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In this paper, our study is divided into two parts. The first part involves analyzing a coupled system of beam deflection type that involves nonlinear equations with sequential Caputo derivatives. The also system incorporates the Caputo derivatives in the initial conditions, which adds a layer of complexity and realism to the problem. We focus on proving the existence of a unique solution for this system, and highlighting the robustness and applicability of fractional derivatives in modeling complex physical phenomena. In the second part of the paper, we employ conformable fractional derivatives, as defined by Khalil, to examine another system consisting of two coupled evolution equations. By the Tanh method, we derive new progressive waves. The connection between these two parts lies in the use of fractional calculus to extend and enhance classical problems. [ABSTRACT FROM AUTHOR]

In this paper, the authors study the existence of positive solutions to the fractional boundary value problem at resonance - (D a + α , ρ x) (t) = f (t , x (t) , D a + α - 1 , ρ x (t)) , t ∈ (a , b) , x (a) = 0 , x (b) = ∫ a b x (t) d A (t) , where 1 < α ≤ 2 , and D a + α , ρ is a Katugampola fractional derivative, which generalizes the Riemann–Liouville and Hadamard fractional derivatives, and ∫ a b x (t) d A (t) denotes a Riemann–Stieltjes integral of x with respect to A, where A is a function of bounded variation. Coincidence degree theory is applied to obtain existence results. This appears to be the first work in the literature to deal with a resonant fractional differential equation with a Katugampola fractional derivative. Examples are given to illustrate the application of their results. [ABSTRACT FROM AUTHOR]

In this paper, we study a coupled system of beam deflection type that involves nonlinear equations with sequential Caputo fractional derivatives. Under flexible/fixed end-conditions, two main theorems on the existence and uniqueness of solutions are proved by using two fixed point theorems. Some examples are discussed to illustrate the applications of the existence and uniqueness of solution results. Another main result on the Ulam–Hyers stability of solutions for the introduced system is also discussed. Some examples of stability are discussed. New travelling wave solutions are obtained for another conformable coupled system of beam type that has a connection with the first considered system. A conclusion follows at the end. [ABSTRACT FROM AUTHOR]

In this paper, the solvability of some inverse problems for a nonlocal analogue of a fourth-order parabolic equation was studied. For this purpose, a nonlocal analogue of the biharmonic operator was introduced. When defining this operator, transformations of the involution type were used. In a parallelepiped, the eigenfunctions and eigenvalues of the Dirichlet type problem for a nonlocal biharmonic operator were studied. The eigenfunctions and eigenvalues for this problem were constructed explicitly and the completeness of the system of eigenfunctions was proved. Two types of inverse problems on finding a solution to the equation and its righthand side were studied. In the two problems, both of the righthand terms depending on the spatial variable and the temporal variable were obtained by using the Fourier variable separation method or reducing it to an integral equation. The theorems for the existence and uniqueness of the solution were proved. [ABSTRACT FROM AUTHOR]

This paper is dedicated to investigating the existence of solutions to the initial value problem (IVP) for a coupled system of Ψ-Hilfer hybrid fractional differential equations (FDEs) and boundary value problem (BVP) for a coupled system of Ψ-Hilfer hybrid FDEs. Analysis of the current paper depends on the two fixed point theorems involving three operators characterized on Banach algebra. In the view of an application, we provided useful examples to exhibit the effectiveness of our achieved results. [ABSTRACT FROM AUTHOR]

This paper deals with a general class of nonlinear fractional differential equations with p-Laplacian operator that involves some φ sequential Caputo derivatives. New criteria on the existence and uniqueness of solutions are established. The stability analysis in the sense of Ulam Hyers is discussed. An illustrative example is presented. [ABSTRACT FROM AUTHOR]

This research paper focuses on investigating the solvability of the Riemann-Liouville differential equation with nonlocal integral condition, study the existence of solutions in the class of continuous functions, we use the technique of the Schauder fixed point Theorem. We drive sufficient conditions for a uniqueness and the continuous dependence on some functions. Additionally, we delve into the study of the Hyers-Ulam stability. Finally, we given an examples are provided to illustrate our results. [ABSTRACT FROM AUTHOR]

