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This paper considers the regional boundary observability problem for semilinear time-fractional systems. The main objective is to reconstruct the initial state on a subregion of the boundary of the evolution domain of the considered fractional system using the output equation. We proceed by providing a link between the regional boundary observability of the considered semilinear system on the desired boundary subregion, and the regional observability of its linear part, in a well chosen subregion of the evolution domain. By adding some assumptions on the nonlinear term appearing in the considered system, we give the main theorem that allows us to reconstruct the initial state in the well-chosen subregion using the Hilbert uniqueness method (HUM). From it, we recover the initial state on the boundary subregion. Finally, we provide a numerical example that backs up the theoretical results presented in this paper with a satisfying reconstruction error. [ABSTRACT FROM AUTHOR]

In this work, we suggest a new method for solving linear multi-term time-fractional wave-diffusion equations, which is named the modified fractional reduced differential transform method (m-FRDTM). The importance of this technique is that it suggests a solution for a multi-term time-fractional equation. Very few techniques have been proposed to solve this type of equation, as will be shown in this paper. To show the effectiveness and efficiency of this proposed method, we introduce two different applications in two-term fractional differential equations. The three-dimensional and two-dimensional plots for different values of the fractional derivative are depicted to compare our results with the exact solutions. [ABSTRACT FROM AUTHOR]

In this paper, we study fractional subdiffusion fourth parabolic equations containing Caputo and Caputo-Fabrizio operators. The main results of the paper are presented in two parts. For the first part with the Caputo derivative, we focus on the global and local well-posedness results. We study the global mild solution for biharmonic heat equation with Caputo derivative in the case of globally Lipschitz source term. A new weighted space is used for this case. We then proceed to give the results about the local existence in the case of locally Lipschitz source term. To overcome the intricacies of the proofs, we applied Lp − Lq estimate for biharmonic heat semigroup, Banach fixed point theory, some estimates for Mittag-Lefler functions and Wright functions, and also Sobolev embeddings. For the second result involving the Cahn-Hilliard equation with the Caputo-Fabrizio operator, we first show the local existence result. In addition, we first provide that the connections of the mild solution between the Cahn-Hilliard equation in the case 0 < α < 1 and α = 1. This is the first result of investigating the Cahn-Hilliard equation with this type of derivative. The main key of the proof is based on complex evaluations involving exponential functions, and some embeddings between Lp spaces and Hilbert scales spaces. [ABSTRACT FROM AUTHOR]

This paper is concerned with the numerical approximations for the variable coefficient fourth-order fractional sub-diffusion equations subject to the second Dirichlet boundary conditions. We construct two effective difference schemes with second order accuracy in time by applying the second order approximation to the time Caputo derivative and the sum-of-exponentials approximation. By combining the discrete energy method and the mathematical induction method, the proposed methods proved to be unconditional stable and convergent. In order to overcome the possible singularity of the solution near the initial stage, a difference scheme based on non-uniform mesh is also given. Some numerical experiments are carried out to support our theoretical results. The results indicate that the our two main schemes has the almost same accuracy and the fast scheme can reduce the storage and computational cost significantly. [ABSTRACT FROM AUTHOR]

In this paper, we propose a fast and accurate numerical method for solving the time-fractional mobile–immobile advection–dispersion equation with Caputo derivative of order α for. The main advantage of this method is its order of accuracy, which is 3 - α . We discretize the problem in space by the Fourier spectral method and time by the fourth-order exponential time-differencing Runge–Kutta method and a high-order approximation for Caputo derivative. We eliminate the first and second spatial derivatives of the well-known equation by the properties of the Fourier spectral method. We also prove the convergence of this method. Several numerical examples are provided to illustrate the efficiency and accuracy of the proposed method. We show that the temporal convergence rates for different values of α are more than two and conclude the effectiveness of the method for solving the well-known equation. [ABSTRACT FROM AUTHOR]

