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Journal articles on the topic 'Radial problem'

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Author: Grafiati

Published: 14 March 2023

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1

Herrero, M. A., and J. J. L. Velázquez. "Radial solutions of a semilinear elliptic problem." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 118, no. 3-4 (1991): 305–26. http://dx.doi.org/10.1017/s0308210500029115.

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SynopsisWe analyse the set of nonnegative, global, and radial solutions (radial solutions, for short) of the equationwhere 0 < p < 1, and is a radial and almost everywhere nonnegative function. We show that radial solutions of (E) exist if f(r) = o(r2p/1−1−p) or if f(r) ≈ cr2p/1−p as r → ∞, whereWhen f(r) = c*r2p/1−p + h(r) with h(r) = o(r2p/1−p) as r → ∞, radial solutions continue to exist if h(r) is sufficiently small at infinity. Existence, however, breaks down if h(r) > 0,Whenever they exist, radial solutions are characterised in terms of their asymptotic behaviour as r → ∞.

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2

BRIHAYE, Y., and PIOTR KOSINSKI. "QUASI-EXACTLY SOLVABLE RADIAL DIRAC EQUATIONS." Modern Physics Letters A 13, no. 18 (June 14, 1998): 1445–52. http://dx.doi.org/10.1142/s0217732398001522.

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In the background of a central Coulomb potential, the Schrödinger and Dirac equations lead to exactly solvable spectral problems. When the Schrödinger–Coulomb equation is supplemented by a Harmonic potential, the corresponding spectral problem still possesses a finite number of algebraic solutions: it is quasi-exactly solvable. In this letter we analyze the spectral problem corresponding to the Dirac–Coulomb problem supplemented by a linear radial potential and we show that it also leads to quasi-exactly solvable equations.

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3

Khamis, H. J. "Buffon's Needle Problem on Radial Lines." Mathematics Magazine 64, no. 1 (February 1, 1991): 56. http://dx.doi.org/10.2307/2690457.

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4

Khamis, H. J. "Buffon's Needle Problem on Radial Lines." Mathematics Magazine 64, no. 1 (February 1991): 56–58. http://dx.doi.org/10.1080/0025570x.1991.11977575.

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5

Pavlova, O. S., and A. R. Frenkin. "Radial Schrödinger equation: The spectral problem." Theoretical and Mathematical Physics 125, no. 2 (November 2000): 1506–15. http://dx.doi.org/10.1007/bf02551010.

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6

Giuffrè, Sofia, and Attilio Marcianò. "On the Existence of Radial Solutions to a Nonconstant Gradient-Constrained Problem." Symmetry 14, no. 7 (July 11, 2022): 1423. http://dx.doi.org/10.3390/sym14071423.

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In this paper, we study a variational problem with nonconstant gradient constraints. Several aspects related to problems with gradient constraints have been studied in the literature and have seen new developments in recent years. In the case of constant gradient constraint, the problem is the well-known elastic–plastic torsion problem. A relevant issue in this type of problem is the existence of Lagrange multipliers. Here, we consider the equivalent Lagrange multiplier formulation of a nonconstant gradient-constrained problem, and we investigate the class of solutions having a radial symmetry. We rewrite the problem in the radial symmetry case, and we analyse the different situations that may arise. In particular, in the planar case, we derive a condition characterizing the free boundary and obtain the explicit radial solution to the problem and the Lp Lagrange multiplier. Some examples support the results.

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7

Akella, Maruthi R., and Roger A. Broucke. "Anatomy of the Constant Radial Thrust Problem." Journal of Guidance, Control, and Dynamics 25, no. 3 (May 2002): 563–70. http://dx.doi.org/10.2514/2.4917.

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8

Kohonen, M., O. Teerenhovi, T. Terho, J. Laurikka, and M. Tarkka. "Non-harvestable radial artery. A bilateral problem?" Interactive CardioVascular and Thoracic Surgery 7, no. 5 (October 1, 2008): 797–800. http://dx.doi.org/10.1510/icvts.2007.172569.

