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Journal articles on the topic 'Partial suitable solutions'

Author: Grafiati

Published: 21 December 2024

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1

Yan, Xiaodong. "Partial Regularity of Suitable Weak Solutions of Complex Ginzburg Landau Equations." Communications in Partial Differential Equations 24, no. 11-12 (January 1999): 390–93. http://dx.doi.org/10.1080/03605309908821501.

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2

He, Cheng, and Zhouping Xin. "Partial regularity of suitable weak solutions to the incompressible magnetohydrodynamic equations." Journal of Functional Analysis 227, no. 1 (October 2005): 113–52. http://dx.doi.org/10.1016/j.jfa.2005.06.009.

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3

Ren, Wei, Yanqing Wang, and Gang Wu. "Partial regularity of suitable weak solutions to the multi-dimensional generalized magnetohydrodynamics equations." Communications in Contemporary Mathematics 18, no. 06 (September 14, 2016): 1650018. http://dx.doi.org/10.1142/s0219199716500188.

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In this paper, we are concerned with the partial regularity of the suitable weak solutions to the fractional MHD equations in [Formula: see text] for [Formula: see text]. In comparison with the work of the 3D fractional Navier–Stokes equations obtained by Tang and Yu in [Partial regularity of suitable weak solutions to the fractional Navier–Stokes equations, Comm. Math. Phys. 334 (2015) 1455–1482], our results include their endpoint case [Formula: see text] and the external force belongs to a more general parabolic Morrey space. Moreover, we prove some interior regularity criteria just via the scaled mixed norm of the velocity for the suitable weak solutions to the fractional MHD equations.

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4

Tang, Lan, and Yong Yu. "Partial Regularity of Suitable Weak Solutions to the Fractional Navier–Stokes Equations." Communications in Mathematical Physics 334, no. 3 (September 19, 2014): 1455–82. http://dx.doi.org/10.1007/s00220-014-2149-z.

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5

Wang, Xiaoming, Shehbaz Ahmad Javed, Abdul Majeed, Mohsin Kamran, and Muhammad Abbas. "Investigation of Exact Solutions of Nonlinear Evolution Equations Using Unified Method." Mathematics 10, no. 16 (August 19, 2022): 2996. http://dx.doi.org/10.3390/math10162996.

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In this article, an analytical technique based on unified method is applied to investigate the exact solutions of non-linear homogeneous evolution partial differential equations. These partial differential equations are reduced to ordinary differential equations using different traveling wave transformations, and exact solutions in rational and polynomial forms are obtained. The obtained solutions are presented in the form of 2D and 3D graphics to study the behavior of the analytical solution by setting out the values of suitable parameters. The acquired results reveal that the unified method is a suitable approach for handling non-linear homogeneous evolution equations.

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6

Zhang, Huan, Yin Zhou, and Yuhua Long. "Results on multiple nontrivial solutions to partial difference equations." AIMS Mathematics 8, no. 3 (2022): 5413–31. http://dx.doi.org/10.3934/math.2023272.

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<abstract><p>In this paper, we consider the existence and multiplicity of nontrivial solutions to second order partial difference equation with Dirichlet boundary conditions by Morse theory. Given suitable conditions, we establish multiple results that the problem admits at least two nontrivial solutions. Moreover, we provide five examples to illustrate applications of our theorems.</p></abstract>

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7

Jiu, Quansen, and Yanqing Wang. "Remarks on partial regularity for suitable weak solutions of the incompressible magnetohydrodynamic equations." Journal of Mathematical Analysis and Applications 409, no. 2 (January 2014): 1052–65. http://dx.doi.org/10.1016/j.jmaa.2013.07.052.

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8

Matsuzawa, Tadato. "Partial regularity and applications." Nagoya Mathematical Journal 103 (October 1986): 133–43. http://dx.doi.org/10.1017/s0027763000000623.

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The problem to determine the Gevrey index of solutions of a given hypoelliptic partial differential equation seems to be not yet well investigated. In this paper, we shall show the Gevrey indices of solutions of the equations of Grushin type, [6], are determined by a rather simple application of a straightforward extension of the results given in [7], [8] and [13]. For simplicity to construct left parametrices in the operator valued sense, we shall consider the equations under the stronger condition than that of [6] (cf. Condition 1 of Section 3). Typical examples of Grushin type are given by which will be discussed in Section 4. We remark that our approach may be compared with the one to a similar problem discussed in [17] by using suitable L2-estimates constructed in [16].

