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This paper focuses on the analysis of a coupled system governed by a Caputo-fractional derivative with q-integral-coupled boundary conditions. This system is particularly relevant in modeling multi-atomic systems, including scenarios involving adsorbed atoms or clusters on crystalline surfaces, surface–atom scattering, and atomic friction. To investigate this system, we introduce an operator that exhibits fixed points corresponding to the solutions of the problem, effectively transforming the system into an equivalent fixed-point problem. We established the necessary conditions for the existence and uniqueness of solutions using the Leray–Schauder nonlinear alternative and the Banach contraction mapping principle, respectively. Stability results in the Ulam sense for the coupled system are also discussed, along with a sensitivity analysis of the range parameters. To support the validity of their findings, we provide illustrative examples. Overall, the paper offers a thorough examination and analysis of the considered coupled system, making important contributions to the understanding of multi-atomic systems and their mathematical modeling. [ABSTRACT FROM AUTHOR]

In this paper, we investigate a class of multi-term implicit fractional differential equation with boundary conditions. The application of the Schauder fixed point theorem and the Banach fixed point theorem allows us to establish the criterion for a solution that exists for the given equation, and the solution is unique. Afterwards, we give the criteria of Ulam–Hyers stability and Ulam–Hyers–Rassias stability. Additionally, we present an example to illustrate the practical application and effectiveness of the results. [ABSTRACT FROM AUTHOR]

This paper considers a nonlinear impulsive fractional boundary value problem, which involves a ψ -Caputo-type fractional derivative and integral. Combining critical point theory and fractional calculus properties, such as the semigroup laws, and relationships between the fractional integration and differentiation, new multiplicity results of infinitely many solutions are established depending on some simple algebraic conditions. Finally, examples are also presented, which show that Caputo-type fractional models can be more accurate by selecting different kernels for the fractional integral and derivative. [ABSTRACT FROM AUTHOR]

Fractional calculus provides some fractional operators for us to model different real-world phenomena mathematically. One of these important study fields is the mathematical model of the elastic beam changes. More precisely, in this paper, based on the behavior patterns of an elastic beam, we consider the generalized sequential boundary value problems of the Navier difference equations by using the post-quantum fractional derivatives of the Caputo-like type. We discuss on the existence theory for solutions of the mentioned (p ; q) -difference Navier problems in two single-valued and set-valued versions. We use the main properties of the (p ; q) -operators in this regard. Application of the fixed points of the ρ - θ -contractions along with the endpoints of the multi-valued functions play a fundamental role to prove the existence results. Finally in two examples, we validate our models and theoretical results by giving numerical models of the generalized sequential (p ; q) -difference Navier problems. [ABSTRACT FROM AUTHOR]

This paper deals with a singular two dimensional initial boundary value problem for a Caputo time fractional parabolic equation supplemented by Neumann and non-local boundary conditions. The well posedness of the posed problem is demonstrated in a fractional weighted Sobolev space. The used method based on some functional analysis tools has been successfully showed its efficiency in proving the existence, uniqueness and continuous dependence of the solution upon the given data of the considered problem. More precisely, for proving the uniqueness of the solution of the posed problem, we established an energy inequality for the solution from which we deduce the uniqueness. For the existence, we proved that the range of the operator generated by the considered problem is dense. [ABSTRACT FROM AUTHOR]

In this paper, we consider the existence of positive solutions for a singular tempered fractional equation with a p-Laplacian operator. By constructing a pair of suitable upper and lower solutions of the problem, some new results on the existence of positive solutions for the equation including singular and nonsingular cases are established. The asymptotic behavior of the solution is also derived, which falls in between two known curves. The interesting points of this paper are that the nonlinearity of the equation may be singular in time and space variables and the corresponding operator can have a singular kernel. [ABSTRACT FROM AUTHOR]

In this paper, we provide existence and uniqueness results for the initial value problems associated with mixed Riemann–Liouville/Caputo differential equations in the real domain. We show that, under appropriate conditions in a fractional order, solutions are always square-integrable on the finite interval under consideration. The results are valid for equations that have sign-indefinite leading terms and measurable coefficients. Existence and uniqueness theorem results are also provided for two-point boundary value problems in a closed interval. [ABSTRACT FROM AUTHOR]