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9

Nesterov, A. V., V. G. Niz’ev, and A. L. Sokolov. "Transformation problem for radiation with radial polarization." Optics and Spectroscopy 90, no. 6 (June 2001): 923–27. http://dx.doi.org/10.1134/1.1380793.

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10

Gerry, Christopher C., and J. Kiefer. "Radial coherent states for the Coulomb problem." Physical Review A 37, no. 3 (February 1, 1988): 665–71. http://dx.doi.org/10.1103/physreva.37.665.

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11

Rouaki, Mohamed. "Nodal radial solutions for a superlinear problem." Nonlinear Analysis: Real World Applications 8, no. 2 (April 2007): 563–71. http://dx.doi.org/10.1016/j.nonrwa.2005.12.010.

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12

Fonda, Alessandro, and Anna Chiara Gallo. "Radial periodic perturbations of the Kepler problem." Celestial Mechanics and Dynamical Astronomy 129, no. 3 (May 25, 2017): 257–68. http://dx.doi.org/10.1007/s10569-017-9769-5.

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13

Guo, Kai, Hu Ye, Honglin Chen, and Xin Gao. "A New Method for Absolute Pose Estimation with Unknown Focal Length and Radial Distortion." Sensors 22, no. 5 (February 25, 2022): 1841. http://dx.doi.org/10.3390/s22051841.

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Estimating the absolute pose of a camera is one of the key steps for computer vision. In some cases, especially when using a wide-angle or zoom lens, the focal length and radial distortion also need to be considered. Therefore, in this paper, an efficient and robust method for a single solution is proposed to estimate the absolute pose for a camera with unknown focal length and radial distortion, using three 2D–3D point correspondences and known camera position. The problem is decomposed into two sub-problems, which makes the estimation simpler and more efficient. The first sub-problem is to estimate the focal length and radial distortion. An important geometric characteristic of radial distortion, that the orientation of the 2D image point with respect to the center of distortion (i.e., principal point in this paper) under radial distortion is unchanged, is used to solve this sub-problem. The focal length and up to four-order radial distortion can be determined with this geometric characteristic, and it can be applied to multiple distortion models. The values with no radial distortion are used as the initial values, which are close to the global optimal solutions. Then, the sub-problem can be efficiently and accurately solved with the initial values. The second sub-problem is to determine the absolute pose with geometric linear constraints. After estimating the focal length and radial distortion, the undistorted image can be obtained, and then the absolute pose can be efficiently determined from the point correspondences and known camera position using the undistorted image. Experimental results indicate this method’s accuracy and numerical stability for pose estimation with unknown focal length and radial distortion in synthetic data and real images.

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14

Wang, Weidong, Heping Chen, and Yuanyuan Zhang. "Busemann-Petty problem for the i-th radial Blaschke-Minkowski homomorphisms." Filomat 32, no. 19 (2018): 6819–27. http://dx.doi.org/10.2298/fil1819819w.

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Schuster introduced the notion of radial Blaschke-Minkowski homomorphism and considered its Busemann-Petty problem. In this paper, we further study the Busemann-Petty problem for the radial Blaschke-Minkowski homomorphisms and give the affirmative and negative forms of Busemann-Petty problem for the i-th radial Blaschke-Minkowski homomorphisms.

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15

Prajapat, J., and G. Tarantello. "On a class of elliptic problems in R2: symmetry and uniqueness results." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 131, no. 4 (August 2001): 967–85. http://dx.doi.org/10.1017/s0308210500001219.

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In the plane R2, we classify all solutions for an elliptic problem of Liouville type involving a (radial) weight function. As a consequence, we clarify the origin of the non-radially symmetric solutions for the given problem, as established by Chanillo and Kiessling.For a more general class of Liouville-type problems, we show that, rather than radial symmetry, the solutions always inherit the invariance of the problem under inversion with respect to suitable circles. This symmetry result is derived with the help of a 'shrinking-sphere' method.