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9

Yang, Jiaqi. "Partially regular weak solutions to the fractional Navier–Stokes equations with the critical dissipation." Journal of Mathematical Physics 63, no. 11 (November 1, 2022): 111501. http://dx.doi.org/10.1063/5.0088047.

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We show that there exist partially regular weak solutions of Navier–Stokes equations with fractional dissipation [Formula: see text] in the critical case of [Formula: see text], which satisfy certain local energy inequalities and whose singular sets have a locally finite two-dimensional parabolic Hausdorff measure. Actually, this problem had been studied by Chen and Wei [Discrete Contin. Dyn. Syst. 36(10), 5309–5322 (2016)]; in this paper, they established the partial regularity of suitable weak solutions for [Formula: see text]. A point is that they admitted the existence of suitable weak solutions but did not give the proof. It should be noted that, when [Formula: see text], the existence of suitable weak solutions is not trivial due to the possible lack of compactness. To overcome this difficulty, we shall use a parabolic concentration-compactness theorem introduced by Wu [Arch. Ration. Mech. Anal. 239(3), 1771–1808 (2021)]. For the partial regularity theory, we will apply the idea of Chen and Wei.

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10

Pla, Francisco, and Henar Herrero. "Reduced Basis Method Applied to Eigenvalue Problems from Convection." International Journal of Bifurcation and Chaos 29, no. 03 (March 2019): 1950028. http://dx.doi.org/10.1142/s0218127419500287.

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The reduced basis method is a suitable technique for finding numerical solutions to partial differential equations that must be obtained for many values of parameters. This method is suitable when researching bifurcations and instabilities of stationary solutions for partial differential equations. It is necessary to solve numerically the partial differential equations along with the corresponding eigenvalue problems of the linear stability analysis of stationary solutions for a large number of bifurcation parameter values. In this paper, the reduced basis method has been used to solve eigenvalue problems derived from the linear stability analysis of stationary solutions in a two-dimensional Rayleigh–Bénard convection problem. The bifurcation parameter is the Rayleigh number, which measures buoyancy. The reduced basis considered belongs to the eigenfunction spaces derived from the eigenvalue problems for different types of solutions in the bifurcation diagram depending on the Rayleigh number. The eigenvalue with the largest real part and its corresponding eigenfunction are easily calculated and the bifurcation points are correctly captured. The resulting matrices are small, which enables a drastic reduction in the computational cost of solving the eigenvalue problems.

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11

Gong, Huajun, Changyou Wang, and Xiaotao Zhang. "Partial Regularity of Suitable Weak Solutions of the Navier--Stokes--Planck--Nernst--Poisson Equation." SIAM Journal on Mathematical Analysis 53, no. 3 (January 2021): 3306–37. http://dx.doi.org/10.1137/19m1292011.

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12

Xu, Xiangsheng. "Local Partial Regularity Theorems for Suitable Weak Solutions of a Class of Degenerate Systems." Applied Mathematics and Optimization 34, no. 3 (March 1, 1996): 299–324. http://dx.doi.org/10.1007/s002459900031.

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13

Xu, Xiangsheng. "Local partial regularity theorems for suitable weak solutions of a class of degenerate systems." Applied Mathematics & Optimization 34, no. 3 (November 1996): 299–324. http://dx.doi.org/10.1007/bf01182628.

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14

Jiu, Quansen, Yanqing Wang, and Gang Wu. "Partial Regularity of the Suitable Weak Solutions to the Multi-dimensional Incompressible Boussinesq Equations." Journal of Dynamics and Differential Equations 28, no. 2 (May 6, 2016): 567–91. http://dx.doi.org/10.1007/s10884-016-9536-4.

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15

Tang, Lan, and Yong Yu. "Erratum to: Partial Regularity of Suitable Weak Solutions to the Fractional Navier–Stokes Equations." Communications in Mathematical Physics 335, no. 2 (February 20, 2015): 1057–63. http://dx.doi.org/10.1007/s00220-015-2289-9.