In the current paper, we are concerned with the existence and uniqueness of mild solutions to a Cauchy problem involving a time-space fractional diffusion equation with an exponential semilinear source. By using the iteration method and some L p − L q -type estimates of fundamental solutions associated with the Mittag-Leffler function, we study the well-posedness of the problem in two different cases corresponding to two assumptions on the Cauchy data. On the one hand, when considering initial data in L p ( R N) ∩ L ∞ ( R N) , the problem possesses a local-in-time solution. On the other hand, we obtain a global existence result for a mild solution with small data in an Orlicz space. [ABSTRACT FROM AUTHOR]

As we know one of the most important equations which have many applications in various areas of physics, mathematics, and financial markets, is the Sturm–Liouville equation. In this paper, by using the α - ψ -contraction technique in fixed point theory and employing some functional inequalities, we study the existence of solutions of the partial fractional hybrid case of generalized Sturm–Liouville-Langevin equations under partial boundary value conditions. Towards the end, we present two examples with numerical and graphical simulation to illustrate our main results. [ABSTRACT FROM AUTHOR]

The generalized time‐fractional Fisher's equation is a substantial model for illustrating the system's dynamics. Studying effective numerical methods for this equation has considerable scientific importance and application value. In that direction, this paper presents designing and analyzing a high‐order numerical scheme for the generalized time‐fractional Fisher's equation. The time‐fractional derivative is taken in the Caputo sense and approximated using Euler backward discretization. The quasilinearization technique is used to linearize the problem, and then a compact finite difference scheme is considered for discretizing the equation in space direction. Our numerical method is convergent of Ok2−α+h4$$ O\left({k}^{2-\alpha }+{h}^4\right) $$, where h$$ h $$ and k$$ k $$ are step sizes in spatial and temporal directions, respectively. Three problems are tested numerically by implementing the proposed technique, and the acquired results reveal that the proposed method is suitable for solving this problem. [ABSTRACT FROM AUTHOR]

This paper addresses the numerical solution of nonlinear time-fractional Fisher equations via local meshless method combined with explicit difference scheme. This procedure uses radial basis functions to compute space derivatives while Caputo derivative scheme utilizes for time-fractional integration to semi-discretize the model equations. To assess the accuracy, maximum error norm is used. In order to validate the proposed method, some non-rectangular domains are also considered. [ABSTRACT FROM AUTHOR]

This paper investigates a class of the time-fractional diffusion-wave equation (TFDWE), which incorporates a fractional derivative in the Caputo sense of order α + 1 where 0 < α < 1. Initially, the original problem is transformed into a new model that incorporates the fractional Riemann–Liouville integral. Subsequently, a more generalized model is considered and investigated, which includes the generalized fractional integral with memory kernel. The paper establishes the existence and uniqueness of a numerical solution for the problem. We introduce a discretization technique for approximating the generalized fractional Riemann–Liouville integral and study its fundamental properties. Based on this technique, we propose a second-order numerical scheme for approximating the generalized model. The proof of the stability of the proposed difference scheme when considered in terms of the L 2 norm is established. Furthermore, a difference scheme is devised for solving a nonlinear generalized model with time delay. The convergence and stability analysis of this particular case are established using the discrete Gronwall inequality. Numerical examples are provided to evaluate the effectiveness and reliability of the proposed methods and to support our theoretical analysis. [ABSTRACT FROM AUTHOR]

The study of nonlinear phenomena associated with physical phenomena is a hot topic in the present era. The fundamental aim of this paper is to find the iterative solution for generalized quintic complex Ginzburg–Landau (GCGL) equation using fractional natural decomposition method (FNDM) within the frame of fractional calculus. We consider the projected equations by incorporating the Caputo fractional operator and investigate two examples for different initial values to present the efficiency and applicability of the FNDM. We presented the nature of the obtained results defined in three distinct cases and illustrated with the help of surfaces and contour plots for the particular value with respect to fractional order. Moreover, to present the accuracy and capture the nature of the obtained results, we present plots with different fractional order, and these plots show the essence of incorporating the fractional concept into the system exemplifying nonlinear complex phenomena. The present investigation confirms the efficiency and applicability of the considered method and fractional operators while analyzing phenomena in science and technology. [ABSTRACT FROM AUTHOR]