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16

Shivanian, Elyas. "Pseudospectral Meshless Radial Point Hermit Interpolation Versus Pseudospectral Meshless Radial Point Interpolation." International Journal of Computational Methods 17, no. 07 (May 7, 2019): 1950023. http://dx.doi.org/10.1142/s0219876219500233.

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This paper develops pseudospectral meshless radial point Hermit interpolation (PSMRPHI) and pseudospectral meshless radial point interpolation (PSMRPI) in order to apply to the elliptic partial differential equations (PDEs) held on irregular domains subject to impedance (convective) boundary conditions. Elliptic PDEs in simplest form, i.e., Laplace equation or Poisson equation, play key role in almost all kinds of PDEs. On the other hand, impedance boundary conditions, from their application in electromagnetic problems, or convective boundary conditions, from their application in heat transfer problems, are nearly more complicated forms of the boundary conditions in boundary value problems (BVPs). Based on this problem, we aim also to compare PSMRPHI and PSMRPI which belong to more influence type of meshless methods. PSMRPI method is based on a combination of meshless methods and spectral collocation techniques. The point interpolation method with the help of radial basis functions is used to construct shape functions which act as basis functions in the frame of PSMRPI and PSMRPHI methods. While the latter one has been rarely used in applications, we observe that is more accurate and reliable than PSMRPI method.

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17

Morabito, Filippo. "Radial and non-radial solutions to an elliptic problem on annular domains in Riemannian manifolds with radial symmetry." Journal of Differential Equations 258, no. 5 (March 2015): 1461–93. http://dx.doi.org/10.1016/j.jde.2014.11.004.

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18

Izzo, Dario, and Francesco Biscani. "Explicit Solution to the Constant Radial Acceleration Problem." Journal of Guidance, Control, and Dynamics 38, no. 4 (April 2015): 733–39. http://dx.doi.org/10.2514/1.g000116.

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19

Quarta, Alessandro A., and Giovanni Mengali. "New Look to the Constant Radial Acceleration Problem." Journal of Guidance, Control, and Dynamics 35, no. 3 (May 2012): 919–29. http://dx.doi.org/10.2514/1.54837.

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20

Bernal, Francisco, and Manuel Kindelan. "Radial basis function solution of the Motz problem." Engineering Computations 27, no. 5 (July 20, 2010): 606–20. http://dx.doi.org/10.1108/02644401011050903.

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21

Iaia, Joseph A. "Radial solutions to a p-laplacian dirichlet problem." Applicable Analysis 58, no. 3-4 (November 1995): 335–50. http://dx.doi.org/10.1080/00036819508840381.

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22

Ma, Limin, and Zongmin Wu. "Radial basis functions method for parabolic inverse problem." International Journal of Computer Mathematics 88, no. 2 (November 28, 2010): 384–95. http://dx.doi.org/10.1080/00207160903452236.

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23

Fraile, J. M., and E. Schiavi. "Exact radial solutions for a nonlinear eigenvalue problem." Applied Mathematics Letters 13, no. 5 (July 2000): 67–72. http://dx.doi.org/10.1016/s0893-9659(00)00035-5.

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24

Govorkov, A. B. "The problem of radial excitations of light mesons." Zeitschrift f�r Physik C Particles and Fields 32, no. 3 (September 1986): 405–16. http://dx.doi.org/10.1007/bf01551838.

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25

Enguiça, Ricardo, and Luís Sanchez. "Radial solutions for a nonlocal boundary value problem." Boundary Value Problems 2006 (2006): 1–18. http://dx.doi.org/10.1155/bvp/2006/32950.

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26

Hwang, J. S., and Peter Lappan. "On a problem of Berman concerning radial limits." Proceedings of the American Mathematical Society 95, no. 1 (January 1, 1985): 155. http://dx.doi.org/10.1090/s0002-9939-1985-0796466-x.

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27

Lahiri, A., P. Kumar Roy, and B. Bagchi. "Supersymmetry in atomic physics and the radial problem." Journal of Physics A: Mathematical and General 20, no. 12 (August 21, 1987): 3825–32. http://dx.doi.org/10.1088/0305-4470/20/12/030.

