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The article focuses on solving multidimensional fractional Fokker–Planck equations via unbiased density formulas for anomalous diffusion processes. Topics include the use of tools from Malliavin calculus in its construction being elementary to implement, solution method captures the cusping behaviors of the solution accurately, and the mathematical explanation being given for many interesting subtleties in the behavior of density functions of anomalous diffusion processes.

Generalized second Einstein relation is obtained from a Fokker–Planck equation for a wide class of space-dependent coefficients. The relation is investigated for population growth models. [ABSTRACT FROM AUTHOR]

The primary objective of this research is to develop and analyze a robust computational method based on exponential B splines for solving fractional sub-diffusion equations. The fractional operator includes the Mittag-Leffler function of one parameter in the form of a kernel that is non-local and non-singular in nature. The current approach is based on an effective finite difference method for discretizing in time, and the exponential B-spline functions for discretizing in space. The proposed scheme is proven to be unconditionally stable and convergent. Also, the unique solvability of the method is established. Numerical simulations conducted for multiple test examples validate the agreement between the obtained theoretical results and the corresponding numerical outcomes. [ABSTRACT FROM AUTHOR]

This paper gives the construction and analysis of a new numerical method to solve variable order time fractional mobile-immobile transport model. The proposed numerical method is based on Crank-Nicolson approach with fourth order accurate compact exponential scheme for spatial discretization. We prove that the new scheme is uniquely solvable and unconditionally stable. Furthermore, some numerical results are presented to check the effectiveness of the proposed scheme and consistency of computational results with the theoretical findings. [ABSTRACT FROM AUTHOR]

This paper presented the formulation and solution of the time fractional q-deformed tanhGordon equation, a new extension to the traditional tanh-Gordon equation using fractional calculus, and a q-deformation parameter. This extension aimed to better model physical systems with violated symmetries. The approach taken involved the controlled Picard method combined with the Laplace transform technique and the Caputo fractional derivative to find solutions to this equation. Our results indicated that the method was effective and highlighted our approach in addressing this equation. We explored both the existence and the uniqueness of the solution, and included various 2D and 3D graphs to illustrate how different parameters affect the solution’s behavior. This work aimed to contribute to the theoretical framework of mathematical physics and has potential applications across multiple interdisciplinary fields. [ABSTRACT FROM AUTHOR]

In the current contribution, we handle the inverse problem of identifying and reconstructing a space-depend source term in a degenerate time-fractional parabolic equation from a final observed data. By the use the optimal control framework, the inverse problem is first reformulated as an optimization problem. After discussing the well-posedness of the state problem, we prove the existence of a minimizer and establish the first-order necessary condition. This last one is used to deduce our main results concerning the uniqueness and stability. In the numerical aspect, we use the decent gradient method to solve the associated optimization problem. Some numerical examples are included to illustrate the efficiency of the algorithm. [ABSTRACT FROM AUTHOR]

This research explored optical soliton solutions for the (2+1)-dimensional generalized fractional Kundu-Mukherjee-Naskar equation (gFKMNE), which is a nonlinear model for explaining pulse transmission in communication structures and optical fibers. Two enhanced variants of (G'/G)- expansion method were employed, namely, extended (G'/G)-expansion method and the generalized (r + G'/G)-expansion method, based on the wave transformation of the model into integer-order nonlinear ordinary differential equations (NODEs). By assuming a series-form solution for the resultant NODEs, these strategic methods further translated them into a system of nonlinear algebraic equations. Solving these equations provided optical soliton solutions for gFKMNE using the Maple-13 tool. Through 3D and contour visuals, it was revealed that the constructed soliton solutions are periodically arranged in the optical medium, forming dark soliton lattices. These dark soliton lattices are significant in several domains, such as optical signal processing, optical communications, and nonlinear optics. [ABSTRACT FROM AUTHOR]

