-based polymer production. Full article

The paper investigates semilinear Hilfer fractional systems. A symmetric fractional derivative and its properties are discussed. A symmetrized model for these systems is proposed and examined. A bounded nonlinear function $f$ is applied, depending on the time as well as on the state. The Laplace transformation is used to derive the solution formula for the systems under consideration. The primary contribution of the paper is the formulation and proof of controllability criteria for symmetrized Hilfer systems. To deepen the understanding of the dynamics of such systems, the concept of reflection symmetries is introduced with a detailed analysis of their essential features, including projection functions and a reflection operator. Furthermore, a decomposition of the symmetric Hilfer fractional derivative is presented, utilizing the projection function and reflection operator. This decomposition not only provides a controllability condition for symmetrized Hilfer systems but also clarifies the relationship between the system’s trajectory across subintervals. Two illustrative examples are presented to demonstrate the computational and practical significance of the theoretical results. Full article

Special functions have been widely used in fractional calculus, particularly for addressing the symmetric behavior of the function. This paper provides improved delta Mittag–Leffler and exponential functions to establish new types of fractional difference operators in the setting of Riemann–Liouville and Liouville–Caputo. We give some properties of these discrete functions and use them as the kernel of the new fractional operators. In detail, we propose the construction of the new fractional sums and differences. We also find the Laplace transform of them. Finally, the relationship between the Riemann–Liouville and Liouville–Caputo operators are examined to verify the feasibility and effectiveness of the new fractional operators. Full article

We analyze the error estimates of a fully discrete scheme for solving a semilinear stochastic subdiffusion problem driven by integrated fractional Gaussian noise with a Hurst parameter $H\in (0,1)$. The covariance operator Q of the stochastic fractional Wiener process satisfies $\parallel A^{-\rho }{Q}^{1/2}{\parallel }_{HS}<\infty$ for some $\rho \in [0,1)$, where ${\parallel ·\parallel }_{HS}$ denotes the Hilbert–Schmidt norm. The Caputo fractional derivative and Riemann–Liouville fractional integral are approximated using Lubich’s convolution quadrature formulas, while the noise is discretized via the Euler method. For the spatial derivative, we use the spectral Galerkin method. The approximate solution of the fully discrete scheme is represented as a convolution between a piecewise constant function and the inverse Laplace transform of a resolvent-related function. By using this convolution-based representation and applying the Burkholder–Davis–Gundy inequality for fractional Gaussian noise, we derive the optimal convergence rates for the proposed fully discrete scheme. Numerical experiments confirm that the computed results are consistent with the theoretical findings. Full article

This paper addresses the extension of Martinez–Kaabar (MK) fractal–fractional calculus (for simplicity, in this research work, it is referred to as MK calculus) to the field of integral transformations, with applications to some solutions to integral equations. A new notion of Laplace transformation, named MK Laplace transformation, is proposed, which incorporates the MK $\alpha ,\gamma$-integral operator into classical Laplace transformation. Laplace transformation is very applicable in mathematical physics problems, especially symmetrical problems in physics, which are frequently seen in quantum mechanics. Symmetrical systems and properties can be helpful in applications of Laplace transformations, which can help in providing an effective computational tool for solving such problems. The main properties and results of this transformation are discussed. In addition, the MK Laplace transformation method is constructed and applied to the non-integer-order first- and second-kind Volterra integral equations, which exhibit a fractal effect. Finally, the MK Abel integral equation’s solution is also investigated via this technique. Full article

Inspired by prior research on fractional calculus, we introduce new fractional integral and derivative operators: the $({\rho }_1,{\rho }_2,k_1,k_2,\phi )$-proportional integral and the $({\rho }_1,{\rho }_2,k_1,k_2,\phi )$-proportional Hilfer fractional derivative. Numerous previous studied fractional integrals and derivatives can be considered as particular instances of the novel operators introduced above. Some properties of the $({\rho }_1,{\rho }_2,k_1,k_2,\phi )$-proportional integral are discussed, including mapping properties, the generalized Laplace transform of the $({\rho }_1,{\rho }_2,k_1,k_2,\phi )$-proportional integral and $({\rho }_1,{\rho }_2,k_1,k_2,\phi )$-proportional Hilfer fractional derivative. The results obtained suggest that the most comprehensive formulation of this fractional calculus has been achieved. Under the guidance of the findings from earlier sections, we investigate the existence of mild solutions for the $({\rho }_1,{\rho }_2,k_1,k_2,\phi )$-proportional Hilfer fractional Cauchy problem. An illustrative example is provided to demonstrate the main results. Full article

Inherent constant Q attenuation can be described using fractional Laplacian operators. Typically, the fractional Laplacian viscoacoustic or viscoelastic wave equations are addressed utilizing the staggered-grid pseudo-spectral (SGPS) method. However, this approach results in time numerical dispersion errors due to the low-order finite difference approximation. In order to address these time-stepping errors, a k-space-based temporal compensating scheme is established to solve the first-order viscoacoustic wave equation. This scheme offers the advantage of being nearly free from grid dispersion for homogeneous media and enhances simulation stability. Numerical examples indicate that the proposed k-space scheme aligns well with analytical solutions for homogeneous media. Additionally, this method demonstrates excellent applicability for complex models and is more efficient due to its ability to adopt a larger time step compared with conventional staggered-grid pseudo-spectral methods. Full article

In this article, a diffusion component in an SIR model is introduced, and its impact is analyzed using fractional calculus. We have included the diffusion component in the SIR model. in order to illustrate the variations. Here, we have applied the general fractional derivative to analyze the impact. The Laplace decomposition technique is employed to obtain the numerical outcomes of the model. In order to observe the effect of the diffusion component in the SIR model, graphical solutions are also displayed. Full article