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28

Sartori, G., and G. Valente. "The radial problem in gauge field theory models." Annals of Physics 319, no. 2 (October 2005): 286–325. http://dx.doi.org/10.1016/j.aop.2005.04.016.

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29

Jacobsen, Jon, and Klaus Schmitt. "The Liouville–Bratu–Gelfand Problem for Radial Operators." Journal of Differential Equations 184, no. 1 (September 2002): 283–98. http://dx.doi.org/10.1006/jdeq.2001.4151.

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30

Greco, Antonio. "Radial symmetry and uniqueness for an overdetermined problem." Mathematical Methods in the Applied Sciences 24, no. 2 (2001): 103–15. http://dx.doi.org/10.1002/1099-1476(20010125)24:2<103::aid-mma200>3.0.co;2-f.

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31

Chuathong, Nissaya. "A Modified RBF Collocation Method for Solving the Convection-Diffusion Problems." Abstract and Applied Analysis 2023 (January 20, 2023): 1–10. http://dx.doi.org/10.1155/2023/8727963.

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The main purposes of this study are to propose the modified radial basis function (RBF) collocation method using a hybrid radial basis function to solve the convection-diffusion problems numerically and to choose the optimal shape parameter of radial basis functions. The modified numerical scheme is tested on a benchmark problem with varying shape parameters. The root mean square error and maximum error are used to validate the accuracy and efficiency of the method. The proposed method can be a good alternative to the radial basis function collocation method to improve accuracy and results.

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32

Hsu, Ming-Hung. "Dynamic Analysis of Wind Turbine Blades Using Radial Basis Functions." Advances in Acoustics and Vibration 2011 (August 17, 2011): 1–11. http://dx.doi.org/10.1155/2011/973591.

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Wind turbine blades play important roles in wind energy generation. The dynamic problems associated with wind turbine blades are formulated using radial basis functions. The radial basis function procedure is used to transform partial differential equations, which represent the dynamic behavior of wind turbine blades, into a discrete eigenvalue problem. Numerical results demonstrate that rotational speed significantly impacts the first frequency of a wind turbine blade. Moreover, the pitch angle does not markedly affect wind turbine blade frequencies. This work examines the radial basis functions for dynamic problems of wind turbine blade.

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33

GODIN, PAUL. "THE LIFESPAN OF SOLUTIONS OF EXTERIOR RADIAL QUASILINEAR CAUCHY–NEUMANN PROBLEMS." Journal of Hyperbolic Differential Equations 05, no. 03 (September 2008): 519–46. http://dx.doi.org/10.1142/s0219891608001581.

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We consider smooth solutions of radial exterior Cauchy–Neumann problems with small initial data for radial quasilinear wave equations in three space dimensions, when the size of the initial data tends to 0. We obtain rather precise information on the lifespan, analogous with well known Cauchy problem results of Hörmander and John.

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34

Andreev, Vladimir I., and Anatoliy S. Avershyev. "Two-Dimensional Problem of Moisture Elasticity of Inhomogeneous Spherical Array with Cavity." Applied Mechanics and Materials 580-583 (July 2014): 812–15. http://dx.doi.org/10.4028/www.scientific.net/amm.580-583.812.

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The numerical-analytical solution of the axisymmetric problem of moisture-elasticity for hollow spherical soil massive is provided. We consider the steady-state water transfer. Inhomogeneity of the array due to the dependence of the deformation modulus on the soil moisture. Stress state of the array caused by forced deformations and uneven pressure rebuff of the soil. The method for solving non-axisymmetric problems with radial inhomogeneity in polar coordinates is described. Diagrams on the stress fields in homogeneous and inhomogeneous formulations of the problem are provided. It describes a method of calculating radial inhomogeneous axisymmetric problem of theory of elasticity for thick-walled spherical shell.