In this manuscript, our work was about a qualitative study for a class of multi-complex orders nonlinear fractional differential equations (FDEs). Our methodology utilized the topological degree theory and studied a novel operator tailored for non-singular FDEs with T-Riemann-Liouville (T-RL) fractional order derivatives. The primary objective was to prove the existence and uniqueness of solutions for both initial and boundary value problems within the intricated framework. With the help of topological degree theory, we contributed to a wider understanding of the structural aspects governing the behavior of the considered FDE. The novel operator proposing for non-singular FDEs added a unique dimension to our analytical problem, offering a versatile and effective means of addressing the challenges posed by these complex systems for their theoretical analysis. For the practical implications of our theoretical framework, we presented two concrete examples that reinforced and elucidated the key concepts introduced. These examples underscored our approach's viability and highlighted its potential applications in diverse scientific and engineering domains. Through this comprehensive exploration, we aimed to contribute significantly to advancing the theoretical foundation related to the study of multi-complex nonlinear FDEs. Moreover, a fixedtime terminal sliding mode control (TSMC) has been developed. This proposed control strategy for eliminating leukemic cells while maintaining normal cells was based on a chemotherapeutic treatment that was not only effective but also widely acknowledged to be safe. This strategy was evaluated using the fixed-time Lyapunov stability theory, and simulations were included to illustrate its performance in terms of tracking and convergence. [ABSTRACT FROM AUTHOR]

This paper introduces a modified version of the Variational Iteration Method, incorporating P-transformation. We propose a novel semi-analytical technique named the modified variational iteration method for addressing fractional difierential equations featuring tempered Liouville- Caputo derivatives. The modified variational iteration method emerges as a highly eficient and powerful mathematical tool, ofiering exact or approximate solutions for a diverse range of real-world problems in en- gineering and the natural sciences, specifically those expressed through difierential equations. To assess its efiectiveness and accuracy, we scru- tinize the modified variational iteration method by applying it to three problems related to the heat-like multidimensional difiusion equation with a fractional time derivative in a tempered Liouville-Caputo form [ABSTRACT FROM AUTHOR]

In this paper, Computing the eigenvalues of the Conformable Sturm-Liouville Problem (CSLP) of order 2α, 1/2 < α; ≤ 1, and dirichlet boundary conditions is considered. For this aim, CSLP is discretized to obtain a matrix eigenvalue problem (MEP) using finite element method with fractional shape functions. Then by a method based on the asymptotic form of the eigenvalues, we correct the eigenvalues of MEP to obtain efficient approximations for the eigenvalues of CSLP. Finally, some numerical examples to show the efficiency of the proposed method are given. Numerical results show that for the nth eigenvalue, the correction technique reduces the error order from O(n4h²) to O(n²h²). [ABSTRACT FROM AUTHOR]

In this paper, we consider a fractional Nernst-Planck-Poisson-Navier-Stokes system in ℝ³. First, we obtain a priori estimates by using energy estimates. Then, we construct an iterative solution sequence by solving the approximate problem and obtaining the local existence and uniqueness of the classical solution. Finally, combining the local existence with a priori estimates, the global existence and uniqueness of the classical solution with small initial data are obtained. [ABSTRACT FROM AUTHOR]

Fractional Fokker-Planck equations are generalizations of ordinary Fokker-Planck equations that include fractional derivative operators to encapsulate trapping effects in the anomalously diffusing population they describe. They do not admit analytical solutions in general and there are significant drawbacks to current numerical approximation methods. Using Malliavin calculus, we establish formulas in the form of a mathematical expectation for the probability density functions of anomalous diffusion processes, which are known to solve fractional Fokker-Planck equations. These density formulas lend themselves easily to computation via Monte Carlo sampling and are easy to implement. Moreover, they have the capability to deal with a general model of anomalous diffusion, including the case where the parent process of the anomalous diffusion has a Lévy jump component. The numerical solutions obtained are inherently unbiased, which allows them to form the basis for reliable inferences about the population dynamics. We outline an easy-to-use implementation scheme including methods for generating sample paths of anomalous diffusion processes driven by the inverse stable, inverse tempered stable, and inverse gamma subordinators. We illustrate the effectiveness and accuracy of our method and the physical insight it brings in both univariate and multivariate settings. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to establish a numerical method for solving time-fractional diffusion equations. To obtain the numerical solution, a binary reproducing kernel space is defined, and the orthonormal basis is constructed based on Legendre polynomials in this space. In order to find the ε-approximation solution of time-fractional diffusion equations, which is defined in this paper, the algorithm is designed using the constructed orthonormal basis. Some numerical examples are analyzed to illustrate the procedure and confirm the performance of the proposed method. The results faithfully reveal that the presented method is considerably accurate and effective, as expected. [ABSTRACT FROM AUTHOR]

