Solutions Modulo p of Gauss–Manin Differential Equations for Multidimensional Hypergeometric Integrals and Associated Bethe Ansatz*Mathematics* **2017**, *5*(4), 52; doi:10.3390/math5040052 (registering DOI) - 17 October 2017**Abstract **

We consider the Gauss–Manin differential equations for hypergeometric integrals associated with a family of weighted arrangements of hyperplanes moving parallel to themselves. We reduce these equations modulo a prime integer p and construct polynomial solutions of the new differential equations as p-analogs of

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We consider the Gauss–Manin differential equations for hypergeometric integrals associated with a family of weighted arrangements of hyperplanes moving parallel to themselves. We reduce these equations modulo a prime integer p and construct polynomial solutions of the new differential equations as p-analogs of the initial hypergeometric integrals. In some cases, we interpret the p-analogs of the hypergeometric integrals as sums over points of hypersurfaces defined over the finite field Fp. This interpretation is similar to the classical interpretation by Yu. I. Manin of the number of points on an elliptic curve depending on a parameter as a solution of a Gauss hypergeometric differential equation. We discuss the associated Bethe ansatz.
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Euclidean Submanifolds via Tangential Components of Their Position Vector Fields*Mathematics* **2017**, *5*(4), 51; doi:10.3390/math5040051 - 16 October 2017**Abstract **

The position vector field is the most elementary and natural geometric object on a Euclidean submanifold. The position vector field plays important roles in physics, in particular in mechanics. For instance, in any equation of motion, the position vector x (*t*)

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The position vector field is the most elementary and natural geometric object on a Euclidean submanifold. The position vector field plays important roles in physics, in particular in mechanics. For instance, in any equation of motion, the position vector x (*t*) is usually the most sought-after quantity because the position vector field defines the motion of a particle (i.e., a point mass): its location relative to a given coordinate system at some time variable t. This article is a survey article. The purpose of this article is to survey recent results of Euclidean submanifolds associated with the tangential components of their position vector fields. In the last section, we present some interactions between torqued vector fields and Ricci solitons.
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The Stability of Parabolic Problems with Nonstandard *p*(*x*, *t*)-Growth*Mathematics* **2017**, *5*(4), 50; doi:10.3390/math5040050 - 12 October 2017**Abstract **

In this paper, we study weak solutions to the following nonlinear parabolic partial differential equation ${\partial}_{t}u-\mathrm{div}a(x,t,\nabla u)+{\lambda (|u|}^{p(x,t)-2}$

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In this paper, we study weak solutions to the following nonlinear parabolic partial differential equation ${\partial}_{t}u-\mathrm{div}a(x,t,\nabla u)+{\lambda (|u|}^{p(x,t)-2}u)=0\phantom{\rule{3.33333pt}{0ex}}\mathrm{in}\phantom{\rule{3.33333pt}{0ex}}{\Omega}_{T},$ where $\lambda \ge 0$ and ${\partial}_{t}u$ denote the partial derivative of *u* with respect to the time variable *t*, while $\nabla u$ denotes the one with respect to the space variable *x*. Moreover, the vector-field $a(x,t,\xb7)$ satisfies certain nonstandard $p(x,t)$ -growth and monotonicity conditions. In this manuscript, we establish the existence of a unique weak solution to the corresponding Dirichlet problem. Furthermore, we prove the stability of this solution, i.e., we show that two weak solutions with different initial values are controlled by these initial values.
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An Optimal Control Approach for the Treatment of Solid Tumors with Angiogenesis Inhibitors*Mathematics* **2017**, *5*(4), 49; doi:10.3390/math5040049 - 10 October 2017**Abstract **

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Cancer is a disease of unregulated cell growth that is estimated to kill over 600,000 people in the United States in 2017 according to the National Institute of Health. While there are several therapies to treat cancer, tumor resistance to these therapies is