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35

Liu, Gao-Lian. "The Radial Equilibrium Problem of Flow in Wave Machinery." Proceedings of the Institution of Mechanical Engineers, Part A: Journal of Power and Energy 207, no. 1 (February 1993): 23–30. http://dx.doi.org/10.1243/pime_proc_1993_207_004_02.

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The present paper deals with the radial equilibrium problem of gas flow at the inlet and outlet of a wave rotor theoretically, presenting a method of solution. The salient feature of this method is that, in contrast to turbomachinery, the outlet flow parameters are related to those at inlet by the state characteristic (compatibility) equations of unsteady rotor flow. The numerical example has shown that the radial equilibrium effect plays a very important role in the design and performance of wave machinery and hence it is suggested that a complete gas dynamic design procedure of a wave machine should include two parts: (a) solution of the one-dimensional unsteady relative flow in rotor at the mean radius; (b) solution of the radial equilibrium problem of gas flow at the rotor inlet and outlet.

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36

Zheng, Xin Wei, Zhi Yu Wang, Zi Gang Chen, Yu Sun, Bing Han, and Xue You Wen. "Feasibility for the Application of the Temperature Difference Computing Method Based on the Simplified Solution of the Radial Temperature for a Boiler Steam Drum Wall to a Supercharged Boiler." Advanced Materials Research 383-390 (November 2011): 6383–90. http://dx.doi.org/10.4028/www.scientific.net/amr.383-390.6383.

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The radial temperature difference calculation of a steam drum wall is an important step in the process of calculating the low-cycle fatigue lifetime of a steam drum. So the German Standard TRD301 recommended a temperature difference computing method based on the simplified solution for the radial temperature of a steam drum wall. In order to validate the feasibility for the application of this computing method to solving the same problem of a supercharged boiler, on the basis of the reasonable simplifications to the unstable heat conduction problem across a steam drum wall, its heat-conduction equations and definite conditions were determined, then the analytical solution for the radial temperature of a steam drum wall, together with its approximate and simplified solutions, is derived. The running example of one super-charged boiler was calculated and analyzed by use of the radial temperature difference computing methods based on the approximate and simplified solutions respectively. The com-puted results show that the German Standard computing method of TRD301 has great limitations in the same problem calculation of a supercharged boiler, and it can’t apply to the researches on the same problem and its relevant problems of a supercharged boiler as a universal computing method.

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37

Shen, Zhaoqiang, Li Zeng, Changcheng Gong, Yumeng Guo, Yuanwei He, and Zhaojun Yang. "Exterior computed tomography image reconstruction based on anisotropic relative total variation in polar coordinates." Journal of X-Ray Science and Technology 30, no. 2 (March 15, 2022): 343–64. http://dx.doi.org/10.3233/xst-211042.

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In computed tomography (CT) image reconstruction problems, exterior CT is an important application in industrial non-destructive testing (NDT). Different from the limited-angle problem that misses part of the rotation angle, the rotation angle of the exterior problem is complete, but for each rotation angle, the projection data through the central region of the object cannot be collected, so that the exterior CT problem is ill-posed inverse problem. The results of traditional reconstruction methods like filtered back-projection (FBP) and simultaneous algebra reconstruction technique (SART) have artifacts along the radial direction edges for exterior CT reconstruction. In this study, we propose and test an anisotropic relative total variation in polar coordinates (P-ARTV) model for addressing the exterior CT problem. Since relative total variation (RTV) can effectively distinguish edges from noises, and P-ARTV with different weights in radial and tangential directions can effectively enhance radial edges, a two-step iteration algorithm was developed to solve the P-ARTV model in this study. The fidelity term and the regularization term are solved in Cartesian and polar coordinate systems, respectively. Numerical experiments show that our new model yields better performance than the existing state-of-the-art algorithms.

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38

Zhang, Hui, Zixin Liu, and Jun Zhang. "An efficient finite element method based on dimension reduction scheme for a fourth-order Steklov eigenvalue problem." Open Mathematics 20, no. 1 (January 1, 2022): 666–81. http://dx.doi.org/10.1515/math-2022-0032.