This study focuses on developing efficient numerical techniques for solving the fractional Keller-Segel (KS) model, which is critical in explaining chemotaxis events. Within the Caputo operator framework, the study applied two unique methodologies: The Aboodh residual power series method (ARPSM) and the Aboodh transform iteration method (ATIM). These approaches were used to find precise solutions to the fractional KS equation, resulting in a better understanding of chemotactic behavior in biological systems. The comparative examination of the ARPSM and ATIM revealed their distinct strengths and applications in solving complicated fractional models. The work advances numerical approaches for fractional differential equations and improves our understanding of chemotaxis dynamics using a precise modeling approach. [ABSTRACT FROM AUTHOR]

In this paper, a mixed high order finite difference scheme-Padé approximation method is applied to obtain numerical solution of the Riesz space fractional advection-dispersion equation(RSFADE). This method is based on the high order finite difference scheme that derived from fractional centered difference and Padé approximation method for space and time integration, respectively. The stability analysis of the proposed method is discussed via theoretical matrix analysis. Numerical experiments are presented to confirm the theoretical results of the proposed method. [ABSTRACT FROM AUTHOR]

Addressing the boundary value problems of fractional-order differential equations hold significant importance due to their applications in various fields. The aim of this paper was to approximate solutions for a class of boundary value problems involving Caputo fractional-order differential equations employing the AA-iterative scheme. Moreover, the stability and data dependence results of the iterative scheme were given for a certain class of mappings. Finally, a numerical experiment was illustrated to support the results presented herein. The results presented in this paper extend and unify some well-known comparable results in the existing literature. [ABSTRACT FROM AUTHOR]

One kind of stochastic delay differential equation in which the delay term is dependent on a proportion of the current time is the pantograph stochastic differential equation. Electric current collection, nonlinear dynamics, quantum mechanics, and electrodynamics are among the phenomena modeled using this equation. A key idea in physics and mathematics is the well-posedness of a differential equation, which guarantees that the solution to the problem exists and is a unique and meaningful solution that relies continuously on the initial condition and the value of the fractional derivative. Ulam-Hyers stability is a property of equations that states that if a function is approximately satisfying the equation, then there exists an exact solution that is close to the function. Inspired by these findings, in this research work, we established the Ulam-Hyers stability and well-posedness of solutions of pantograph fractional stochastic differential equations (PFSDEs) in the framework of conformable derivatives. In addition, we provided examples to analyze the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we consider an inverse source problem with nonlocal boundary conditions for the heat equation involving multi-term time-fractional derivatives. We determine a source term independent of the space variable, and the temperature distribution from the energy measurement. We reduce the solution of the inverse problem to finding solutions to two problems. The well-posedness of each problem is shown using the generalized Fourier method. [ABSTRACT FROM AUTHOR]

We investigate the criteria for both the existence and uniqueness of solutions within a nonlinear coupled system of Hilfer-Hadamard sequential fractional differential equations featuring varying orders. This system is complemented by nonlocal coupled Hadamard fractional integral boundary conditions. The desired outcomes are attained through the application of well-established fixed-point theorems. It is underscored that the fixed-point approach serves as an effective method for establishing both the existence and uniqueness of solutions to boundary value problems. The results obtained are further demonstrated and validated through illustrative examples. [ABSTRACT FROM AUTHOR]

This paper discusses the existence of solutions for fractional differential equations with nonlocal boundary conditions (NFDEs) under essential assumptions. The boundary conditions incorporate a point 0 ≤ c < d and fixed points at the end of the interval [0, d]. For i = 0, 1, the boundary conditions are as follows: ai, bi > 0, a0p(c) = -b0p(d), a1p'(c) = -b1p'(d). Furthermore, the research aims to expand the usability and comprehension of these results to encompass not just NFDEs but also classical fractional differential equations (FDEs) by using the Krasnoselskii fixedpoint theorem and the contraction principle to improve the completeness and usefulness of the results in a wider context of fractional differential equations. We offer examples to demonstrate the results we have achieved. [ABSTRACT FROM AUTHOR]