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Cancer is a disease of unregulated cell growth that is estimated to kill over 600,000 people in the United States in 2017 according to the National Institute of Health. While there are several therapies to treat cancer, tumor resistance to these therapies is a concern. Drug therapies have been developed that attack proliferating endothelial cells instead of the tumor in an attempt to create a therapy that is resistant to resistance in contrast to other forms of treatment such as chemotherapy and radiation therapy. In this study, a two-compartment model in terms of differential equations is presented in order to determine the optimal protocol for the delivery of anti-angiogenesis therapy. Optimal control theory is applied to the model with a range of anti-angiogenesis doses to determine optimal doses to minimize tumor volume at the end of a two week treatment and minimize drug toxicity to the patient. Applying a continuous optimal control protocol to our model of angiogenesis and tumor cell growth shows promising results for tumor control while minimizing the toxicity to the patients. By investigating a variety of doses, we determine that the optimal angiogenesis inhibitor dose is in the range of 10–20 mg/kg. In this clinically useful range of doses, good tumor control is achieved for a two week treatment period. This work shows that varying the toxicity of the treatment to the patient will change the optimal dosing scheme but tumor control can still be achieved.
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Least-Squares Solution of Linear Differential Equations*Mathematics* **2017**, *5*(4), 48; doi:10.3390/math5040048 - 8 October 2017**Abstract **

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This study shows how to obtain least-squares solutions to initial value problems (IVPs), boundary value problems (BVPs), and multi-value problems (MVPs) for nonhomogeneous linear differential equations (DEs) with nonconstant coefficients of any order. However, without loss of generality, the approach has been applied

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This study shows how to obtain least-squares solutions to initial value problems (IVPs), boundary value problems (BVPs), and multi-value problems (MVPs) for nonhomogeneous linear differential equations (DEs) with nonconstant coefficients of any order. However, without loss of generality, the approach has been applied to second-order DEs. The proposed method has two steps. The first step consists of writing a *constrained expression*, that has the DE constraints embedded. These kind of expressions are given in terms of a new unknown function, $g\left(t\right)$ , and they satisfy the constraints, no matter what $g\left(t\right)$ is. The second step consists of expressing $g\left(t\right)$ as a linear combination of *m* independent known basis functions. Specifically, orthogonal polynomials are adopted for the basis functions. This choice requires rewriting the DE and the constraints in terms of a new independent variable, $x\in [-1,+1]$ . The procedure leads to a set of linear equations in terms of the unknown coefficients of the basis functions that are then computed by least-squares. Numerical examples are provided to quantify the solutions’ accuracy for IVPs, BVPs and MVPs. In all the examples provided, the least-squares solution is obtained with machine error accuracy.
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New Analytical Technique for Solving a System of Nonlinear Fractional Partial Differential Equations*Mathematics* **2017**, *5*(4), 47; doi:10.3390/math5040047 - 25 September 2017**Abstract **

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This paper introduces a new analytical technique (NAT) for solving a system of nonlinear fractional partial differential equations (NFPDEs) in full general set. Moreover, the convergence and error analysis of the proposed technique is shown. The approximate solutions for a system of NFPDEs

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This paper introduces a new analytical technique (NAT) for solving a system of nonlinear fractional partial differential equations (NFPDEs) in full general set. Moreover, the convergence and error analysis of the proposed technique is shown. The approximate solutions for a system of NFPDEs are easily obtained by means of Caputo fractional partial derivatives based on the properties of fractional calculus. However, analytical and numerical traveling wave solutions for some systems of nonlinear wave equations are successfully obtained to confirm the accuracy and efficiency of the proposed technique. Several numerical results are presented in the format of tables and graphs to make a comparison with results previously obtained by other well-known methods.
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The Catastrophe of Electric Vehicle Sales*Mathematics* **2017**, *5*(3), 46; doi:10.3390/math5030046 - 17 September 2017**Abstract **

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Electric vehicles have undergone a recent faddy trend in the United States and Europe, and several recent publications trumpet the continued rise of electric vehicles citing steadily-climbing monthly vehicle sales. The broad purpose of this study is to examine this optimism with some

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Electric vehicles have undergone a recent faddy trend in the United States and Europe, and several recent publications trumpet the continued rise of electric vehicles citing steadily-climbing monthly vehicle sales. The broad purpose of this study is to examine this optimism with some degree of mathematical rigor. Specifically, the methodology will use *catastrophe* theory to explore the possibility of a sudden, seemingly-unexplainable crash in vehicle sales. The study begins by defining optimal system equations that well-model the available sales data. Next, these optimal models are used to investigate the potential response to a *slow dynamic* acting on the relatively faster dynamic of the optimal system equations. Catastrophe theory indicates a potential sudden crash in sales when a slow dynamic is at-work. It is noteworthy that the prediction can be made even while sales are increasing.
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Fusion Estimation from Multisensor Observations with Multiplicative Noises and Correlated Random Delays in Transmission*Mathematics* **2017**, *5*(3), 45; doi:10.3390/math5030045 - 4 September 2017**Abstract **