In this paper we define the tempered-fractal derivative e-λt d/dtβ(f(t)eλt) and study the initial-value problem of the delay tempered-fractal differential equation e-λt d/dtβ(x(t)eλt) = f(t,x(ϕ(t))), a,e., t∈(0,T], x(0) = x0. We discuss the existence of at least one solution x ∈ C[0, T]. The Uniqueness of the solution will be proved. The continuous dependence on the initial data x0, the delay function ϕ and on the function f is proved. The Hyers - Ulam stability of the problem itself will be established. Finally, some examples will be given to verify our results. [ABSTRACT FROM AUTHOR]

The main objective of this study is to determine the existence and uniqueness of solutions to the fractional Black–Scholes equation. The solution to the fractional Black–Scholes equation is expressed as an infinite series of converging Mittag-Leffler functions. The method used to discover the new solution to the fractional Black–Scholes equation was the Daftardar-Geiji method. Additionally, the Picard–Lindelöf theorem was utilized for the existence and uniqueness of its solution. The fractional derivative employed was the Caputo operator. The search for a solution to the fractional Black–Scholes equation was essential due to the Black–Scholes equation's assumptions, which imposed relatively tight constraints. These included assumptions of a perfect market, a constant value of the risk-free interest rate and volatility, the absence of dividends, and a normal log distribution of stock price dynamics. However, these assumptions did not accurately reflect market realities. Therefore, it was necessary to formulate a model, particularly regarding the fractional Black–Scholes equation, which represented more market realities. The results obtained in this paper guaranteed the existence and uniqueness of solutions to the fractional Black–Scholes equation, approximate solutions to the fractional Black–Scholes equation, and very small solution errors when compared to the Black–Scholes equation. The novelty of this article is the use of the Daftardar-Geiji method to solve the fractional Black–Scholes equation, guaranteeing the existence and uniqueness of the solution to the fractional Black–Scholes equation, which has not been discussed by other researchers. So, based on this novelty, the Daftardar-Geiji method is a simple and effective method for solving the fractional Black–Scholes equation. This article presents some examples to demonstrate the application of the Daftardar-Gejji method in solving specific problems. [ABSTRACT FROM AUTHOR]

In this paper, we study new classes of differential equations of fractional order. The first considered problem involves the derivative of Caputo, while the second one involves conformable Khalil derivative. For the first class, we prove an existence and uniqueness result, then, we discuss an example to show the applicability of the result. For the second one, we apply the exp-function method to find new traveling wave solutions for a generalised conformable fractional partial differential equation, then some examples on Ostrovsky and KdV equations are given to illustrate the efficiency and accuracy of the method. Some graphs are plotted and discussed to show more the importance of the obtained results. [ABSTRACT FROM AUTHOR]

The purpose of the present paper is to study the existence of solutions for the following nonhomogeneous singular problem involving the fractional p (x ,.) -Laplace operator { (− Δ) p (x ,.) s u + | u | q (x) − 2 u = g (x) u ϵ − 1 − γ (x) ∓ λ f (x , u) in Ω , u = 0 on ∂ Ω , where Ω is a smooth bounded domain in R N ( N ≥ 3), 0 < s , ϵ < 1 , λ is a positive parameter and γ : Ω ¯ ⟶ (0 , ϵ) is a continuous function, p : Ω ¯ × Ω ¯ ⟶ (1 , ∞) is a bounded, continuous and symmetric function, q : Ω ¯ ⟶ (1 , ∞) is a continuous function, g ∈ L p s ∗ (x) − ϵ p s ∗ (x) + γ (x) − 2 ϵ (Ω) and g (x) > 0 with p s ∗ (x) = N p (x , x) N − s p (x , x) . Here, the nonlinearity f is in C 1 (Ω ¯ × R) and assumed to satisfy suitable assumptions. Using variational methods combined with monotonicity arguments, we obtain the existence of solutions to the problem in a fractional Sobolev space with variable exponent. To our best knowledge, this paper is the first attempt in the study of singular problems involving fractional p (x ,.) -Laplace operators. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of solutions of a nonlocal Cauchy problem for nonlinear fractional integro-differential equations involving generalized Katugampola fractional derivative. By using fixed point theorems, the results are obtained in weighted space of continuous functions. In the last, results are illustrated with suitable examples. [ABSTRACT FROM AUTHOR]