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16

Ladyzhenskaya, O. A., and G. A. Seregin. "On Partial Regularity of Suitable Weak Solutions to the Three-Dimensional NavierStokes equations." Journal of Mathematical Fluid Mechanics 1, no. 4 (December 1, 1999): 356–87. http://dx.doi.org/10.1007/s000210050015.

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17

Han, Pigong, and Cheng He. "Partial regularity of suitable weak solutions to the four-dimensional incompressible magneto-hydrodynamic equations." Mathematical Methods in the Applied Sciences 35, no. 11 (June 5, 2012): 1335–55. http://dx.doi.org/10.1002/mma.2536.

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18

Yu, Huan. "Partial regularity criteria for suitable weak solutions of the three-dimensional liquid crystals flow." Mathematical Methods in the Applied Sciences 39, no. 14 (February 8, 2016): 4196–207. http://dx.doi.org/10.1002/mma.3856.

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19

S, Priyadharshini, and Sadhasivam V. "Forced oscillation of solutions of conformable hybrid elliptic partial differential equations." Journal of Computational Mathematica 6, no. 1 (March 31, 2022): 396–412. http://dx.doi.org/10.26524/cm140.

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In this paper, we investigate the forced oscillation of solutions of conformable hybrid elliptic partial differential equations. We show that, the suitable condition for the infinite sequence of annular domains which gives every solution has a zero. Some examples are given to illustrate the effectiveness of our main result.

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20

Zhao, Zhi-Han, Yong-Kui Chang, and Juan J. Nieto. "Asymptotic Behavior of Solutions to Abstract Stochastic Fractional Partial Integrodifferential Equations." Abstract and Applied Analysis 2013 (2013): 1–8. http://dx.doi.org/10.1155/2013/138068.

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The existence of asymptotically almost automorphic mild solutions to an abstract stochastic fractional partial integrodifferential equation is considered. The main tools are some suitable composition results for asymptotically almost automorphic processes, the theory of sectorial linear operators, and classical fixed point theorems. An example is also given to illustrate the main theorems.

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21

Meyer, J. C., and D. J. Needham. "Extended weak maximum principles for parabolic partial differential inequalities on unbounded domains." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 470, no. 2167 (July 8, 2014): 20140079. http://dx.doi.org/10.1098/rspa.2014.0079.

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In this paper, we establish extended maximum principles for solutions to linear parabolic partial differential inequalities on unbounded domains, where the solutions satisfy a variety of growth/decay conditions on the unbounded domain. We establish a conditional maximum principle, which states that a solution u to a linear parabolic partial differential inequality satisfies a maximum principle whenever a suitable weight function can be exhibited. Our extended maximum principles are then established by exhibiting suitable weight functions and applying the conditional maximum principle. In addition, we include several specific examples, to highlight the importance of certain generic conditions, which are required in the statements of maximum principles of this type. Furthermore, we demonstrate how to obtain associated comparison theorems from our extended maximum principles.

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22

Cuccu, Fabrizio, and Giovanni Porru. "Symmetry of solutions to optimization problems related to partial differential equations." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 136, no. 5 (October 2006): 921–34. http://dx.doi.org/10.1017/s0308210500004807.

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We investigate maxima and minima of some functionals associated with solutions to Dirichlet problems for elliptic equations. We prove existence results and, under suitable restrictions on the data, we show that any maximal configuration satisfies a special system of two equations. Next, we use the moving-plane method to find symmetry results for solutions of a system. We apply these results in our discussion of symmetry for the maximal configurations of the previous problem.

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23

Et al., Enadi. "New Approach for Solving Three Dimensional Space Partial Differential Equation." Baghdad Science Journal 16, no. 3(Suppl.) (September 23, 2019): 0786. http://dx.doi.org/10.21123/bsj.2019.16.3(suppl.).0786.

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This paper presents a new transform method to solve partial differential equations, for finding suitable accurate solutions in a wider domain. It can be used to solve the problems without resorting to the frequency domain. The new transform is combined with the homotopy perturbation method in order to solve three dimensional second order partial differential equations with initial condition, and the convergence of the solution to the exact form is proved. The implementation of the suggested method demonstrates the usefulness in finding exact solutions. The practical implications show the effectiveness of approach and it is easily implemented in finding exact solutions. Finally, all algorithms in this paper are implemented in MATLAB version 7.12.