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In this paper, the information fusion estimation problem is investigated for a class of multisensor linear systems affected by different kinds of stochastic uncertainties, using both the distributed and the centralized fusion methodologies. It is assumed that the measured outputs are perturbed by

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In this paper, the information fusion estimation problem is investigated for a class of multisensor linear systems affected by different kinds of stochastic uncertainties, using both the distributed and the centralized fusion methodologies. It is assumed that the measured outputs are perturbed by one-step autocorrelated and cross-correlated additive noises, and also stochastic uncertainties caused by multiplicative noises and randomly missing measurements in the sensor outputs are considered. At each sampling time, every sensor output is sent to a local processor and, due to some kind of transmission failures, one-step correlated random delays may occur. Using only covariance information, without requiring the evolution model of the signal process, a local least-squares (LS) filter based on the measurements received from each sensor is designed by an innovation approach. All these local filters are then fused to generate an optimal distributed fusion filter by a matrix-weighted linear combination, using the LS optimality criterion. Moreover, a recursive algorithm for the centralized fusion filter is also proposed and the accuracy of the proposed estimators, which is measured by the estimation error covariances, is analyzed by a simulation example.
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Topics of Measure Theory on Infinite Dimensional Spaces*Mathematics* **2017**, *5*(3), 44; doi:10.3390/math5030044 - 29 August 2017**Abstract **

This short review is devoted to measures on infinite dimensional spaces. We start by discussing product measures and projective techniques. Special attention is paid to measures on linear spaces, and in particular to Gaussian measures. Transformation properties of measures are considered, as well

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This short review is devoted to measures on infinite dimensional spaces. We start by discussing product measures and projective techniques. Special attention is paid to measures on linear spaces, and in particular to Gaussian measures. Transformation properties of measures are considered, as well as fundamental results concerning the support of the measure.
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On Minimal Covolume Hyperbolic Lattices*Mathematics* **2017**, *5*(3), 43; doi:10.3390/math5030043 - 22 August 2017**Abstract **

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We study lattices with a non-compact fundamental domain of small volume in hyperbolic space ${\mathbb{H}}^{n}$ . First, we identify the arithmetic lattices in ${\mathrm{Isom}}^{+}{\mathbb{H}}^{n}$ of minimal covolume for even *n* up to 18. Then, we discuss the related problem

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We study lattices with a non-compact fundamental domain of small volume in hyperbolic space ${\mathbb{H}}^{n}$ . First, we identify the arithmetic lattices in ${\mathrm{Isom}}^{+}{\mathbb{H}}^{n}$ of minimal covolume for even *n* up to 18. Then, we discuss the related problem in higher odd dimensions and provide solutions for $n=11$ and $n=13$ in terms of the rotation subgroup of certain Coxeter pyramid groups found by Tumarkin. The results depend on the work of Belolipetsky and Emery, as well as on the Euler characteristic computation for hyperbolic Coxeter polyhedra with few facets by means of the program *CoxIter* developed by Guglielmetti. This work complements the survey about hyperbolic orbifolds of minimal volume.
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On the Uniqueness Results and Value Distribution of Meromorphic Mappings*Mathematics* **2017**, *5*(3), 42; doi:10.3390/math5030042 - 17 August 2017**Abstract **

This research concentrates on the analysis of meromorphic mappings. We derived several important results for value distribution of specific difference polynomials of meromorphic mappings, which generalize the work of Laine and Yang. In addition, we proved uniqueness theorems of meromorphic mappings. The difference

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This research concentrates on the analysis of meromorphic mappings. We derived several important results for value distribution of specific difference polynomials of meromorphic mappings, which generalize the work of Laine and Yang. In addition, we proved uniqueness theorems of meromorphic mappings. The difference polynomials of these functions have the same fixed points or share a nonzero value. This extends the research work of Qi, Yang and Liu, where they used the finite ordered meromorphic mappings.
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On the Duality of Regular and Local Functions*Mathematics* **2017**, *5*(3), 41; doi:10.3390/math5030041 - 9 August 2017**Abstract **