We investigate the error of the (semidiscrete) Galerkin method applied to a semilinear subdiffusion equation in the presence of nonsmooth initial data. The diffusion coefficient is allowed to depend on time. We use the energy method to find optimal bounds on the error under weak natural assumptions on the diffusivity. First, we prove the result for the linear problem and then use the "frozen nonlinearity" technique coupled with various generalizations of Grönwall inequality to carry the result to the semilinear case. The paper ends with numerical illustrations supporting the theoretical results. [ABSTRACT FROM AUTHOR]

The fractional generalized Bagley–Torvik equation (FGB-TE) is a mathematical description of the motion of an immersed plate in a Newtonian fluid. The analytical study of the FGB-TE with uncertain initial conditions and two independent fractional orders is usually complex and difficult. Therefore, it is necessary to develop a proper and effective technique to solve the fuzzy fractional generalized Bagley–Torvik equation (FFGB-TE) analytically. This paper presents the analytical fuzzy solution of the FFGB-TEs based on the concept of the fuzzy Caputo generalized Hukuhara differentiability (g H -differentiability) using the fuzzy Laplace transform (FLT) technique. The closed-form solution of FFGB-TEs is presented for both the homogeneous and non-homogeneous cases in terms of the Mittag-Leffler function (MLF) involving double series. Several significant results are introduced and proven with true reasoning. We illustrate our proposed analytical approach with the help of several demonstrative examples. To enhance the novelty of the proposed work, we solved the FFGB-TE as an application of the motion of an immersed plate and visualized their graphs to support the theoretical results. • Analytical solutions of fuzzy fractional Bagley-Torvik equation are investigated. • The potential solutions are extracted using fuzzy Caputo g H -differentiability. • Closed-form solutions of proposed scheme are presented using Mittag-Leffler function. • Several significant results are presented using the proposed analytical approach. • The real-world problem is solved as an application of the proposed study. [ABSTRACT FROM AUTHOR]

Fractional Bagley-Torvik differential equations can be used to model a variety of natural phenomena in many branches of applied mathematics and engineering in general. The focus of this study is on solving the fractional Bagley-Torvik equations by using sine-cosine wavelet. To this end, the operational matrix of fractional integration is obtained for sine-cosine wavelets. By utilizing this matrix, fractional Bagley-Torvik differential equation is transformed to a system of linear algebraic equations with unknown coefficients, which in turn can readily be solved using numerical solution methods. Test examples are provided to demonstrate the validity and efficiency of this approach. [ABSTRACT FROM AUTHOR]

In this paper, we have studied the seventh‐order time‐fractional Sawada Kotera Ito equation via natural transform decomposition method (NTDM) with singular and nonsingular kernel derivatives. The fractional derivative considered in Caputo, Caputo‐Fabrizio (CF), and Atangana‐Baleanu‐Caputo (ABC). We employed the natural transform with the Adomian decomposition process on time‐fractional Sawada Kotera Ito equation to obtain the solution. To validate the method, we have considered a few examples and compared with the actual results. Numerical results are in accordance with the existing results. [ABSTRACT FROM AUTHOR]

In this study, we employ the effective iterative method to address the fractional Wu-Zhang Equation within the framework of the Caputo Derivative. The effective iterative method offers a practical approach to obtaining approximate solutions for fractional differential equations. We seek to provide insights into its solution and behavior by applying this method to the Wu-Zhang Equation. Through numerical analysis and the presentation of relevant tables and Figures, we demonstrate the accuracy and efficiency of this method in solving the fractional Wu-Zhang Equation. This research contributes to the understanding and solution of fractional-order differential equations and their applications in various scientific and engineering domains. [ABSTRACT FROM AUTHOR]

In this paper, we use both Riemann-Liouville integral and Caputo derivative to investigate a new nonlinear singular differential problem of Lane and Emden type. We note that for the studied singular problem, we will be concerned with some anti periodic conditions. So, we prove a first uniqueness result by application of Banach principle of contraction. Then, we prove a "second" existence result by application of Schaefer theorem. Two illustrative examples are discussed in details to show the applicability of the obtained results. [ABSTRACT FROM AUTHOR]