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24

Kamont, Z., and K. Kropielnicka. "Implicit Difference Inequalities Corresponding to First-Order Partial Differential Functional Equations." Journal of Applied Mathematics and Stochastic Analysis 2009 (March 16, 2009): 1–18. http://dx.doi.org/10.1155/2009/254720.

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We give a theorem on implicit difference functional inequalities generated by mixed problems for nonlinear systems of first-order partial differential functional equations. We apply this result in the investigations of the stability of difference methods. Classical solutions of mixed problems are approximated in the paper by solutions of suitable implicit difference schemes. The proof of the convergence of difference method is based on comparison technique, and the result on difference functional inequalities is used. Numerical examples are presented.

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25

Beirão da Veiga, Hugo, and Jiaqi Yang. "On the partial regularity of suitable weak solutions in the non-Newtonian shear-thinning case." Nonlinearity 34, no. 1 (January 1, 2021): 562–77. http://dx.doi.org/10.1088/1361-6544/abcd06.

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26

Chen, Ya-zhou, Hai-liang Li, and Xiao-ding Shi. "Partial Regularity of Suitable Weak Solutions to the System of the Incompressible Shear-thinning Flow." Acta Mathematicae Applicatae Sinica, English Series 37, no. 2 (April 2021): 348–63. http://dx.doi.org/10.1007/s10255-021-1011-2.

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27

Suzuki, Tomoyuki. "On partial regularity of suitable weak solutions to the Navier–Stokes equations in unbounded domains." manuscripta mathematica 125, no. 4 (January 23, 2008): 471–93. http://dx.doi.org/10.1007/s00229-007-0163-6.

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28

Guo, Boling, and Peicheng Zhu. "Partial Regularity of Suitable Weak Solutions to the System of the Incompressible Non-Newtonian Fluids." Journal of Differential Equations 178, no. 2 (January 2002): 281–97. http://dx.doi.org/10.1006/jdeq.2000.3958.

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29

Wang, Yan Qing, Yi Ke Huang, Gang Wu, and Dao Guo Zhou. "Partial Regularity of Suitable Weak Solutions of the Model Arising in Amorphous Molecular Beam Epitaxy." Acta Mathematica Sinica, English Series 39, no. 11 (November 2023): 2219–46. http://dx.doi.org/10.1007/s10114-023-2458-2.

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30

Chamorro, Diego, and Claudiu Mîndrilă. "A new approach for the regularity of weak solutions of the 3D Boussinesq system." Nonlinearity 37, no. 6 (May 14, 2024): 065019. http://dx.doi.org/10.1088/1361-6544/ad4504.

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Abstract We address here the problem of regularity for weak solutions of the 3D Boussinesq equation. By introducing the new notion of partial suitable solutions, which imposes some conditions over the velocity field only, we show a local gain of regularity for the two variables u → and θ.

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31

van der Walt, Jan Harm. "The Order Completion Method for Systems of Nonlinear PDEs: Solutions of Initial Value Problems." Abstract and Applied Analysis 2013 (2013): 1–12. http://dx.doi.org/10.1155/2013/739462.

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We present an existence result for generalized solutions of initial value problems obtained through the order completion method. The solutions we obtain satisfy the initial condition in a suitable extended sense, and each such solution may be represented in a canonical way through its generalized partial derivatives as nearly finite normal lower semicontinuous function.

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32

Zhu, Qingfeng, and Yufeng Shi. "Mean-Field Forward-Backward Doubly Stochastic Differential Equations and Related Nonlocal Stochastic Partial Differential Equations." Abstract and Applied Analysis 2014 (2014): 1–10. http://dx.doi.org/10.1155/2014/194341.

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Mean-field forward-backward doubly stochastic differential equations (MF-FBDSDEs) are studied, which extend many important equations well studied before. Under some suitable monotonicity assumptions, the existence and uniqueness results for measurable solutions are established by means of a method of continuation. Furthermore, the probabilistic interpretation for the solutions to a class of nonlocal stochastic partial differential equations (SPDEs) combined with algebra equations is given.

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33

Qu, Haidong, Zihang She, and Xuan Liu. "Homotopy Analysis Method for Three Types of Fractional Partial Differential Equations." Complexity 2020 (July 10, 2020): 1–13. http://dx.doi.org/10.1155/2020/7232907.