The paper explores an optimal control problem concerning transport processes in a porous medium. The transport phenomena is governed by diffusion-reaction equations, which is basically a semi-linear parabolic system. An |$L^{2}$| -cost (energy) functional is introduced, and control (distributed) is applied in the porous part of the medium. The primary objective is to characterize a given control to be an optimal control and analyse the relationship between optimal control and the adjoint state. The analysis commences with the microscopic description of the controlled system followed by upscaling the system via periodic homogenization. [ABSTRACT FROM AUTHOR]

In this paper, our study is divided into two parts. The first part involves analyzing a coupled system of beam deflection type that involves nonlinear equations with sequential Caputo derivatives. The also system incorporates the Caputo derivatives in the initial conditions, which adds a layer of complexity and realism to the problem. We focus on proving the existence of a unique solution for this system, and highlighting the robustness and applicability of fractional derivatives in modeling complex physical phenomena. In the second part of the paper, we employ conformable fractional derivatives, as defined by Khalil, to examine another system consisting of two coupled evolution equations. By the Tanh method, we derive new progressive waves. The connection between these two parts lies in the use of fractional calculus to extend and enhance classical problems.

This paper is devoted to studying a complicated implicit obstacle problem involving a nonhomogenous differential operator, called double phase operator, a nonlinear convection term (i.e. a reaction term depending on the gradient), and a multivalued term which is described by Clarke's generalized gradient. We develop a general framework to deliver an existence result for the double phase implicit obstacle problem under consideration. Our proof is based on the Kakutani–Ky Fan fixed point theorem together with the theory of nonsmooth analysis and a surjectivity theorem for multivalued mappings generated by the sum of a maximal monotone multivalued operator and a bounded pseudomonotone mapping. [ABSTRACT FROM AUTHOR]

In this research paper, we study a coupled system of piecewise-order differential equations (DEs) with variable kernel and impulsive conditions. DEs with variable kernel have high flexibility due to the freedom of changing the kernel. We study existence and stability theory and derive sufficient conditions for main results of the proposed problem. We apply Scheafer's fixed point theorem and Banach fixed point theorem for the result of at least one and unique solution, respectively. In addition, stability results based on the Ulam–Hyers concept are derived. Being a coupled system of piecewise fractional-order DEs with variable kernel and impulsive effects, the obtained results have multi-dimension applications. To demonstrate the applications, we apply the derived results to a numerical problem. [ABSTRACT FROM AUTHOR]

In this paper, we initially derive the equivalent fractional integral equation to Ψ -Hilfer hybrid fractional differential equations and through it, we prove the existence of a solution in the weighted space. The paper's primary objective is to obtain estimates on Ψ -Hilfer fractional derivative and utilize it to derive the hybrid fractional differential inequalities involving Ψ -Hilfer fractional derivative. With the assistance of these fractional differential inequalities, we determine the existence of extremal solutions and comparison theorems. [ABSTRACT FROM AUTHOR]

This paper focuses on revisiting and improving the results regarding the existence of a solution to the implicit fractional differential equations (FDEs) given in the following form: DCρω(ϰ)=fϰ,ω(ϰ),DCρω(ϰ)$$ {\mathcal{D}}_C&#x0005E;{\rho}\omega \left(\mathrm{\varkappa}\right)&#x0003D;f\left(\mathrm{\varkappa},\omega \left(\mathrm{\varkappa}\right),{\mathcal{D}}_C&#x0005E;{\rho}\omega \left(\mathrm{\varkappa}\right)\right) $$for all ϰ∈[0,T]$$ \mathrm{\varkappa}\in \left[0,T\right] $$ with the initial condition ω(0)=ω0$$ \omega (0)&#x0003D;{\omega}_0 $$, where 0<ρ<1,DCρ$$ 0<\rho <1,{\mathcal{D}}_C&#x0005E;{\rho } $$ is the Caputo fractional derivative (CFD). Mainly, we aim to weaken the necessary conditions often found in the literature for guaranteeing the existence of solution. [ABSTRACT FROM AUTHOR]