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Abstract In this article, an effective finite element method based on dimension reduction scheme is proposed for a fourth-order Steklov eigenvalue problem in a circular domain. By using the Fourier basis function expansion and variable separation technique, the original problem is transformed into a series of radial one-dimensional eigenvalue problems with boundary eigenvalue. Then we introduce essential polar conditions and establish the discrete variational form for each radial one-dimensional eigenvalue problem. Based on the minimax principle and the approximation property of the interpolation operator, we prove the error estimates of approximation eigenvalues. Finally, some numerical experiments are provided, and the numerical results show the efficiency of the proposed algorithm.

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39

Ramirez, Juan M., and Diana P. Montoya. "A Piecewise Solution to the Reconfiguration Problem by a Minimal Spanning Tree Algorithm." International Journal of Emerging Electric Power Systems 15, no. 5 (October 1, 2014): 419–27. http://dx.doi.org/10.1515/ijeeps-2013-0094.

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Abstract This paper proposes a minimal spanning tree (MST) algorithm to solve the networks’ reconfiguration problem in radial distribution systems (RDS). The paper focuses on power losses’ reduction by selecting the best radial configuration. The reconfiguration problem is a non-differentiable and highly combinatorial optimization problem. The proposed methodology is a deterministic Kruskal’s algorithm based on graph theory, which is appropriate for this application generating only a feasible radial topology. The proposed MST algorithm has been tested on an actual RDS, which has been split into subsystems.

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40

Pérez-Rodríguez, Ricardo. "A Radial Estimation-of-Distribution Algorithm for the Job-Shop Scheduling Problem." International Journal of Applied Metaheuristic Computing 13, no. 1 (January 2022): 1–25. http://dx.doi.org/10.4018/ijamc.292519.

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The job-shop environment has been widely studied under different approaches. It is due to its practical characteristic that makes its research interesting. Therefore, the job-shop scheduling problem continues being attracted to develop new evolutionary algorithms. In this paper, we propose a new estimation of distribution algorithm coupled with a radial probability function. The aforementioned radial function comes from the hydrogen element. This approach is proposed in order to build a competitive evolutionary algorithm for the job-shop scheduling problem. The key point is to exploit the radial probability distribution to construct offspring, and to tackle the inconvenient of the EDAs, i.e., lack of diversity of the solutions and poor ability of exploitation. Various instances and numerical experiments are presented to illustrate, and to validate this novel research. The results, obtained from this research, permits to conclude that using radial probability distributions is an emerging field to develop new and efficient EDAs.

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41

Protektor, Denys, and Iryna Hariachevska. "SOFTWARE FOR SIMULATION OF NON-STATIONARY HEAT TRANSFER IN ANISOTROPIC SOLIDS." Grail of Science, no. 12-13 (May 28, 2022): 356–59. http://dx.doi.org/10.36074/grail-of-science.29.04.2022.059.

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The software «AnisotropicHeatTransfer3D» for the numerical solution of three-dimensional non-stationary heat conduction problems in anisotropic solids of complex domain by meshless method [1] was developed. The numerical solution of the non-stationary anisotropic heat conduction problem in the software «AnisotropicHeatTransfer3D» is based on a combination of the dual reciprocity method [2] with anisotropic radial basis functions [3] and the method of fundamental solutions [4]. The dual reciprocity method with anisotropic radial basis functions is used to obtain particular solution, and the method of fundamental solutions is used to obtain a homogeneous solution of boundary-value problem.

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42

Ji, X., A. Q. Li, and S. J. Zhou. "The Strain Gradient Elasticity Theory in Orthogonal Curvilinear Coordinates and its Applications." Journal of Mechanics 34, no. 3 (December 13, 2016): 311–23. http://dx.doi.org/10.1017/jmech.2016.122.