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In this paper, three types of fractional order partial differential equations, including the fractional Cauchy–Riemann equation, fractional acoustic wave equation, and two-dimensional space partial differential equation with time-fractional-order, are considered, and these models are obtained from the standard equations by replacing an integer-order derivative with a fractional-order derivative in Caputo sense. Firstly, we discuss the fractional integral and differential properties of several functions which are derived from the Mittag-Leffler function. Secondly, by using the homotopy analysis method, the exact solutions for fractional order models mentioned above with suitable initial boundary conditions are obtained. Finally, we draw the computer graphics of the exact solutions, the approximate solutions (truncation of finite terms), and absolute errors in the limited area, which show that the effectiveness of the homotopy analysis method for solving fractional order partial differential equations.

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34

Li, Wen-Tao, Zhao Zhang, Xiang-Yu Yang, and Biao Li. "High-order breathers, lumps and hybrid solutions to the (2+1)-dimensional fifth-order KdV equation." International Journal of Modern Physics B 33, no. 22 (September 10, 2019): 1950255. http://dx.doi.org/10.1142/s0217979219502552.

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In this paper, the (2+1)-dimensional fifth-order KdV equation is analytically investigated. By using Hirota’s bilinear method combined with perturbation expansion, the high-order breather solutions of the fifth-order KdV equation are generated. Then, the high-order lump solutions are also derived from the soliton solutions by a long-wave limit method and some suitable parameter constraints. Furthermore, we extend this method to obtain hybrid solutions by taking long-wave limit for partial soliton solutions. Finally, the dynamic behavior of these solutions is presented in the figures.

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35

Yan, Li, Gulnur Yel, Ajay Kumar, Haci Mehmet Baskonus, and Wei Gao. "Newly Developed Analytical Scheme and Its Applications to the Some Nonlinear Partial Differential Equations with the Conformable Derivative." Fractal and Fractional 5, no. 4 (November 23, 2021): 238. http://dx.doi.org/10.3390/fractalfract5040238.

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This paper presents a novel and general analytical approach: the rational sine-Gordon expansion method and its applications to the nonlinear Gardner and (3+1)-dimensional mKdV-ZK equations including a conformable operator. Some trigonometric, periodic, hyperbolic and rational function solutions are extracted. Physical meanings of these solutions are also presented. After choosing suitable values of the parameters in the results, some simulations are plotted. Strain conditions for valid solutions are also reported in detail.

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36

Khan, Masood. "Flow and heat transfer to Sisko fluid with partial slip." Canadian Journal of Physics 94, no. 8 (August 2016): 724–30. http://dx.doi.org/10.1139/cjp-2016-0040.

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In this paper we study the partial slip effects on the flow and heat transfer of an incompressible non-Newtonian fluid over a nonlinear stretching sheet. The velocity slip boundary condition based on the Sisko constitutive fluid model is introduced. Suitable dimensionless variables are used to convert the governing partial differential equations into ordinary differential equations. Numerical solutions of these equations are obtained by the Runge–Kutta Fehlberg method. Additionally, the exact analytical solutions are presented in some special cases. The computational results for the velocity, temperature, skin-friction coefficient, and Nusselt number are presented in graphical and tabular forms. To validate the numerical results obtained, a comparison is made with the exact analytical solutions. The analysis of the results obtained shows that enhancement in the velocity slip parameter reduces the velocity as well as the momentum boundary layer thickness. However, quite the opposite is true with the temperature and corresponding thermal boundary layer thickness.

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37

Bulut, Hasan and Khalid, Ban Jamal. "Optical Soliton Solutions of Fokas-Lenells Equation via (m + 1/G')- Expansion Method." Journal of Advances in Applied & Computational Mathematics 7 (October 16, 2020): 20–24. http://dx.doi.org/10.15377/2409-5761.2020.07.3.

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In this research paper, we investigate some novel soliton solutions to the perturbed Fokas-Lenells equation by using the (m + 1/G') expansion method. Some new solutions are obtained and they are plotted in two and three dimensions. This technique appears as a suitable, applicable, and efficient method to search for the exact solutions of nonlinear partial differential equations in a wide range. All gained optical soliton solutions are substituted into the FokasLenells equation and they verify it. The constraint conditions are also given.