In this article, we are concerned with a very general integral boundary value problem of Riemann–Liouville derivatives. We will study the problem in Banach space. To be more specific, we are interested in proving the existence of a solution to our problem via the measure of noncompactness and Mönch fixed-point theorem. Our study in Banach space contains two nonlinear terms and two different orders of derivatives, ς and τ , such that ς ∈ 1 , 2 and τ ∈ 0 , ς . Our paper ends with a conclusion. [ABSTRACT FROM AUTHOR]

The main purpose of this paper was to consider new sandwich pairs and investigate the existence of a solution for a new class of fractional differential equations with p-Laplacian via variational methods in ψ -fractional space H p α , β ; ψ (Ω) . The results obtained in this paper are the first to make use of the theory of ψ -Hilfer fractional operators with p-Laplacian. [ABSTRACT FROM AUTHOR]

The results for a new modeling integral boundary value problem (IBVP) using Caputo-Hadamard impulsive fractional integro-differential equations (C-HIFI-DE) with Banach space are investigated, along with the existence and uniqueness of solutions. The Krasnoselskii fixed-point theorem (KFPT) and the Banach contraction principle (BCP) serve as the basis of this unique strategy, and are used to achieve the desired results. We develop the illustrated examples at the end of the paper to support the validity of the theoretical statements. [ABSTRACT FROM AUTHOR]

This paper delves into an examination of the existence, uniqueness, and stability properties of a non-local integro-differential equation featuring the Hilfer fractional derivative with order ω ∈ (1 , 2) for the RLC model. Based on Schaefer's fixed point theorem and Banach's contraction principle, the existence and uniqueness results are established. Furthermore, Ulam–Hyers and Ulam–Hyers–Rassias stability results for the boundary value problem of the RLC model are discussed. To showcase the practicality and efficacy of our theoretical findings, a two-step Lagrange polynomial interpolation method is applied to solve some numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, we derive the coincidence fixed-point and common fixed-point results for ℑ-type mappings satisfying certain contractive conditions and containing fewer conditions imposed on function ℑ with regard to generalized metric spaces (in terms of Jleli Samet). Finally, a fractional boundary value problem is reduced to an equivalent Volterra integral equation, and the existence results of common solutions are obtained with the use of proved fixed-point results. [ABSTRACT FROM AUTHOR]

In this paper, by using the Leggett–Williams fixed-point theorem, we study the existence of positive solutions to fractional differential equations with mixed Riemann–Liouville and quantum fractional derivatives. To prove the effectiveness of our main result, we investigate an interesting example. [ABSTRACT FROM AUTHOR]

In the present paper, we prove common fixed point theorems under different contractive mappings on C ★ -algebra-valued bipolar metric spaces. The results generalize and expand some well-known results from the literature. Examples and applications are given to strengthen our conclusion. Additionally, as an application of our results, we show that some classes of the nonlinear fractional differential equations with the boundary conditions have a unique common solution. [ABSTRACT FROM AUTHOR]

In this paper, we provide sufficient conditions for the existence, uniqueness and global behavior of a positive continuous solution to some nonlinear Riemann-Liouville fractional boundary value problems. The nonlinearity is allowed to be singular at the boundary. The proofs are based on perturbation techniques after reducing the considered problem to the equivalent Fredholm integral equation of the second kind. Some examples are given to illustrate our main results. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the existence and uniqueness of solutions for boundary value problems for Hilfer generalized proportional fractional differential equations with multi-point boundary conditions. Firstly, we consider the scalar case for which the uniqueness result is proved by using Banach's fixed point theorem and the existence results are established via Krasnosel'skiĭ's fixed point theorem and Leray–Schauder nonlinear alternative. We then establish an existence result in the Banach space case based on Mönch's fixed point theorem and the technique of the measure of noncompactness. Examples are constructed to illustrate the application of the main results. We emphasize that, in this paper, we initiate the study of Hilfer generalized proportional fractional boundary value problems of order in (1, 2]. [ABSTRACT FROM AUTHOR]