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AbstractThe strain gradient elasticity theory including only three independent material length scale parameters has been proposed by Zhou et al. to explain the size effect phenomena in micro scales. In this paper, the general formulations of strain gradient elasticity theory in orthogonal curvilinear coordinates are derived, and then are specified for the cylindrical and spherical coordinates for the convenience of applications in cases where orthogonal curvilinear coordinates are suitable. Two basic problems, one is the twist of a cylindrical bar and the other is the radial deformation of a solid sphere, are analyzed under the cylindrical and spherical coordinates, respectively. The results reveal that only the material length scale parameter l2 enters the torsion problem, while completely disappears in the problem of radial deformation of a sphere. The size effect of radial deformation of a solid sphere is controlled by the material length scale parameters l1 and l2. In addition, for the incompressible solid sphere especially, only the material length scale parameter l1 enters this radial deformation problem by neglecting the strain gradient terms associated with hydrostatic strains. Predictably, the present paper offers an alternative avenue for measuring the three independent material length scale parameters from bar twisting and sphere expansion tests.

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43

Liu, Weifang, Sanfei Zhao, Cong Zhang, and Huipo Qiao. "Solving acoustic scattering problem by the meshless method based on radial basis function." MATEC Web of Conferences 288 (2019): 01008. http://dx.doi.org/10.1051/matecconf/201928801008.

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In this paper, radial point collocation method (RPCM) is introduced to solve the acoustic scattering problem. This is a mathematically simple, easy-to-program and truly meshless method, which has been successfully applied to solve the solid mechanics and convection diffusion problems. However, application of this method to investigate acoustic problems, in particle the acoustic scattering problem is relatively new. The main advantage of this method is its mathematically simple, easy to program, and truly meshless. A Hermite-type interpolation method is employed to improve the solution accuracy while the Neumann boundary conditions exist. In addition, acoustic scattering problem is a typical unbounded domain problem, in order to solve it with RPCM, the domain is truncated to a finite region and an artificial boundary condition (ABC) is imposed. Finally, numerical example is presented to validate the accuracy and effectiveness of RPCM. In the future, the extension of RPCM to more complex and practical problems, especially the three-dimensional situations need to be investigated in more detail.

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44

Cerda, Patricio, Leonelo Iturriaga, Sebastián Lorca, and Pedro Ubilla. "Positive radial solutions of a nonlinear boundary value problem." Communications on Pure & Applied Analysis 17, no. 5 (2018): 1765–83. http://dx.doi.org/10.3934/cpaa.2018084.

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45

Ralph, Saxton, and Dongming Wei. "Radial solutions to a nonlinear p-harmonic dirichlet problem." Applicable Analysis 51, no. 1-4 (December 1993): 59–80. http://dx.doi.org/10.1080/00036819308840204.

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46

Costa, D. G., and D. G. de Figueiredo. "Radial solutions for a Dirichlet problem in a ball." Journal of Differential Equations 60, no. 1 (October 1985): 80–89. http://dx.doi.org/10.1016/0022-0396(85)90121-4.

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47

Figalli, A., and F. Maggi. "On the isoperimetric problem for radial log-convex densities." Calculus of Variations and Partial Differential Equations 48, no. 3-4 (October 3, 2012): 447–89. http://dx.doi.org/10.1007/s00526-012-0557-5.

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48

Królikowski, W. "Useful radial equations for the dirac two-body problem." Il Nuovo Cimento A 105, no. 5 (May 1992): 655–62. http://dx.doi.org/10.1007/bf02730771.

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Bonheure, Denis, Massimo Grossi, Benedetta Noris, and Susanna Terracini. "Multi-layer radial solutions for a supercritical Neumann problem." Journal of Differential Equations 261, no. 1 (July 2016): 455–504. http://dx.doi.org/10.1016/j.jde.2016.03.016.

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Mohamed El-Saeed, Menna Allah El-sayed, Amal Farouk Abdel-Gwaad, and Mohamed AbdEl-fattah Farahat. "Solving the capacitor placement problem in radial distribution networks." Results in Engineering 17 (March 2023): 100870. http://dx.doi.org/10.1016/j.rineng.2022.100870.

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