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In this paper, we relate Poisson’s summation formula to Heisenberg’s uncertainty principle. They both express Fourier dualities within the space of tempered distributions and these dualities are also inverse of each other. While Poisson’s summation formula expresses a duality between discretization and periodization,

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In this paper, we relate Poisson’s summation formula to Heisenberg’s uncertainty principle. They both express Fourier dualities within the space of tempered distributions and these dualities are also inverse of each other. While Poisson’s summation formula expresses a duality between discretization and periodization, Heisenberg’s uncertainty principle expresses a duality between regularization and localization. We define regularization and localization on generalized functions and show that the Fourier transform of regular functions are local functions and, vice versa, the Fourier transform of local functions are regular functions.
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Integral Representations of the Catalan Numbers and Their Applications*Mathematics* **2017**, *5*(3), 40; doi:10.3390/math5030040 - 3 August 2017**Abstract **

In the paper, the authors survey integral representations of the Catalan numbers and the Catalan–Qi function, discuss equivalent relations between these integral representations, supply alternative and new proofs of several integral representations, collect applications of some integral representations, and present sums of several

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In the paper, the authors survey integral representations of the Catalan numbers and the Catalan–Qi function, discuss equivalent relations between these integral representations, supply alternative and new proofs of several integral representations, collect applications of some integral representations, and present sums of several power series whose coefficients involve the Catalan numbers.
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Confidence Intervals for Mean and Difference between Means of Normal Distributions with Unknown Coefficients of Variation*Mathematics* **2017**, *5*(3), 39; doi:10.3390/math5030039 - 28 July 2017**Abstract **

This paper proposes confidence intervals for a single mean and difference of two means of normal distributions with unknown coefficients of variation (CVs). The generalized confidence interval (GCI) approach and large sample (LS) approach were proposed to construct confidence intervals for the single

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This paper proposes confidence intervals for a single mean and difference of two means of normal distributions with unknown coefficients of variation (CVs). The generalized confidence interval (GCI) approach and large sample (LS) approach were proposed to construct confidence intervals for the single normal mean with unknown CV. These confidence intervals were compared with existing confidence interval for the single normal mean based on the Student’s t-distribution (small sample size case) and the z-distribution (large sample size case). Furthermore, the confidence intervals for the difference between two normal means with unknown CVs were constructed based on the GCI approach, the method of variance estimates recovery (MOVER) approach and the LS approach and then compared with the Welch–Satterthwaite (WS) approach. The coverage probability and average length of the proposed confidence intervals were evaluated via Monte Carlo simulation. The results indicated that the GCIs for the single normal mean and the difference of two normal means with unknown CVs are better than the other confidence intervals. Finally, three datasets are given to illustrate the proposed confidence intervals.
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Variable Shape Parameter Strategy in Local Radial Basis Functions Collocation Method for Solving the 2D Nonlinear Coupled Burgers’ Equations*Mathematics* **2017**, *5*(3), 38; doi:10.3390/math5030038 - 21 July 2017**Abstract **

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This study aimed at investigating a local radial basis function collocation method (LRBFCM) in the reproducing kernel Hilbert space. This method was, in fact, a meshless one which applied the local sub-clusters of domain nodes for the approximation of the arbitrary field. For

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This study aimed at investigating a local radial basis function collocation method (LRBFCM) in the reproducing kernel Hilbert space. This method was, in fact, a meshless one which applied the local sub-clusters of domain nodes for the approximation of the arbitrary field. For time-dependent partial differential equations (PDEs), it would be changed to a system of ordinary differential equations (ODEs). Here, we intended to decrease the error through utilizing variable shape parameter (VSP) strategies. This method was an appropriate way to solve the two-dimensional nonlinear coupled Burgers’ equations comprised of Dirichlet and mixed boundary conditions. Numerical examples indicated that the variable shape parameter strategies were more efficient than constant ones for various values of the Reynolds number.
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Elimination of Quotients in Various Localisations of Premodels into Models*Mathematics* **2017**, *5*(3), 37; doi:10.3390/math5030037 - 9 July 2017**Abstract **

The contribution of this article is quadruple. It (1) unifies various schemes of premodels/models including situations such as presheaves/sheaves, sheaves/flabby sheaves, prespectra/$\Omega $ -spectra, simplicial topological spaces/(complete) Segal spaces, pre-localised rings/localised rings, functors in categories/strong stacks and, to some extent, functors from a