The Levin‐Nohel integro‐differential equation is one of the famous equations in various fields of science and engineering. It was first studied by Volterra in connection with biological applications. This paper deals with the nonlinear neutral Levin‐Nohel integro‐differential equation with Caputo fractional derivative and variable delays. The Ulam‐Hyers‐Rassias stability, Ulam‐Hyers stability, semi‐Ulam‐Hyers‐Rassias stability are studied. The existence and uniqueness of solutions are established by using Krasnoselskii's fixed point theorem and contraction mapping principle. An illustrative example is also given and that is studied by the help of the Adomian decomposition method. [ABSTRACT FROM AUTHOR]

Waterborne diseases are illnesses caused by pathogenic bacteria that spread through water and have a negative influence on human health. Due to the involvement of most countries in this vital issue, accurate analysis of mathematical models of such diseases is one of the first priorities of researchers. In this regard, in this paper, we turn to a waterborne disease model for solution's existence, HU-stability, and computational analysis. We transform the model to an analogous fractal-fractional integral form and study its qualitative analysis using an iterative convergent sequence and fixed-point technique to see whether there is a solution. We use Lagrange's interpolation to construct numerical algorithms for the fractal-fractional waterborne disease model in terms of computations. The approach is then put to the test in a case study, yielding some interesting outcomes. [ABSTRACT FROM AUTHOR]

In this paper, we apply the reproducing kernel Hilbert space method (RKHSM) for solving third order differential equations with multiple characteristics in a rectangular domain. The exact solution is expressed in a series form. The numerical examples are given to demonstrate the good performance of the presented method. The results obtained indicate that the method is simple and effective. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the existence and uniqueness of solutions for the coupled system of Phi-Caputo fractional differential equations. An illustrative example is included to show the applicability of our results. [ABSTRACT FROM AUTHOR]

In this paper we study the L p Gauss image problem, which is a generalization of the L p Aleksandrov problem and the Gauss image problem in convex geometry. We obtain the existence result for the L p Gauss image problem in two cases (i) p > 0 or (ii) p < 0 with the given even measures. [ABSTRACT FROM AUTHOR]

Through literature retrieval and classification, it can be found that for the fractional delay impulse differential system, the existence and uniqueness of the solution and UHR stability of the fractional delay impulse differential system are rarely studied by using the polynomial function of the fractional delay impulse matrix. In this paper, we firstly introduce a new concept of impulsive delayed Mittag–Leffler type solution vector function, which helps us to construct a representation of an exact solution for the linear impulsive fractional differential delay equations (IFDDEs). Secondly, by using Banach's and Schauder's fixed point theorems, we derive some sufficient conditions to guarantee the existence and uniqueness of solutions of nonlinear IFDDEs. Finally, we obtain the Ulam–Hyers stability (UHs) and Ulam–Hyers–Rassias stability (UHRs) for a class of nonlinear IFDDEs. [ABSTRACT FROM AUTHOR]

In this paper, we first introduce the new concept of an impulsive delayed vector function, which helps us to build the representation of an exact solution for the linear impulsive oscillating differential delay systems (IODDSs). Second, we derive some sufficient conditions to guarantee the existence and uniqueness of solution by Schauder's and Banach's fixed point theorems for a new class of nonlinear IODDSs. Finally, we obtain the Ulam‐Hyers stability (UHs) and Ulam‐Hyers‐Rassias stability (UHRs) for nonlinear IODDSs, and two examples are presented to illustrate the main results of our study. [ABSTRACT FROM AUTHOR]

