• Title, Summary, Keyword: multiple solutions

### A study on multiple load flow solution (다근 조류 계산의 해법에 대하여)

• You, Seok-Koo;Kim, Chan-Yang
• Proceedings of the KIEE Conference
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• pp.27-30
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• 1991
• Recently, the phenomena of voltage instability have become major concern in power system, these phenomena are closely related to what are called multiple load flow solutions and calculation methods on these solutions have developed. The multiple solutions concerned with phenomena are pair solutions in heavy loaded contion. But conventional methods require much run time to find this pair solutions and unusual generation patten in the condition of multiple solutions. In this paper, by using concept of existence condition of multiple solutions, calculation method on the multiple load flow solutions is presented which find the pair solutions in light loaded condintion.

### N-SOLITON SOLUTIONS FOR THE SINE-GORDON EQUATION OF DIFFERENT DIMENSIONS

• Wazwaz, Abdul-Majid
• Journal of applied mathematics & informatics
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• v.30 no.5_6
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• pp.925-934
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• 2012
• In this work the sine-Gordon equation will be examined for multiple soliton solutions. The higher dimensional sine-Gordon equations will be studied for multiple soliton solutions as well. The simplified form of the Hirota's method will be employed to conduct this analytic study.

### A Study on the Qualitative Differences Analysis between Multiple Solutions in Terms of Mathematical Creativity (수학적 창의성 관점에서 다중해법 간의 질적 차이 분석)

• Baek, Dong-Hyeon;Lee, Kyeong-Hwa
• School Mathematics
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• v.19 no.3
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• pp.481-494
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• 2017
• Tasks of multiple solutions have been said to be suitable for the cultivation of mathematical creativity. However, studies on the fact that multiple solutions presented by students are useful or meaningful, and students' thoughts while finding multiple solutions are very short. In this study, we set goals to confirm the qualitative differences among the multiple solutions presented by the students and, if present, from the viewpoint of mathematical creativity. For this reason, after presenting the set of tasks of the two versions to eight mathematically gifted students of the second-grade middle school, we analyzed qualitative differences that appeared among the solutions. In the study, there was a difference among the solution presented first and the solutions presented later, and qualitatively substantial differences in terms of flexibility and creativity. In this regard, it was concluded that the need to account for such qualitative differences in designing and applying multiple solutions should be considered.

### EFFECTIVE CALCULATION METHOD ON THE MULTIPLE LOAD FLOW SOLUTIONS. (효율적인 조류다근 계산법)

• Song, K.Y.;Kim, S.Y.;Choi, S.G.
• Proceedings of the KIEE Conference
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• pp.158-161
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• 1990
• Recently, the phenomena of voltage instability have become major concern in power system. These phenomena are closely related to what are called multiple load flow solutions and calculation methods on these solutions have developed. But conventional methods require much run time. In this paper, by using sign of |J| and weighting factor considering system configuration, fast calculation method on the multiple load flow solutions is presented.

### MULTIPLE PERIODIC SOLUTIONS OF SECOND-ORDER ORDINARY DIFFERENTIAL EQUATIONS ACROSS RESONANCE

• Cai, Hua;Chang, Xiaojun;Zhao, Xin
• Bulletin of the Korean Mathematical Society
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• v.51 no.5
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• pp.1433-1451
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• 2014
• In this paper we study the existence of multiple periodic solutions of second-order ordinary differential equations. New results of multiplicity of periodic solutions are obtained when the nonlinearity may cross multiple consecutive eigenvalues. The arguments are proceeded by a combination of variational and degree theoretic methods.

### A Study on Finding of Simplified Multiple Load Flow Solutions and Evaluating of Voltage Stability (간략조류계산법과 전압안정도 평가예 관한 연구)

• Song, Kil-Young;Kim, Sae-Young
• Proceedings of the KIEE Conference
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• pp.556-558
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• 1995
• This paper presents a new simplified method for finding the multiple load flow solutions and through their solutions the voltage stability can be evaluated. Line flow($P_{ij}$, $Q_{ij}$) may be formulated with the second-order equations for $V_{i}^{2}$ in polar coordinates or two circle equations for $e_{i}$ and $f_{i}$ in rectangular coordinates. Based on this feature, multiple load flow solutions are calculated with simple works, results of multiple load flow solutions are used for sensitivity analysis of voltage stability. Also, in the case that reactive power sources is considered, method of evaluating the voltage stability is introduced. The proposed method was validated to 2-bus and IEEE 6-bus system.

### MULTIPLE SOLUTIONS IN NATURAL CONVECTION BETWEEN TWO HORIZONTAL PLATES WITH SMALL MAGNITUDE NON-UNIFORM TEMPERATURE IN THE UPPER PLATE (위 평판이 작은 불균일 온도를 갖는 두 수평 평판 사이의 자연 대류에서의 다중해)

• Yoo, Joo-Sik
• Journal of computational fluids engineering
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• v.21 no.3
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• pp.64-70
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• 2016
• Multiple solutions in natural convection of water with Pr=7 between two horizontal plates with small magnitude non-uniform temperature distribution in the upper plate is numerically investigated. The dimensionless temperature of upper plate is ${\theta}={\epsilon}sinkx$. Two upright cells are formed over one wave length in the conduction-dominated regime of small Rayleigh number. However, multicellular convection occurs above a critical Rayleigh number for small wave number. When k = 1.5, dual solutions are found and a transition of $6{\rightarrow}4$ eddy flow occurs with decrease of Rayleigh number. When k = 0.75, two, three, four and five multiple solutions are observed. Transitions of $14{\rightarrow}12$, $12{\rightarrow}10$, $10{\rightarrow}8$ and $6{\rightarrow}8$ eddy flow occur with decrease of Rayleigh number.

### EXISTENCE OF MULTIPLE PERIODIC SOLUTIONS FOR SEMILINEAR PARABOLIC EQUATIONS WITH SUBLINEAR GROWTH NONLINEARITIES

• Kim, Wan-Se
• Journal of the Korean Mathematical Society
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• v.46 no.4
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• pp.691-699
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• 2009
• In this paper, we establish a multiple existence result of T-periodic solutions for the semilinear parabolic boundary value problem with sublinear growth nonlinearities. We adapt sub-supersolution scheme and topological argument based on variational structure of functionals.

### MULTIPLE SOLUTIONS FOR A p-LAPLACIAN SYSTEM WITH NONLINEAR BOUNDARY CONDITIONS

• Zhou, Jun;Kim, Chan-Gyun
• Bulletin of the Korean Mathematical Society
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• v.51 no.1
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• pp.99-113
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• 2014
• A nonlinear elliptic problem involving p-Laplacian and nonlinear boundary condition is considered in this paper. By using the method of Nehari manifold, it is proved that the system possesses two nontrivial nonnegative solutions if the parameter is small enough.

### MULTIPLE EXISTENCE OF POSITIVE GLOBAL SOLUTIONS FOR PARAMETERIZED NONHOMOGENEOUS ELLIPTIC EQUATIONS INVOLVING CRITICAL EXPONENTS

• Kim, Wan Se
• East Asian mathematical journal
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• v.30 no.3
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• pp.335-353
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• 2014
• We establish multiple extence of positive solutions for parameterized nonhomogeneous elliptic equations involving critical Sobolev exponent. The approach to the problem is variational method.