This paper presented a new Ruscheweyh fractional derivative of fractional order in the complex conformable calculus sense. We applied the constructed complex conformable Ruscheweyh derivative (CCRD) on a certain base of polynomials (BPs) in different regions of convergence in Fréchet spaces (F-spaces). Accordingly, we investigated the relation between the approximation properties of the resulting base and the original one. Moreover, we deduced the mode of increase (the order and type) and the Tp-property of the polynomial bases defined by the CCRD. Some bases of special polynomials, such as Bessel, Chebyshev, Bernoulli, and Euler polynomials, have been discussed to ensure the validity of the obtained results. [ABSTRACT FROM AUTHOR]

In this article, the generalized Rosenau-Korteweg-de Vries-regularized long wave (GR-KDV-RLW) equation was numerically studied by employing the Fourier spectral collection method to discretize the space variable, while the central finite difference method was utilized for the time dependency. The efficiency, accuracy, and simplicity of the employed methodology were tested by solving eight different cases involving four examples of the single solitary wave with different parameter values, interactions between two solitary waves, interactions between three solitary waves, and evolution of solitons with Gaussian and undular bore initial conditions. The error norms were evaluated for the motion of the single solitary wave. The conservation properties of the GR-KDV-RLW equation were studied by computing the momentum and energy. Additionally, the numerical results were compared with the previous studies in the literature. [ABSTRACT FROM AUTHOR]

Brownian motion and viscoelastic anomalous diffusion in homogeneous environments are intrinsically Gaussian processes. In a growing number of systems, however, non-Gaussian displacement distributions of these processes are being reported. The physical cause of the non-Gaussianity is typically seen in different forms of disorder. These include, for instance, imperfect "ensembles" of tracer particles, the presence of local variations of the tracer mobility in heteroegenous environments, or cases in which the speed or persistence of moving nematodes or cells are distributed. From a theoretical point of view stochastic descriptions based on distributed ("superstatistical") transport coefficients as well as time-dependent generalisations based on stochastic transport parameters with built-in finite correlation time are invoked. After a brief review of the history of Brownian motion and the famed Gaussian displacement distribution, we here provide a brief introduction to the phenomenon of non-Gaussianity and the stochastic modelling in terms of superstatistical and diffusing-diffusivity approaches. [ABSTRACT FROM AUTHOR]

The intent of this work to implement first-integral method to study traveling wave solutions of some space-time nonlinear fractional differential equations (FDEs) and present their graphical simulations for analyzing different wave profiles. We show how a specific properties of Gamma functions and wave transformation can be used to reduce a FDE to an ordinary one. This method works well and reveals distinct exact solutions which are classified into two different types, namely trigonometric function and hyperbolic function solutions. The results are also depicted graphically in both 3D and 2D for different values of associated parameters. The obtained results may be useful to understand ion-acoustic waves in plasma, shallow water waves in seas and the evolution of a wave packet in three dimensions with finite depth on water under weak nonlinearity by the space-time-fractional regularized long wave equation and space-time-fractional Davey–Stewartson equation, respectively. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of a multi-criteria scheduling problem on unrelated processors with machines' learning effect, with the goal of minimizing makespan, machine cost and maximal flow-time simultaneously, which is an NP-hard problem. An improved particle swarm optimization algorithm equipped with the overloaded operators, as well as a procedure of Levy flight, is proposed to generate the Pareto-optimal solutions. The experimental results show that the Levy flight strategy can effectively improve the performance of the algorithm, which can generate more non-dominated solutions, and slightly reduce the execution time of the process. [ABSTRACT FROM AUTHOR]

Mathematics has several uses for operators on bounded symmetric domains of Bergman spaces including complex geometry, functional analysis, harmonic analysis and operator theory. They offer instruments for examining the interaction between complex function theory and the underlying domain geometry. Here, we extend the Atangana-Baleanu fractional differential operator acting on a special type of class of analytic functions with the m-fold symmetry characteristic in a bounded symmetric domain (we suggest the open unit disk). We explore the most significant geometric properties, including convexity and star-likeness. The boundedness in the weighted Bergman and the convex Bergman spaces associated with a bounded symmetric domain is investigated. A dual relations exist in these spaces. The subordination and superordination inequalities are presented. Our method is based on Young's convolution inequality. [ABSTRACT FROM AUTHOR]