In this paper, we consider a coupled system, known as Kobayashi–Warren–Carter system, abbreviated as the KWC system. KWC system consists of an Allen–Cahn type equation and a singular diffusion equation, and it was proposed by [Kobayashi et al, Phys. D, 140,141–150 (2000)] as a possible mathematical model of grain boundary motion. The focus of this work is on the dynamic boundary condition imposed in our KWC system, and the mathematical interest is in a conflicting situation between: the continuity of the transmission condition included in the dynamic boundary condition; and the discontinuity encouraged by the singular diffusion equation. On this basis, we will prove the Main Theorem concerned with the existence of solution to our KWC system with energy-dissipation. Additionally, as a sub-result, we will prove a key-lemma that is to give a certain mathematical interpretation for the conflicting situation. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of solutions for a new tripled system of hybrid differential equations with nonlocal integro multi point boundary conditions by using the φ-Caputo derivatives. The presented results are obtained by using hybrid fixed point theorems for three Dhage operators. An illustrative example is presented for the applicability of the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the boundary value problem of a class of fractional (p, q)-difference equations involving the Riemann–Liouville fractional derivative. Based on the generalization of Banach contraction principle, we obtain a sufficient condition for existence and uniqueness of solutions of the problem. By applying a fixed point theorem in cones, we establish a sufficient condition for the existence of at least one positive solution of the problem. As an application, some examples are presented to illustrate the main results. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the following Kirchhoff equations involving the fractional Laplacian, (0.1) {(a+b[u]s²) (−Δ)su = up + h(x, u, ∇u), x∈Ω, u>0, x∈Ω, u=0, x∉Ω, where Ω is a smooth bounded domain in RN(N ≥ 3), 0 < s <1, a, b>0 and 0 < p < (N+2s)/(N−2s) are constants. Under suitable conditions on h(x,u,∇u), using the defining integral, we carry on a blowing-up and rescaling argument directly on the nonlocal equations and thus obtain a priori estimates on the positive solutions. Moreover, existence results for positive solutions of problem (0.1) are proved by Leray-Schauder degree theory and the above estimates. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the boundary value problem of a class of fractional (p, q)-difference Schrödinger equations. By applying Banach contraction mapping principle and a fixed point theorem in cones, we obtain the existence and uniqueness of solutions for the boundary value problem. An example illustrating the main results is also presented. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence and uniqueness of solutions for a tripled system of fractional differential equations with nonlocal integro multi point boundary conditions by using the p-Laplacian operator and the γ-Caputo derivatives. The presented results are obtained by the two fixed point theorems of Banach and Krasnoselskii. An illustrative example is presented at the end to show the applicability of the obtained results. To the best of our knowledge, this is the first time where such problem is considered. [ABSTRACT FROM AUTHOR]

In this paper, we study a new nonlinear sequential differential prob-lem with nonlocal integral conditions that involve convergent series. The problem involves two fractional order operators: Riemann-Liouville inte-gral, Caputo and Riemann-Liouville derivatives. We prove an existence and uniqueness result. Also, we prove a new existence result. We end our paper by presenting some illustrative examples.

In this paper, we treat the problem of existence and uniqueness of positive solutions for an interesting differential equation in the field of population dynamics. This is the Nicholson's blowflies fractional differential equation with delay using Schauder's fixed point theorem. Moreover, an illustrative example is given to demonstrate the effectiveness of the obtained results. [ABSTRACT FROM AUTHOR]

In this paper, we study a class of fractional q-differential inclusion of order 0 < q < 1 under L¹-Caratheodory with convex-compact valued properties on multifunctions. By the use of existence of fixed point for closed valued contractive multifunction on a complete metric space which has been proved by Covitz and Nadler, we provide the existence of solutions for the inclusion problem via some conditions. Also, we give a couple of examples to elaborate our results and to present the obtained results by some numerical computations. [ABSTRACT FROM AUTHOR]

This paper deals with a fractional boundary value problem involving variable delays. Sufficient conditions for the existence of a unique solution are investigated. Moreover the stability of the unique solution is discussed. A numerical example that emphasizes the importance of the results obtained in this article is also included. [ABSTRACT FROM AUTHOR]