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The contribution of this article is quadruple. It (1) unifies various schemes of premodels/models including situations such as presheaves/sheaves, sheaves/flabby sheaves, prespectra/$\Omega $ -spectra, simplicial topological spaces/(complete) Segal spaces, pre-localised rings/localised rings, functors in categories/strong stacks and, to some extent, functors from a limit sketch to a model category versus the homotopical models for the limit sketch; (2) provides a general construction from the premodels to the models; (3) proposes technics that allow one to assess the nature of the universal properties associated with this construction; (4) shows that the obtained localisation admits a particular presentation, which organises the structural and relational information into bundles of data. This presentation is obtained via a process called an *elimination of quotients* and its aim is to facilitate the handling of the relational information appearing in the construction of higher dimensional objects such as weak $(\omega ,n)$ -categories, weak $\omega $ -groupoids and higher moduli stacks.
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Lattices and Rational Points*Mathematics* **2017**, *5*(3), 36; doi:10.3390/math5030036 - 9 July 2017**Abstract **

In this article, we show how to use the first and second Minkowski Theorems and some Diophantine geometry to bound explicitly the height of the points of rank $N-1$ on transverse curves in ${E}^{N}$ , where *E* is an elliptic

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In this article, we show how to use the first and second Minkowski Theorems and some Diophantine geometry to bound explicitly the height of the points of rank $N-1$ on transverse curves in ${E}^{N}$ , where *E* is an elliptic curve without Complex Multiplication (CM). We then apply our result to give a method for finding the rational points on such curves, when *E* has $\mathbb{Q}$ -rank $\le N-1$ . We also give some explicit examples. This result generalises from rank 1 to rank $N-1$ previous results of S. Checcoli, F. Veneziano and the author.
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Banach Subspaces of Continuous Functions Possessing Schauder Bases*Mathematics* **2017**, *5*(3), 35; doi:10.3390/math5030035 - 24 June 2017**Abstract **

In this article, Müntz spaces ${M}_{\Lambda ,C}$ of continuous functions supplied with the absolute maximum norm are considered. An existence of Schauder bases in Müntz spaces ${M}_{\Lambda ,C}$ is investigated. Moreover, Fourier series approximation of functions in Müntz spaces

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In this article, Müntz spaces ${M}_{\Lambda ,C}$ of continuous functions supplied with the absolute maximum norm are considered. An existence of Schauder bases in Müntz spaces ${M}_{\Lambda ,C}$ is investigated. Moreover, Fourier series approximation of functions in Müntz spaces ${M}_{\Lambda ,C}$ is studied.
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Lie Symmetries, Optimal System and Invariant Reductions to a Nonlinear Timoshenko System*Mathematics* **2017**, *5*(2), 34; doi:10.3390/math5020034 - 17 June 2017**Abstract **

Lie symmetries and their Lie group transformations for a class of Timoshenko systems are presented. The class considered is the class of nonlinear Timoshenko systems of partial differential equations (PDEs). An optimal system of one-dimensional sub-algebras of the corresponding Lie algebra is derived.

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Lie symmetries and their Lie group transformations for a class of Timoshenko systems are presented. The class considered is the class of nonlinear Timoshenko systems of partial differential equations (PDEs). An optimal system of one-dimensional sub-algebras of the corresponding Lie algebra is derived. All possible invariant variables of the optimal system are obtained. The corresponding reduced systems of ordinary differential equations (ODEs) are also provided. All possible non-similar invariant conditions prescribed on invariant surfaces under symmetry transformations are given. As an application, explicit solutions of the system are given where the beam is hinged at one end and free at the other end.
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An Analysis on the Fractional Asset Flow Differential Equations*Mathematics* **2017**, *5*(2), 33; doi:10.3390/math5020033 - 16 June 2017**Abstract **

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The asset flow differential equation (AFDE) is the mathematical model that plays an essential role for planning to predict the financial behavior in the market. In this paper, we introduce the fractional asset flow differential equations (FAFDEs) based on the Liouville-Caputo derivative. We

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The asset flow differential equation (AFDE) is the mathematical model that plays an essential role for planning to predict the financial behavior in the market. In this paper, we introduce the fractional asset flow differential equations (FAFDEs) based on the Liouville-Caputo derivative. We prove the existence and uniqueness of a solution for the FAFDEs. Furthermore, the stability analysis of the model is investigated and the numerical simulation is accordingly performed to support the proposed model.
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