This work proposes a semi-analytical new hybrid approach, so-called differential -transform method (D TM), to evaluate the behavior of n-space dimensional fractional-nonlinear hyperbolic-like wave equations, where time-fractional derivative is considered in Caputo format. The D TM is the hybrid method in which projected differential transform is implemented after imposing the recently introduced integral transform, i.e., so-called transform [W. Zhao and S. Maitama, J. Appl. Anal. Comput. 10, 1223 (2020)]. The efficiency and applicability of the proposed D TM had been tested by considering three different test examples of the Caputo time-fractional nonlinear hyperbolic-like wave equations in terms of absolute error norms, and the different order D TM solutions are compared with exact solution behaviors and the existing results, for the large time level τ ∈ [ 0 , 1 0 ]. In addition, the convergence analysis of D TM is studied theoretically and verified it numerically as well as graphically, which confirms that the numerical experiments via D TM for distinct fractional orders support the theoretical findings excellently, and the presented D TM results converge to their exact solution behavior, very fast. The evaluated series approximations are expressed in the compact form of Mittag-Leffler functions. [ABSTRACT FROM AUTHOR]

In this article, the Elzaki homotopy perturbation transform method (EHPTM) is profusely employed to discover the approximate solutions of fractional‐order (FO) heat‐like equations. To show this, we first establish the Elzaki transform in the context of the Atangana–Baleanu fractional derivative in the Caputo sense (ABC) and then extend it to heat‐like equations. Our suggested approach has been reinforced by convergence and error analysis. The validity of the novel technique is tested with the aid of some illustrative examples. Comparative analysis has been established for both fractional and integer‐order solutions. EHPTM is considered to be an appropriate and convenient approach for solving FO time‐dependent linear and nonlinear partial differential problems. Plots and tables are being used to reveal the findings. The relatively high validity and reliability of the present approach are also reflected by the comparative solution analysis by means of statistical analysis. [ABSTRACT FROM AUTHOR]

In this article, we studied the time‐fractional Burgers–Huxley equation using the natural transform decomposition method. The fractional operator is treated in the Caputo, Caputo–Fabrizio, and Atangana–Baleanu senses. We employed the natural transform with the Adomian decomposition process on time‐fractional Burgers–Huxley equation to obtain the solution. To establish the uniqueness and convergence of the accomplished solution, the Banach's fixed point theorem is used. The obtained findings are visually shown in two‐ and three‐dimensional graphs for various fractional orders. To illustrate the efficacy of the method under discussion, numerical simulations are provided. The proposed solution captures the behavior of the reported findings for various fractional orders. A comparative study was conducted to ascertain the proposed method's correctness. The findings of this study establish that the technique investigated is both efficient and accurate for solving nonlinear fractional differential equations that arise in science and technology. [ABSTRACT FROM AUTHOR]

The article deals with an optimal control problem for the model system described by the one-dimensional inhomogeneous diffusion-wave equation that is a generalization of the wave equation to the case when the time derivative is replaced with the fractional Caputo derivative. In the general case, we consider both boundary and distributed controls which are Lebesgue -summable functions, with and . We state and study the two types of optimal control problems: The problem of finding a minimal norm control for a given control time and the performance problem of finding a control that brings the system to a given state in the minimal time for a given constraint on the control norm. The study bases on using an exact solution to the diffusion-wave equation, which allows us to reduce the optimal control problem to an infinite-dimensional -moment problem. We also examine the similar finite-dimensional -moment problem that uses an approximate solution to the diffusion-wave equation and analyze the well-posedness and solvability of this problem. Also, we exhibit some example of calculating the boundary control by using the finite-dimensional -moment problem. [ABSTRACT FROM AUTHOR]