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38

Meng, Fanwei. "A New Approach for Solving Fractional Partial Differential Equations." Journal of Applied Mathematics 2013 (2013): 1–5. http://dx.doi.org/10.1155/2013/256823.

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We propose a new approach for solving fractional partial differential equations based on a nonlinear fractional complex transformation and the general Riccati equation and apply it to solve the nonlinear time fractional biological population model and the (4+1)-dimensional space-time fractional Fokas equation. As a result, some new exact solutions for them are obtained. This approach can be suitable for solving fractional partial differential equations with more general forms than the method proposed by S. Zhang and H.-Q. Zhang (2011).

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39

Wang, Shuang, and Dingbian Qian. "Subharmonic Solutions of Indefinite Hamiltonian Systems via Rotation Numbers." Advanced Nonlinear Studies 21, no. 3 (July 17, 2021): 557–78. http://dx.doi.org/10.1515/ans-2021-2134.

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Abstract We investigate the multiplicity of subharmonic solutions for indefinite planar Hamiltonian systems J ⁢ z ′ = ∇ ⁡ H ⁢ ( t , z ) {Jz^{\prime}=\nabla H(t,z)} from a rotation number viewpoint. The class considered is such that the behaviour of its solutions near zero and infinity can be compared two suitable positively homogeneous systems. Our approach can be used to deal with the problems in absence of the sign assumption on ∂ ⁡ H ∂ ⁡ x ⁢ ( t , x , y ) {\frac{\partial H}{\partial x}(t,x,y)} , uniqueness and global continuability for the solutions of the associated Cauchy problems. These systems may also be resonant. By the use of an approach of rotation number, the phase-plane analysis of the spiral properties of large solutions and a recent version of Poincaré–Birkhoff theorem for Hamiltonian systems, we are able to extend previous multiplicity results of subharmonic solutions for asymptotically semilinear systems to indefinite planar Hamiltonian systems.

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40

Dusunceli, Faruk. "New Exponential and Complex Traveling Wave Solutions to the Konopelchenko-Dubrovsky Model." Advances in Mathematical Physics 2019 (February 11, 2019): 1–9. http://dx.doi.org/10.1155/2019/7801247.

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The Konopelchenko-Dubrovsky (KD) system is presented by the application of the improved Bernoulli subequation function method (IBSEFM). First, The KD system being Nonlinear partial differential equations system is transformed into nonlinear ordinary differential equation by using a wave transformation. Last, the resulting equation is successfully explored for new explicit exact solutions including singular soliton, kink, and periodic wave solutions. All the obtained solutions in this study satisfy the Konopelchenko-Dubrovsky model. Under suitable choice of the parameter values, interesting two- and three-dimensional graphs of all the obtained solutions are plotted.

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41

Phillips, Alexander M., Michael J. Wright, Isabelle Riou, Stephen Maddox, Simon Maskell, and Jason F. Ralph. "Position fixing with cold atom gravity gradiometers." AVS Quantum Science 4, no. 2 (June 2022): 024404. http://dx.doi.org/10.1116/5.0095677.

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This paper proposes a position fixing method for autonomous navigation using partial gravity gradient solutions from cold atom interferometers. Cold atom quantum sensors can provide ultra-precise measurements of inertial quantities, such as acceleration and rotation rates. However, we investigate the use of pairs of cold atom interferometers to measure the local gravity gradient and to provide position information by referencing these measurements against a suitable database. Simulating the motion of a vehicle, we use partial gravity gradient measurements to reduce the positional drift associated with inertial navigation systems. Using standard open source global gravity databases, we show stable navigation solutions for trajectories of over 1000 km.

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42

Zhang, Zhijun. "Two Classes of Nonlinear Singular Dirichlet Problems with Natural Growth: Existence and Asymptotic Behavior." Advanced Nonlinear Studies 20, no. 1 (February 1, 2020): 77–93. http://dx.doi.org/10.1515/ans-2019-2054.