A scalar nonlinear impulsive differential equation with a delay and generalized proportional Caputo fractional derivatives (IDGFDE) is investigated. The linear boundary value problem (BVP) for the given fractional differential equation is set up. The explicit form of the unique solution of BVP in the special linear case is obtained. This formula is a generalization of the explicit solution of the case without any delay as well as the case of Caputo fractional derivatives. Furthermore, this integral form of the solution is used to define a special proportional fractional integral operator applied to the determination of a mild solution of the studied BVP for IDGFDE. The relation between the defined mild solution and the solution of the BVP for the IDGFDE is discussed. The existence and uniqueness results for BVP for IDGFDE are proven. The obtained results in this paper are a generalization of several known results. [ABSTRACT FROM AUTHOR]

This paper studies a new class of instantaneous and non-instantaneous impulsive boundary value problem involving the generalized ψ -Caputo fractional derivative with a weight. Depending on critical point theorems and some properties of ψ -Caputo-type fractional integration and differentiation, the variational construction and multiplicity result of solutions are established. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence and uniqueness of minimizers of a fractional variational problem generalized from the energy functional associated with a cantilever beam under a uniformly distributed load. We apply the fractional Euler–Lagrange condition to formulate the minimization problem as a boundary value problem and obtain existence and uniqueness results in both L 2 and L ∞ settings. Additionally, we characterize the continuous dependence of the minimizers on varying loads in the energy functional. Moreover, an approximate solution is derived via the homotopy perturbation method, which is numerically demonstrated in various examples. The results show that the deformations are larger for smaller orders of the fractional derivative. [ABSTRACT FROM AUTHOR]

Chemical graph theory (CGT) is a field of mathematical science that applies classical graph theory to chemical structures and processes. Chemical graphs are the principal data format used in cheminformatics to illustrate chemical interactions. Several researchers have addressed boundary-value problems using star graphs. Star graphs were used since their method requires a central point linked to other vertices but not to itself. Our objective is to expand the mechanism by introducing the idea of an isobutane graph that has the chemical formula C 4 H 10 and CAS number 75-28-5. By using the appropriate fixed point theory findings, this paper investigates the existence of solutions to fractional boundary value problems of Caputo type on such graphs. Additionally, two examples are provided to strengthen our important conclusions. [ABSTRACT FROM AUTHOR]

In this paper, we study the initial boundary value problem for a fractional viscoelastic equation of the Kirchhoff type. In suitable functional spaces, we define a potential well. In the framework of the potential well theory, we obtain the global existence of solutions by using the Galerkin approximations. Moreover, we derive the asymptotic behavior of solutions by means of the perturbed energy method. Our main results provide sufficient conditions for the qualitative properties of solutions in time. [ABSTRACT FROM AUTHOR]

This survey paper is concerned with some of the most recent results on Lyapunov-type inequalities for fractional boundary value problems involving a variety of fractional derivative operators and boundary conditions. Our work deals with Caputo, Riemann-Liouville, ψ -Caputo, ψ -Hilfer, hybrid, Caputo-Fabrizio, Hadamard, Katugampola, Hilfer-Katugampola, p-Laplacian, and proportional fractional derivative operators. [ABSTRACT FROM AUTHOR]

This paper proposes a three–dimensional (3D) local boundary element model based on meshless moving least squares (MLS) method for ultrasonic wave propagation fractional order boundary value problems of functionally graded anisotropic (FGA) fiber-reinforced plates. The problem domain is split into several circular sub-domains. The nodal points are randomly distributed across the examined region. Each node is the focal point of a circular sub-domain that encircles it. The Laplace-transform approach is used to solve dynamic issues. In the local weak form of the governing equations for the converted quantities, a unit test function is utilized. The Gauss divergence theorem to the weak-form is used to produce local boundary-domain integral equations. A meshless approximation is achieved using the MLS method. To find time-dependent solutions, an inverse Laplace-transform approach is used. The effects of the fractional order parameter, functionally graded material, anisotropy, and the time characteristic of the laser pulse are investigated. The proposed method's validity and performance are demonstrated for a two-dimensional problem with excellent agreement with the finite element method. [ABSTRACT FROM AUTHOR]