As a natural generalization of the discrete-time random walk, the continuous-time random walk (CTRW) has been applied to stochastic models with random dynamics in various fields. In this paper, we show that the deterministic alternation of two unbiased CTRWs can lead to a phenomenon similar to the Parrondo paradox, in which the asymptotic mean drift of the combined CTRW becomes positive or negative depending on the parameter values. This extends the case in which the paradox occurs due to the random combination of two CTRWs with memory shown by Montero [Phys. Rev. E 84 (2011) 051139]. [ABSTRACT FROM AUTHOR]

Importance of studying Brinkman–Navier–Stokes flow under the influence of electric and magnetic fields lies in its relevance to fundamental physical phenomena, its applications in various fields of science and engineering and its potential for technological advancements. The study of Brinkman–Navier–Stokes flow under the influence of electric and magnetic fields, along with additional factors such as Joule heating, fractional derivatives and convection, represents a multifaceted and challenging problem in fluid dynamics and applied mathematics. In this study, we explore the combined effects of Joule heating due to electric current passing through the fluid and fractional derivatives that describe nonlocal behaviors. The fractional formulation leads to a relaxation mechanism that exhibits delay and recollection of the fluid motion. The organization of direct temperature and concentration changes in MHD flow is made possible by this formalization. A finite difference/finite element method is used to calculate the flow dynamics issue and fractionally linked fields. The physical factors explain how the topic under examination is relevant. [ABSTRACT FROM AUTHOR]

In this article, we propose two numerical schemes for solving the time-fractional heat equation (TFHE). The proposed methods are based on applying the collocation and tau spectral methods. We introduce and employ a new set of basis functions: The unified Chebyshev polynomials (UCPs) of the first and second kinds. We establish some new theoretical results regarding the new UCPs. We employ these results to derive the proposed algorithms and analyze the convergence of the proposed double expansion. Furthermore, we compute specific integer and fractional derivatives of the UCPs in terms of their original UCPs. The derivation of these derivatives will be the fundamental key to deriving the proposed algorithms. We present some examples to verify the effciency and applicability of the proposed algorithms. [ABSTRACT FROM AUTHOR]

This paper was concerned with the existence and uniqueness results for a coupled system of nonlinear generalized fractional differential equations supplemented with a new class of nonlocal coupled multipoint boundary conditions containing Riemann-Stieltjes and generalized fractional integrals. The nonlinearities in the given system depend on the unknown functions as well as their lower order generalized fractional derivatives. We made use of the Leray-Schauder alternative and Banach contraction mapping principle to obtain the desired results. An illustrative example was also discussed. The paper concluded with some interesting observations. [ABSTRACT FROM AUTHOR]

In this paper, α -deformed mechanics was presented with the α -deformed translation symmetry based on α -addition and α -subtraction. The α -deformed Langevin equation is also discussed. Finally, the anomalous diffusion relation is derived from the α -deformed Langevin equation. [ABSTRACT FROM AUTHOR]

We analyze the motion of a particle governed by a generalized Langevin equation with fractional derivative, nonlocal dissipative force and linear external force. We consider the dissipative memory kernel given by the exponential and power-law functions. For these cases, one can obtain exact results for the relaxation function. The long-time behavior of this function is also investigated. Our results show that the inertial term has no significant influence on the long-time behavior. [ABSTRACT FROM AUTHOR]

During the inflationary regime, the expansion of the Universe is driven by a scalar field ϕ(t) which may be in thermal contact with the radiation fluid. In this work, we study the influence of the thermal bath assuming that it is responsible for the stochastic evolution of the inflaton field. Assuming that the fluctuation dynamics is described by a Langevin-type equation of motion, a large set of analytical solutions including white and colored noises are derived. It is found that even in the case of white noise the field experience an anomalous diffusion. Such results may be important for studying thermally induced initial density perturbations in inflationary cosmologies, mainly in the framework of warm inflation. [ABSTRACT FROM AUTHOR]

We present a stabilizer-free weak Galerkin finite element method (SFWG-FEM) with polynomial reduction on a quasi-uniform mesh in space and Alikhanov's higher order L2-1σ scheme for discretization of the Caputo fractional derivative in time on suitable graded meshes for solving time-fractional subdiffusion equations. Typical solutions of such problems have a singularity at the starting point since the integer-order temporal derivatives of the solution blow up at the initial point. Optimal error bounds in H¹ norm and L² norm are proven for the semi-discrete numerical scheme. Furthermore, we have obtained the values of user-chosen mesh grading constant r, which gives the optimal convergence rate in time for the fully discrete scheme. The optimal rate of convergence of order O(hk+1 + M-2) in the L∞(L²)-norm has been established. We give several numerical examples to confirm the theory presented in this work. [ABSTRACT FROM AUTHOR]