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AbstractThis paper is concerned with the existence, uniqueness and asymptotic behavior of classical solutions to two classes of models {-\triangle u\pm\lambda\frac{|\nabla u|^{2}}{u^{\beta}}=b(x)u^{-\alpha}}, {u>0}, {x\in\Omega}, {u|_{\partial\Omega}=0}, where Ω is a bounded domain with smooth boundary in {\mathbb{R}^{N}}, {\lambda>0}, {\beta>0}, {\alpha>-1}, and {b\in C^{\nu}_{\mathrm{loc}}(\Omega)} for some {\nu\in(0,1)}, and b is positive in Ω but may be vanishing or singular on {\partial\Omega}. Our approach is largely based on nonlinear transformations and the construction of suitable sub- and super-solutions.

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43

Si, Xinhui, Lili Yuan, Limei Cao, Liancun Zheng, Yanan Shen, and Lin Li. "Perturbation solutions for a micropolar fluid flow in a semi-infinite expanding or contracting pipe with large injection or suction through porous wall." Open Physics 14, no. 1 (January 1, 2016): 231–38. http://dx.doi.org/10.1515/phys-2016-0029.

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AbstractWe investigate an unsteady incompressible laminar micropolar flow in a semi-infinite porous pipe with large injection or suction through a deforming pipe wall. Using suitable similarity transformations, the governing partial differential are transformed into a coupled nonlinear singular boundary value problem. For large injection, the asymptotic solutions are constructed using the Lighthill method, which eliminates singularity of solution in the high order derivative. For large suction, a series expansion matching method is used. Analytical solutions are validated against the numerical solutions obtained by Bvp4c.

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44

Ma, Ruyun, Zhongzi Zhao, and Dongliang Yan. "Connected components of positive solutions of biharmonic equations with the clamped plate conditions in two dimensions." Electronic Journal of Differential Equations, Special Issue 01 (November 3, 2021): 239–53. http://dx.doi.org/10.58997/ejde.sp.01.m1.

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This article concerns the clamped plate equation $$\displaylines{ \Delta^2 u=\lambda a(x)f(u), \quad \text{in } \Omega,\cr u=\frac {\partial u}{\partial \nu}= 0 \quad \text{on } \partial \Omega, }$$ where \(\Omega\) is a bounded domain in \(\mathbb{R}^2\) of class \(C^{4, \alpha}\), \(a\in C(\bar \Omega, (0, \infty))\), \(f: [0, \infty)\to [0,\infty)\) is a locally H\"older continuous function with exponent \(\alpha\), and \(\lambda\) is a positive parameter. We show the existence of S-shaped connected component of positive solutions under suitable conditions on the nonlinearity. Our approach is based on bifurcation techniques. For more information see https://ejde.math.txstate.edu/special/m1/c5/abstr.html

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45

Kamont, Zdzisław, and Adam Nadolski. "Functional Differential Inequalities with Unbounded Delay." gmj 12, no. 2 (June 2005): 237–54. http://dx.doi.org/10.1515/gmj.2005.237.

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Abstract We prove that a function of several variables satisfying a functional differential inequality with unbounded delay can be estimated by a solution of a suitable initial problem for an ordinary functional differential equation. As a consequence of the comparison theorem we obtain a Perron-type uniqueness result and a result on continuous dependence of solutions on given functions for partial functional differential equations with unbounded delay. We consider classical solutions on the Haar pyramid.

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STEIN, O., and M. WINKLER. "Amorphous molecular beam epitaxy: global solutions and absorbing sets." European Journal of Applied Mathematics 16, no. 6 (October 21, 2005): 767–98. http://dx.doi.org/10.1017/s0956792505006315.

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The parabolic equation \[u_t + u_{xxxx} + u_{xx} = - (|u_x|^\alpha)_{xx}, \qquad \alpha>1\], is studied under the boundary conditions $u_x|_{\partial\Omega}=u_{xxx}|_{\partial\Omega}=0$ in a bounded real interval $\Omega$. Solutions from two different regularity classes are considered: It is shown that unique mild solutions exist locally in time for any $\alpha>1$ and initial data $u_0 \in W^{1,q}(\Omega)$ ($q>\alpha$), and that they are global if $\alpha \le \frac{5}{3}$. Furthermore, from a semidiscrete approximation scheme global weak solutions are constructed for $\alpha < \frac{10}{3}$, and for suitable transforms of such solutions the existence of a bounded absorbing set in $L^1(\Omega)$ is proved for $\alpha \in [2,\frac{10}{3})$. The article closes with some numerical examples which do not only document the roughening and coarsening phenomena expected for thin film growth, but also illustrate our results about absorbing sets.