In the last few decades, it has become increasingly clear that fractional calculus always plays a very significant role in various branches of applied sciences. For this reason, fractional partial differential equations (FPDEs) are of more importance to model the different physical processes in nature more accurately. Therefore, the analytical or numerical solutions to these problems are taken into serious consideration and several techniques or algorithms have been developed for their solution. In the current work, the idea of fractional calculus has been used, and fractional Fornberg Whitham equation (FFWE) is represented in its fractional view analysis. A well-known method which is residual power series method (RPSM), is then implemented to solve FFWE. The RPSM results are discussed through graphs and tables which conform to the higher accuracy of the proposed technique. The solutions at different fractional orders are obtained and shown to be convergent toward an integer-order solution. Because the RPSM procedure is simple and straightforward, it can be extended to solve other FPDEs and their systems. [ABSTRACT FROM AUTHOR]

This work aims at a new semi-analytical method called the variational iteration transform method for investigating fractional-order Emden–Fowler equations. The Shehu transformation and the iterative method are applied to achieve the solution of the given problems. The proposed method has the edge over other techniques as it does not required extra calculations. Some numerical problems are used to test the validity of the suggested method. The solution obtained has demonstrated that the proposed technique has a higher level of accuracy. The proposed method is capable of tackling various nonlinear fractional-order problems due to its simple implementation. [ABSTRACT FROM AUTHOR]

Necessary and sufficient stability and instability conditions are reviewed and extended for multi-term homogeneous linear fractional differential equations with Caputo derivatives and constant coefficients. A comprehensive review of the state of the art regarding the stability analysis of two-term and three-term fractional-order differential equations is provided, which is then extended to the case of four-term fractional-order differential equations. The stability and instability properties are characterized with respect to the coefficients of the multi-term fractional differential equations, leading to both fractional-order-dependent and fractional-order-independent characterizations. In the general case, fractional-order-independent stability and instability properties are described for fractional-order differential equations with an arbitrary number of fractional derivatives. [ABSTRACT FROM AUTHOR]

We derive a fundamental solution E to a space-fractional diffusion problem on the half-line. The equation involves the Caputo derivative. We establish properties of E as well as formulas for solutions to the Dirichlet and fixed slope problems in terms of convolution of E with data. We also study integrability of derivatives of solutions given in this way. We present conditions, which are sufficient for uniqueness of solutions. Finally, we show the infinite speed of signal propagation. [ABSTRACT FROM AUTHOR]

The authors solved telegraph equation by q-homotopy analysis method (q-HAM) [[33]] and modified adomian decomposition method (MADM) and reduced differential transform method (RDTM) have been implemented to solve second order telegraph equation [[34]]. Keywords: B-spline collocation method; Caputo derivative; fractional partial differential equation (FPDE); L2 and L norms; stability analysis; time fractional inhomogeneous nonlinear telegraph equation EN B-spline collocation method Caputo derivative fractional partial differential equation (FPDE) L2 and L norms stability analysis time fractional inhomogeneous nonlinear telegraph equation 735 749 15 08/09/22 20220801 NES 220801 1 Introduction Fractional calculus and particularly fractional partial differential equations (FPDEs) have gained considerable popularity over the past few decades due to their applicability in various fields of science and engineering. B-spline collocation method, Caputo derivative, fractional partial differential equation (FPDE), L2 and L norms, time fractional inhomogeneous nonlinear telegraph equation, stability analysis. [Extracted from the article]

An algorithm based on a class of the Chebyshev polynomials family called the fifth-kind Chebyshev polynomials (FCPs) is introduced to solve the multi-term variable-order time-fractional diffusion-wave equation (MVTFD-WE). Appeared fractional derivative operators in these equations are of the Caputo type. Coupling FCPs and the collocation method leads to reduce the MVTFD-WE to a system of algebraic equations. The convergence of the proposed scheme is investigated in a weighted Sobolev space via obtaining error bounds for approximate solutions which shows the method error tends to zero if the number of terms of the series solution is selected sufficiently large. The applicability and efficiency of the suggested method are examined through several illustrative examples. [ABSTRACT FROM AUTHOR]

In this paper, we study local and nonlocal boundary value problems for hyperbolic equations of general form with variable coefficients and a Caputo fractional derivative. To study the stated problem, a certain fractional-order functional space is introduced. The problem posed is reduced to an integral equation, and the existence of its solution is proved using an a priori estimate. [ABSTRACT FROM AUTHOR]