In this present paper, we study the difference method for solving a boundary value problem of the Caputo type q-fractional differential equation. This method is based on the numerical quadrature of the q-fractional derivative and the q-Taylor expansion of related function. We first derive the truncation error boundness of O (▵ x n 2) -order and prove the existence and uniqueness of the numerical solution. Then, we prove the stability of the numerical solution and give the error estimation. Numerical experiments finally verify the validity of the theoretical analysis. [ABSTRACT FROM AUTHOR]

Numerical methods for spectral space-fractional elliptic equations are studied. The boundary value problem is defined in a bounded domain of general geometry, W - Rd, d 2 f1, 2, 3g. Assuming that the finite difference method (FDM) or the finite element method (FEM) is applied for discretization in space, the approximate solution is described by the system of linear algebraic equations Aau = f, a 2 (0, 1). Although matrix A 2 RN-N is sparse, symmetric and positive definite (SPD), matrix Aa is dense. The recent achievements in the field are determined by methods that reduce the original non-local problem to solving k auxiliary linear systems with sparse SPD matrices that can be expressed as positive diagonal perturbations of A. The present study is in the spirit of the BURA method, based on the best uniform rational approximation ra,k(t) of degree k of ta in the interval [0, 1]. The introduced additive BURA-AR and multiplicative BURA-MR methods follow the observation that the matrices of part of the auxiliary systems possess very different properties. As a result, solution methods with substantially improved computational complexity are developed. In this paper, we present new theoretical characterizations of the BURA parameters, which gives a theoretical justification for the new methods. The theoretical estimates are supported by a set of representative numerical tests. The new theoretical and experimental results raise the question of whether the almost optimal estimate of the computational complexity of the BURA method in the form O(N log2 N) can be improved. [ABSTRACT FROM AUTHOR]

In this study, we focus on the stability analysis of the RLC model by employing differential equations with Hadamard fractional derivatives. We prove the existence and uniqueness of solutions using Banach's contraction principle and Schaefer's fixed point theorem. To facilitate our key conclusions, we convert the problem into an equivalent integro-differential equation. Additionally, we explore several versions of Ulam's stability findings. Two numerical examples are provided to illustrate the applications of our main results. We also observe that modifications to the Hadamard fractional derivative lead to asymmetric outcomes. The study concludes with an applied example demonstrating the existence results derived from Schaefer's fixed point theorem. These findings represent novel contributions to the literature on this topic, significantly advancing our understanding. [ABSTRACT FROM AUTHOR]

This work addresses the qualitative analysis of a novel non-linear coupled system of fractional differential problems (FDPs) using Caputo and Liouville–Riemann fractional derivatives. Fractional calculus has demonstrated significant applicability across various fields, including financial systems, optimal control, epidemiological models, chaotic systems, and engineering. The proposed model builds on existing research by formulating a non-linear coupled fractional boundary value problem with mixed boundary conditions. The primary advantages of our method include its ability to capture the dynamics of complex systems more accurately and its flexibility in handling different types of fractional derivatives. The model's solution was derived using advanced mathematical techniques, and the results confirmed the existence and uniqueness of the solutions. This approach not only generalizes classical differential equation methods but also offers a robust framework for modeling real-world phenomena governed by fractional dynamics. The study concludes with the validation of the theoretical findings through illustrative examples, highlighting the method's efficacy and potential for further applications. [ABSTRACT FROM AUTHOR]

The present paper is devoted to the existence of solution for the Hybrid differential inclusions of the second type. Here, we present the inclusion problem with two multi-valued maps. In addition, it is considered with nonlocal integral boundary condition η (0) ∈ ∫ 0 σ Δ s , η (s) d s , where Δ is a multi-valued map. Relative compactness of the set ∫ 0 σ Δ s , η (s) d s in L 2 (0 , ε) , R is used to justify the condensing condition for some created operators. Fixed point theorems connected with the weak compactness manner is utilized to explore the results throughout this paper. [ABSTRACT FROM AUTHOR]

There are very few papers that talk about the global convergence of iterative methods with the help of Banach spaces. The main purpose of this paper is to discuss the global convergence of third order iterative method. The convergence analysis of this method is proposed under the assumptions that Fréchet derivative of first order satisfies continuity condition of the Hölder. Finally, we consider some integral equation and boundary value problem (BVP) in order to illustrate the suitability of theoretical results. [ABSTRACT FROM AUTHOR]