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Kocak, Huseyin. "Kink and anti-kink wave solutions for the generalized KdV equation with Fisher-type nonlinearity." An International Journal of Optimization and Control: Theories & Applications (IJOCTA) 11, no. 2 (April 2, 2021): 123–27. http://dx.doi.org/10.11121/ijocta.01.2021.00973.

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This paper proposes a new dispersion-convection-reaction model, which is called the gKdV-Fisher equation, to obtain the travelling wave solutions by using the Riccati equation method. The proposed equation is a third-order dispersive partial differential equation combining the purely nonlinear convective term with the purely nonlinear reactive term. The obtained global and blow-up solutions, which might be used in the further numerical and analytical analyses of such models, are illustrated with suitable parameters.

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Alesemi, Meshari, Jameelah S. Al Shahrani, Naveed Iqbal, Rasool Shah, and Kamsing Nonlaopon. "Analysis and Numerical Simulation of System of Fractional Partial Differential Equations with Non-Singular Kernel Operators." Symmetry 15, no. 1 (January 13, 2023): 233. http://dx.doi.org/10.3390/sym15010233.

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The exact solution to fractional-order partial differential equations is usually quite difficult to achieve. Semi-analytical or numerical methods are thought to be suitable options for dealing with such complex problems. To elaborate on this concept, we used the decomposition method along with natural transformation to discover the solution to a system of fractional-order partial differential equations. Using certain examples, the efficacy of the proposed technique is demonstrated. The exact and approximate solutions were shown to be in close contact in the graphical representation of the obtained results. We also examine whether the proposed method can achieve a quick convergence with a minimal number of calculations. The present approaches are also used to calculate solutions in various fractional orders. It has been proven that fractional-order solutions converge to integer-order solutions to problems. The current technique can be modified for various fractional-order problems due to its simple and straightforward implementation.

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Zhang, Huan, and Yuhua Long. "Multiple Existence Results of Nontrivial Solutions for a Class of Second-Order Partial Difference Equations." Symmetry 15, no. 1 (December 20, 2022): 6. http://dx.doi.org/10.3390/sym15010006.

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In this paper, we consider the existence and multiplicity of nontrivial solutions for discrete elliptic Dirichlet problems Δ12u(i−1,j)+Δ22u(i,j−1)=−f((i,j),u(i,j)),(i,j)∈Ω,u(i,0)=u(i,T2+1)=0i∈Z(1,T1),u(0,j)=u(T1+1,j)=0j∈Z(1,T2), which have a symmetric structure. When the nonlinearity f(·,u) is resonant at both zero and infinity, we construct a variational functional on a suitable function space and turn the problem of finding nontrivial solutions of discrete elliptic Dirichlet problems to seeking nontrivial critical points of the corresponding functional. We establish a series of results based on the existence of one, two or five nontrivial solutions under reasonable assumptions. Our results depend on the Morse theory and local linking.

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Cibotaru, Dorina, Marie N. Celestin, Michael P. Kane, and Florin M. Musteata. "Comparison of liquid–liquid extraction, microextraction and ultrafiltration for measuring free concentrations of testosterone and phenytoin." Bioanalysis 14, no. 4 (February 2022): 195–204. http://dx.doi.org/10.4155/bio-2021-0249.

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Aim: The purpose of the study was to find methods suitable for measuring the free concentrations of testosterone and phenytoin. Materials & methods: Sample solutions of the compounds in buffer and human albumin were processed using liquid–liquid extraction, microextraction and ultrafiltration and analyzed by LC–MS/MS. Results: Liquid–liquid extraction with dibutyl phthalate provided complete extraction from buffer solutions and partial extraction from albumin samples. Spintip C18 devices provided exhaustive extraction from buffer and albumin samples. Spintip C8 devices offered complete extraction from buffer and approximately 50% recovery from albumin samples. Centrifree ultrafiltration devices showed high recovery of free concentrations from all the samples, while Amicon and Nanosep devices provided partial recovery. Conclusion: Spintip C8 and Centrifree devices proved useful for measuring free concentrations.

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