In this work, we study a posteriori error analysis of a general class of fractional initial value problems and fractional boundary value problems. A Petrov-Galerkin spectral method is adopted as the discretization technique in which the generalized Jacobi functions are utilized as basis functions for constructing efficient spectral approximations. The unique solvability of the weak problems is established by verifying the Babuška-Brezzi inf-sup condition. Then, we introduce some residual-type a posteriori error estimators, and deduce their efficiency and reliability in properly weighted Sobolev space. Numerical examples are given to illustrate the performance of the obtained error estimators. [ABSTRACT FROM AUTHOR]

In this work, a novel Hilfer cotangent fractional derivative is presented. This derivative combines the characteristics of the Riemann-Liouville cotangent fractional derivative and the Caputo cotangent fractional derivative. The essential properties of the newly introduced derivative are discussed. By utilizing this derivative, a nonlinear fractional differential problem with a nonlocal initial condition is investigated, and its equivalence to a cotangent Volterra integral equation is demonstrated. The uniqueness and existence of solutions are established by employing fixed-point theorems. Additionally, two illustrative examples are provided to illustrate the obtained results. [ABSTRACT FROM AUTHOR]

In Ouéedraogo A. et al (cf. [30]), it is provided existence and uniqueness results of L¹-renormalized entropy solution for the Cauchy problem associated to the following vast class of nonlinear anisotropic degenerate parabolic-hyperbolic equations involving a nonlocal diffusion term: ∂t u + ∇∙F(u) - ∑i,j=1N ∂xixj² Aij (u)-Lμ [u] = f(u) in Q = (0; T) xRN with T > 0 and N ≥ 1. Our goal is to complement this previous work with a continuous dependence result of the L¹-solution with respect to the data set (F; a; μ; f; u0). The strategy is to follow the approach developed by Karlsen and Ulusoy in [28]. However, we must manage the diculties due to the fact that we are working in the whole space RN with an only integrable initial datum u0 and the term source f depends on the unknown function u. [ABSTRACT FROM AUTHOR]

The barrier escaping problem of non-Ohmic damping Brownian particles diffusing over the saddle point of a three-dimensional potential energy surface is studied by using the reactive flux method. Analytical expression of the transmission coefficient κ (t) is obtained for the first time to reveal the time-dependent barrier passage of the system. Numerical analyses on the stationary transmission coefficient κ s t are compared with those obtained in the previous one- and two-dimensional studies. It is found that κ st varies non-monotonically with the change of the non-Ohmic exponent δ , indicating a similar trend as well as the low-dimensional results. However comparatively, although sometimes it seems to be not the best choice for the particles to accomplish the process of diffusion, the three-dimensional barrier passage is still fairly satisfactory in most cases of the non-Ohmic damping. [ABSTRACT FROM AUTHOR]

The q-deformed Sinh-Gordon equation extends the classical Sinh-Gordon equation by incorporating a deformation parameter q. It provides a framework for studying nonlinear phenomena and soliton dynamics in the presence of quantum deformations, leading to intriguing mathematical structures and potential applications in diverse areas of physics. In this work, we imply the homotopy analysis method, to obtain approximate solutions for the proposed equation, the error estimated from the obtained solutions reflects the efficiency of the solving method. The solutions were presented in the form of 2D and 3D graphics. The graphics clarify the impact of a set of parameters on the solution, including the deformation parameter q, as well as the effect of time and the fractional order derivative. [ABSTRACT FROM AUTHOR]

The aim of this paper is to establish the existence of a sequence of infinitely many small energy solutions to nonlocal problems of Kirchhoffi type involving Hardy potential. To this end, we used the Dual Fountain Theorem as a key tool. In particular, we describe this multiplicity result on a class of the Kirchhoffi coefficient and the nonlinear term which diffier from previous related works. To the best of our belief, the present paper is the first attempt to obtain the multiplicity result for nonlocal problems of Kirchhoffi type involving Hardy potential by utilizing the Dual Fountain Theorem. [ABSTRACT FROM AUTHOR]