In this paper, we study a general system of fractional hybrid differential equations with a nonlinear ϕp-operator, and prove the existence of solution, uniqueness of solution and Hyers-Ulam stability. We use the Caputo fractional derivative in this system so that our system is more general and complex than other nonlinear systems studied before. To establish the results, Green functions are used to transform the considered hybrid boundary problem into a system of fractional integral equations. Then, with the help of the topological degree theorem, we derive some sufficient conditions that ensure the existence and uniqueness of solutions for the proposed system. Finally, an example is presented to show the validity and correctness of the obtained results. [ABSTRACT FROM AUTHOR]

We study an integro differential problem of Lane-Emden type that involves three phi Caputo derivatives. We begin by proving an existence results by means of Schauder theorem. Then, we investigate the niqueness of solution using Banach contraction principle. At the end, one example is discussed. [ABSTRACT FROM AUTHOR]

We show existence of a positive solution for the Kirchhoff equation with small tension forces and general nonlinearities, including singular terms. We use an approximation scheme of Galerkin. [ABSTRACT FROM AUTHOR]

In this paper, we initiate the solvability and stability for a class of singular fractional (p; q)-difference equations. First, we obtain an existence theorem of solution for the fractional (p; q)-difference equation. Then, by using a fractional (p; q)-Gronwall inequality, some stability criteria of solution are established, which also implies the uniqueness of solution. [ABSTRACT FROM AUTHOR]

This study presents the first comprehensive numerical simulation of heat and mass transfer in fractal-like mixed convective nanofluid flows. The flow of non-Newtonian nanofluids over flat and oscillating sheets is modelled mathematically, and a finite difference scheme is used to solve this model. The two-stage scheme can tackle fractal and fractal stochastic mathematical models of partial differential equations. The consistency in the mean square is proved, and Fourier series stability analysis is adopted to find stability conditions for fractal stochastic partial differential equation. The scheme is applied to solve the unsteady Casson nanofluid flow over the flat and oscillatory sheet, which affects thermal radiation, heat source, and chemical reaction. The existence of the solution is also provided for the Navier-Stokes equation of the considered flow model using fractal time derivative. The graph illustrates that the proposed fractal technique achieves faster convergence than the Crank-Nicolson approach. Applications in energy systems, materials science, and environmental engineering are just a few of the domains that could benefit from a better understanding of mixed convective nanofluid flows with fractal features, and that is what this research study hopes to accomplish. Scientists and engineers may better develop efficient and environmentally friendly systems by simulating and analyzing these complicated processes with the suggested finite difference technique. [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]

In this paper, we study a class of nonlocal singular differential equations of fractional orders. The considered class involves Caputo derivatives and Riemann-Liouville integral in its linearities. It is a more general problem for which the question of existence and uniqueness of solutions is investigated. For this class, the question of existence of at least one solution is also discussed. At the end, an illustrative example is presented. [ABSTRACT FROM AUTHOR]

In this paper, the solvability of some inverse problems for a nonlocal analogue of a fourth-order parabolic equation was studied. For this purpose, a nonlocal analogue of the biharmonic operator was introduced. When defining this operator, transformations of the involution type were used. In a parallelepiped, the eigenfunctions and eigenvalues of the Dirichlet type problem for a nonlocal biharmonic operator were studied. The eigenfunctions and eigenvalues for this problem were constructed explicitly and the completeness of the system of eigenfunctions was proved. Two types of inverse problems on finding a solution to the equation and its righthand side were studied. In the two problems, both of the righthand terms depending on the spatial variable and the temporal variable were obtained by using the Fourier variable separation method or reducing it to an integral equation. The theorems for the existence and uniqueness of the solution were proved.

• We discuss the existence of solution of stochastic functional equation with the help of condensing operators and measure of noncompactness. • This functional integral equations contains several equations in this area. • This type of equations help us to represent many real life nonlinear problems. • We provide three examples to illustrate our main results. In this paper, the existence of the solution of a class of nonlinear stochastic functional integral equations is investigated. Such that the concept of measures of noncompactness, and Petryshyn's fixed point theorem in Banach space has been used. In addition to proving the theorems of the existence of the solution, by solving some examples, it is shown that the method is efficient. We also give an example that satisfies the conditions of our main theorem, while the conditions described in some other papers do not. [ABSTRACT FROM AUTHOR]