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Second-Order Linear Equations
Published in Steven G. Krantz, Differential Equations, 2022
Variation of parameters is a method for producing a particular solution to a nonhomogeneous equation by exploiting the (usually much simpler to find) solutions to the associated homogeneous equation.
Student perceptions of a guided inquiry approach to a service-taught ordinary differential equations course
Published in International Journal of Mathematical Education in Science and Technology, 2023
Diarmaid Hyland, Paul van Kampen, Brien Nolan
The learning outcomes for the module state that students will encounter some definitions and theorems, several solution techniques, and problems involving ODEs in context during the module. The module could be described as a methods course, however, with the focus very much on the latter two outcomes. Following a brief recap on differential and integral calculus, students encounter the separation of variables and integrating factor methods in the opening weeks of the twelve-week module. They use these techniques to solve first order ODEs in a variety of contexts (for example, Newton’s Law of Cooling), including many initial value problems. Students are then introduced to second order linear constant coefficient and Euler-Cauchy equations (both homogeneous and inhomogeneous). These equations are also presented in context (damped oscillators), where students use the method of undetermined coefficients or variation of parameters.
Optimization of reinforced concrete beams under local mechanical and corrosive damage
Published in Engineering Optimization, 2022
Ashot Tamrazyan, Anatoly V. Alekseytsev
The general diagram of the search is shown in Figure 5. In block 1, information on the finite element model of structures is formed; this comprises loading, kinematic constraints, data about the materials, including their deformation curves, and the multitude of variable parameters assumed for selection in the process of finding a solution. In block 2, on the basis of the adapted genetic optimization algorithm for reinforced concrete frame structures, the variation of parameters for cross-sections, grades of materials and reinforcement is implemented. In addition, the constraints described in Section 2.3 are considered. Moreover, in compliance with the accepted model of corrosive damage, the discrete interval of various time-points is fixed. At each moment of time for the chosen concrete grade, the dimensions of the corrosion spot are measured, and its permanent displacement is chosen along the length of the structure. In addition, corrosive damage can be accounted for jointly with the conditions of progressive fracture avoidance. In this case, reinforcement is designed allowing for the avoidance of brittle failure under the condition , , where are the height of the compressed zone and the boundary height of the compressed zone for ductile failure, respectively; and is the height of the cross-section. When carrying out the calculations in block 3 (see Figure 5), the results for each moment of time remain unchanged. If the constraints for the design moment of time (corrosion time) are satisfied, then the structure's variant is placed in block 4, which is edited under the elitism principle (Tamrazyan and Alekseytsev 2019). If not, that moment of time is fixed at a point when corrosion would not affect the progressive collapse of a structure.
Increasing effects of Coriolis force on the cupric oxide and silver nanoparticles based nanofluid flow when thermal radiation and heat source/sink are significant
Published in Waves in Random and Complex Media, 2022
Thirupathi Thumma, N. Ameer Ahammad, Kharabela Swain, Isaac Lare Animasauan, S. R. Mishra
The governing equation of three-dimensional Maxwell nanofluid flow in Ref. [24], Ref. [25] shows blood carrying gold nanoparticles on a curved surface, Ref. [26] shows the dynamics of a Burgers nanofluid subject to an induced magnetic field, while Ref. [27,28] shows the dynamics of a ternary-hybrid nanofluid, Ref. [29] shows the dynamics of a Maxwell nanofluid, and Ref. [30] shows the motion of ethylene glycol and water on a convectively heated surface, mixed convection hybridized micropolar nanofluid in Ref. [31], bio–convective hybrid nanofluid in Ref. [32], flow of couple stress Casson nanofluid in Ref. [33], and rotating Maxwell nanofluid in Ref. [34] has been solved numerically with the intention to explore the transport phenomenon and advice experts dealing with such fluids in the industry. The technique of Nonlinear Variation of Parameters was used due to its distinct advantage over the method of Undetermined Coefficients and because it has no prior conditions to be satisfied. It is a well-known fact that VPM yields a particular solution, provided that the associated homogeneous equation can be solved. However, Wang and Liu [35] created a formula within the scope of change of parameters for impulsive differential equations. The method of modification of parameters, according to Moore and Ertürk [36], the technique called VPM is a traditional numerical procedures for obtaining the solution of nonlinear BVPs because it is a potentially efficient and accurate alternative to semi-analytical approach. The standard form of nonlinear differential equations is. Where is the non-homogeneous function, is the nonlinear terms, is the higher-order (m) derivative which is linear, and is the remainder terms present in the differential equation. The general iterative solution of VPM is represented as, Where is the initial function defined as However, with the help of initial/boundary conditions, the values of unknown constants say, ’s which are obtained and the coefficients of Wronskian technique i.e. is The stated equations may be expressed as follows using the normal VPM approach outlined above: