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Mathematics, Volume 5, Issue 4 (December 2017)

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Research

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Open AccessArticle New Analytical Technique for Solving a System of Nonlinear Fractional Partial Differential Equations
Mathematics 2017, 5(4), 47; doi:10.3390/math5040047
Received: 24 August 2017 / Revised: 19 September 2017 / Accepted: 20 September 2017 / Published: 25 September 2017
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Abstract
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. Full article
(This article belongs to the Special Issue Operators of Fractional Calculus and Their Applications)
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Open AccessArticle Least-Squares Solution of Linear Differential Equations
Mathematics 2017, 5(4), 48; doi:10.3390/math5040048
Received: 30 July 2017 / Revised: 11 September 2017 / Accepted: 29 September 2017 / Published: 8 October 2017
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Abstract
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 ( t ) , and they satisfy the constraints, no matter what g ( t ) is. The second step consists of expressing g ( t ) 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 [ 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. Full article
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Open AccessArticle An Optimal Control Approach for the Treatment of Solid Tumors with Angiogenesis Inhibitors
Mathematics 2017, 5(4), 49; doi:10.3390/math5040049
Received: 24 June 2017 / Revised: 14 September 2017 / Accepted: 29 September 2017 / Published: 10 October 2017
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Abstract
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. Full article
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Open AccessArticle The Stability of Parabolic Problems with Nonstandard p(x, t)-Growth
Mathematics 2017, 5(4), 50; doi:10.3390/math5040050
Received: 17 September 2017 / Revised: 5 October 2017 / Accepted: 8 October 2017 / Published: 12 October 2017
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Abstract
In this paper, we study weak solutions to the following nonlinear parabolic partial differential equation tudiva(x,t,u)+λ(|u|p(x,t)2
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In this paper, we study weak solutions to the following nonlinear parabolic partial differential equation t u div a ( x , t , u ) + λ ( | u | p ( x , t ) 2 u ) = 0 in Ω T , where λ 0 and t u denote the partial derivative of u with respect to the time variable t, while u denotes the one with respect to the space variable x. Moreover, the vector-field a ( x , t , · ) 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. Full article
Open AccessFeature PaperArticle 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)
Received: 22 September 2017 / Revised: 13 October 2017 / Accepted: 13 October 2017 / Published: 17 October 2017
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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. Full article

Review

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Open AccessReview Euclidean Submanifolds via Tangential Components of Their Position Vector Fields
Mathematics 2017, 5(4), 51; doi:10.3390/math5040051
Received: 6 September 2017 / Revised: 6 September 2017 / Accepted: 10 October 2017 / Published: 16 October 2017
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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. Full article
(This article belongs to the Special Issue Differential Geometry)

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