We investigate a class of boundary value problems (BVPs) involving an impulsive fractional integro-differential equation (IF-IDE) with the Caputo–Hadamard fractional derivative (C-HFD). We employ some fixed-point theorems (FPTs) to study the existence of this fractional BVP and its unique solution. The boundary conditions (BCs) established in this study are of a more general type and can be reduced to numerous specific examples by defining the parameters involved in the conditions. In this way, we extend some recent nice results. At the end, we use an example to verify our results. [ABSTRACT FROM AUTHOR]

In this study, we present a novel approach for the numerical solution of high-order ODEs and MTVOFDEs with BCs. Our method leverages a class of GSJPs that possess the crucial property of satisfying the given BCs. By establishing OMs for both the ODs and VOFDs of the GSJPs, we integrate them into the SCM, enabling efficient and accurate numerical computations. An error analysis and convergence study are conducted to validate the efficacy of the proposed algorithm. We demonstrate the applicability and accuracy of our method through eight numerical examples. Comparative analyses with prior research highlight the improved accuracy and efficiency achieved by our approach. The recommended approach exhibits excellent agreement between approximate and precise results in tables and graphs, demonstrating its high accuracy. This research contributes to the advancement of numerical methods for ODEs and MTVOFDEs with BCs, providing a reliable and efficient tool for solving complex BVPs with exceptional accuracy. [ABSTRACT FROM AUTHOR]

In this paper, we establish the existence of solutions to a nonlinear boundary value problem (BVP) of variable order at resonance. The main theorem in this study is proved with the help of generalized intervals and piecewise constant functions, in which we convert the mentioned Caputo BVP of fractional variable order to an equivalent standard Caputo BVP at resonance of constant order. In fact, to use the Mawhin's continuation technique, we have to transform the variable order BVP into a constant order BVP. We prove the existence of solutions based on the existing notions in the coincidence degree theory and Mawhin's continuation theorem (MCTH). Finally, an example is provided according to the given variable order BVP to show the correctness of results. [ABSTRACT FROM AUTHOR]

Fractional calculus (FC) generalizes the concepts of derivative and integral to non-integer orders. To suppress the chaotic behavior two control strategies are proposed: a time delay feedback controller; and a mixed controller, which includes a time delay feedback controller and a fractional-order PD HT ht controller. In "Fractals Parrondo's paradox in alternated superior complex system" [[9]] a kind of fractals Parrondo's paradoxial phenomenon "deiconnected + diconnected = connected" in an alternated superior complex system HT ht , HT ht is addressed. [Extracted from the article]

This paper is concerned with the existence of solutions for a fully coupled Riemann–Stieltjes, integro-multipoint, boundary value problem of Caputo-type sequential fractional differential equations. The given system is studied with the aid of the Leray–Schauder alternative and contraction mapping principle. A numerical example illustrating the abstract results is also presented. [ABSTRACT FROM AUTHOR]

The present paper presents a study on a problem with a fractional integro-differentiation operator in the boundary condition for an equation with a partial Riemann-Liouville fractional derivative. The unique solvability of the problem is proved. In the hyperbolic part of the considered domain, the functional equation is solved by the iteration method. The problem is reduced to solving the Volterra integro-differential equation. [ABSTRACT FROM AUTHOR]

In this paper, a new approach is developed to solve a class of first-order fractional initial value problems. The present class is of practical interest in engineering science. The results are based on the Riemann–Liouville fractional derivative. It is shown that the dual solution can be determined for the considered class. The first solution is obtained by means of the Laplace transform and expressed in terms of the Mittag–Leffler functions. The second solution was determined through a newly developed approach and given in terms of exponential and trigonometric functions. Moreover, the results reduce to the ordinary version as the fractional-order tends to unity. Characteristics of the dual solution are discussed in detail. Furthermore, the advantages of the second solution over the first one is declared. It is revealed that the second solution is real at certain values of the fractional-order. Such values are derived theoretically and accordingly, and the behavior of the real solution is shown through several plots. The present analysis may be introduced for obtaining the solution in a straightforward manner for the first time. The developed approach can be further extended to include higher-order fractional initial value problems of oscillatory types. [ABSTRACT FROM AUTHOR]

In this paper, we are interested in an inverse geometric problem for the three-dimensional Laplace equation to recover an inner boundary of an annular domain. This work is based on the method of fundamental solutions (MFS) by imposing the boundary Cauchy data in a least-square sense and minimisation of the objective function. This approach can also be considered with noisy boundary Cauchy data. The simplicity and efficiency of this method is illustrated in several numerical examples. [ABSTRACT FROM AUTHOR]

Our goal is to address the question of existence and uniqueness of a positive continuous solution to some semipositone fractional boundary value problems on the half-line. Global estimates on this solution are given. This kind of problems, where the nonlinearity is allowed to be sign-changing, are often difficult to solve analytically and becomes more challenging specially when we are looking for positive solutions. The main result is obtained by means of the properties of the Green function and fixed point theorem. [ABSTRACT FROM AUTHOR]

Exact conditions for the existence of the unique solution of a boundary value problem for linear fractional functional differential equations related to ς -nonpositive operators are established. The exact solvability conditions are based on the a priori estimation method. All theoretical investigations are illustrated by an example of the pantograph-type model from electrodynamics. [ABSTRACT FROM AUTHOR]

In this paper, the solvability of a system of nonlinear Caputo fractional differential equations at resonance is considered. The interesting point is that the state variable x ∈ R n and the effect of the coefficient matrices matrices B and C of boundary value conditions on the solvability of the problem are systematically discussed. By using Mawhin coincidence degree theory, some sufficient conditions for the solvability of the problem are obtained. [ABSTRACT FROM AUTHOR]

In this paper, a new approach to find exact solutions is carried out for a generalized unsteady magnetohydrodynamic transport of a rate-type fluid near an unbounded upright plate, which is analyzed for ramped-wall temperature and velocity with constant concentration. The vertical plate is suspended in a porous medium and encounters the effects of radiation. An innovative definition of the time-fractional operator in power-law-kernel form is implemented to hypothesize the constitutive mass, energy, and momentum equations. The Laplace integral transformation technique is applied on a dimensionless form of governing partial differential equations by introducing some non-dimensional suitable parameters to establish the exact expressions in terms of special functions for ramped velocity, temperature, and constant-concentration fields. In order to validate the problem, the absence of the mass Grashof parameter led to the investigated solutions obtaining good agreement in existing literature. Additionally, several system parameters were used, such as as magnetic value M, Prandtl value P r , Maxwell parameter λ , dimensionless time τ , Schmidt number " S c ", fractional parameter α , andMass and Thermal Grashof numbers G m and G r , respectively, to examine their impacts on velocity, wall temperature, and constant concentration. Results are also discussed in detail and demonstrated graphically via Mathcad-15 software. A comprehensive comparative study between fractional and non-fractional models describes that the fractional model elucidate the memory effects more efficiently. [ABSTRACT FROM AUTHOR]

Fractional-order boundary value problems are used to model certain phenomena in chemistry, physics, biology, and engineering. However, some of these models do not meet the existence and uniqueness required in the mainstream of mathematical processes. Therefore, in this paper, the existence, stability, and uniqueness for the solution of the coupled system of the Caputo-type sequential fractional differential equation, involving integral boundary conditions, was discussed, and investigated. Leray–Schauder's alternative was applied to derive the existence of the solution, while Banach's contraction principle was used to examine the uniqueness of the solution. Moreover, Ulam–Hyers stability of the presented system was investigated. It was found that the theoretical-related aspects (existence, uniqueness, and stability) that were examined for the governing system were satisfactory. Finally, an example was given to illustrate and examine certain related aspects. [ABSTRACT FROM AUTHOR]

In this paper, fractional-order Bernoulli wavelets based on the Bernoulli polynomials are constructed and applied to evaluate the numerical solution of the general form of Caputo fractional order diffusion wave equations. The operational matrices of ordinary and fractional derivatives for Bernoulli wavelets are set via fractional Riemann–Liouville integral operator. Then, these wavelets and their operational matrices are utilized to reduce the nonlinear fractional problem to a set of algebraic equations. For solving the obtained system of equations, Galerkin and collocation spectral methods are employed. To demonstrate the validity and applicability of the presented method, we offer five significant examples, including generalized Cattaneo diffusion wave and Klein–Gordon equations. The implementation of algorithms exposes high accuracy of the presented numerical method. The advantage of having compact support and orthogonality of these family of wavelets trigger having sparse operational matrices, which reduces the computational time and CPU requirements. [ABSTRACT FROM AUTHOR]

In this paper, electrochemical impedance responses of subdiffusive phase transition materials are calculated and analyzed for one-dimensional cell with reflecting and absorbing boundary conditions. The description is based on the generalization of the diffusive Warburg impedance within the fractional phase field approach utilizing the time-fractional Cahn–Hilliard equation. The driving force in the model is the chemical potential of ions, that is described in terms of the phase field allowing us to avoid additional calculation of the activity coefficient. The derived impedance spectra are applied to describe the response of supercapacitors with polyaniline/carbon nanotube electrodes. [ABSTRACT FROM AUTHOR]

This paper is devoted to generalizing the standard system of Navier boundary value problems to a fractional system of coupled sequential Navier boundary value problems by using terms of the Caputo derivatives. In other words, for the first time, we design a multi-term fractional coupled system of Navier equations under the fractional boundary conditions. The existence theory is studied regarding solutions of the given coupled sequential Navier boundary problems via the Krasnoselskii's fixed-point theorem on two nonlinear operators. Moreover, the Banach contraction principle is applied to investigate the uniqueness of solution. We then focus on the Hyers–Ulam-type stability of its solution. Furthermore, the approximate solutions of the proposed coupled fractional sequential Navier system are obtained via the generalized differential transform method. Lastly, the results of this research are supported by giving simulated examples. [ABSTRACT FROM AUTHOR]

In this paper, the existence and uniqueness of solutions for a coupled system of ψ-Hilfer type sequential fractional differential equations supplemented with nonlocal integro-multi-point boundary conditions is investigated. The presented results are obtained via the classical Banach and Krasnosel'skiĭ's fixed point theorems and the Leray–Schauder alternative. Examples are included to illustrate the effectiveness of the obtained results. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence results for nonlinear fractional q-difference equations with two different fractional orders supplemented with the Dirichlet boundary conditions. Our main existence results are obtained by applying the contraction mapping principle and Krasnoselskii's fixed point theorem. An illustrative example is also discussed. [ABSTRACT FROM AUTHOR]

In the article, affine-periodic boundary value problem involving fractional derivative is considered. Existence of solutions to a Caputo-type fractional differential inclusion is researched by some fixed-point theorems and set-valued analysis theory. Specifically, we consider two cases in which the multifunction has convex values and nonconvex values, respectively. [ABSTRACT FROM AUTHOR]

The existence of a positive solution to a system of nonlinear semipositone Hadamard fractional BVP with the p-Laplacian operator is examined in this research. The boundary value problem's associated Green's function and some of its properties are first obtained. Additionally, the existence results are established using the nonlinear alternative of the Leray–Schauder theorem and the Guo–Krasnosel'skii fixed-point theorem. [ABSTRACT FROM AUTHOR]

A nonlocal boundary value problem for the fractional version of the Rayleigh–Stokes equation, well-known in fluid dynamics, is studied. Namely, the condition u (x , T) = β u (x , 0) + φ (x) , where β is an arbitrary real number, is proposed instead of the initial condition. If β = 0 , then we have the inverse problem in time, called the backward problem. It is well-known that the backward problem is ill-posed in the sense of Hadamard. If β = 1 , then the corresponding non-local problem becomes well-posed in the sense of Hadamard, and moreover, in this case a coercive estimate for the solution can be established. The aim of this work is to find values of the parameter β , which separates two types of behavior of the semi-backward problem under consideration. We prove the following statements: if β ≥ 1 , or β < 0 , then the problem is well-posed; if β ∈ (0 , 1) , then depending on the eigenvalues of the elliptic part of the equation, for the existence of a solution an additional condition on orthogonality of the right-hand side of the equation and the boundary function to some eigenfunctions of the corresponding elliptic operator may emerge. [ABSTRACT FROM AUTHOR]