Geometrical solutions of some quadratic equations with non-real roots

This note gives geometrical/graphical methods of finding solutions of the quadratic equation ax ² + bx + c = 0, a p 0, with non-real roots. Three different cases which give rise to non-real roots of the quadratic equation have been discussed. In case I a geometrical construction and its proof for finding the solutions of the quadratic equation ax ² + bx + c = 0, a p 0, when a,b,c ] R, the set of real numbers, are presented. Case II deals with the geometrical solutions of the quadratic equation ax ² + bx + c = 0, a p 0, when b ] R, the set of real numbers; and a,c ] C, the set of complex numbers. Finally, the solutions of the quadratic equation ax ² + bx + c = 0, a p 0, when a,c ] R, the set of real numbers, and b ] C, the set of complex numbers, are presented in case III.     

The solution of cubic equations by iteration

Riassunto L'autore tratta la convergenza del metodo iterativo diLin nel caso di un'equazione cubica. Stabilisce una relazione fra la convergenza e la grandezza delle parti reali delle radici, e dimostra che una radice puo essere sempre trovata in questo modo.     

The solution of some integral equations

Using some elementary generalisations of the Erdélyi-Kober operators of fractional integration the solutions of some general integral equations are obtained in concise forms from which the solutions of several well known examples of the equations are easily found.     

On approximating the roots of some transcendental equations

A study is made of an approximation by means of algebraic numbers of fixed degree of the roots of the equation P(z, a^z)=0, a A, a 0, 1, where P(x, y) is a polynomial with integral rational coefficients.Translated from Matematicheskie Zametki, Vol. 7, No. 2, pp. 203–210, February, 1970.     

具有多参数的奇摄动非线性边值问题的摄动解

讨论含多个参数的高阶非线性方程的摄动解,在适当的条件下,先构造出外部解,再根据不同的边界层,利用伸展变量和幂级数展开式理论,构造问题的形式渐近解,最后利用微分不等式理论证明渐近解的一致有效性和渐近形态,把奇摄动非线性问题中的参数推广到多个参数.     

Relations between roots and coefficients of cubic equations with one root negative the reciprocal of another

Under predetermined conditions on the roots and coefficients, necessary and sufficient conditions relating the coefficients of a given cubic equation x ³?+?ax ²?+?bx?+?c?=?0 can be established so that the roots possess desired properties. In this note, the condition for one root of a cubic equation to be the negative reciprocal of another one is obtained. Given that the coefficients a, b, c of the cubic equation are in arithmetical or geometrical progression, further conditions are deived for one root to be the negative reciprocal of another. These results provide useful means for checking calculated roots of cubic equations and could serve the needs of teachers and students of Mathematical Sciences in tertiary institutions when the solution of cubic equations are first studied.     

Iterative solution of some nonlinear differential equations

We propose an iterative method to solve some non-linear ordinary differential equations. Comparing on the Mathieu, van der Pol and Hill equation of fourth order, we see that this method is much more efficient than the well known methods by Lyapunov or Picard.     

Generalized solution of some parabolic equations with a random drift

Some stochastic partial differential equations arising from a turbulent transport model are studied using Hida's theory of Brownian functionals. For the spatially homogeneous case, the solutions are constructed as a regular or generalized Brownian functional, depending on a small parameter. The regularity property of such solutions is also determined. However, for the spatially nonhomogeneous equations, only generalized solutions in a series form involving iterated singular Wiener integrals are found.This work was supported in part by NSF Grant DMS-87-02236.     

Numerical solution of fractional differential equations using cubic B-spline wavelet collocation method

Physical processes with memory and hereditary properties can be best described by fractional differential equations due to the memory effect of fractional derivatives. For that reason reliable and efficient techniques for the solution of fractional differential equations are needed. Our aim is to generalize the wavelet collocation method to fractional differential equations using cubic B-spline wavelet. Analytical expressions of fractional derivatives in Caputo sense for cubic B-spline functions are presented. The main characteristic of the approach is that it converts such problems into a system of algebraic equations which is suitable for computer programming. It not only simplifies the problem but also speeds up the computation. Numerical results demonstrate the validity and applicability of the method to solve fractional differential equation.     

On the absolute continuity of the roots of some algebraic equations

We prove that for every polynomial of degreem≤3, with coefficients depending smoothly on a real parametert, it is possible to select am-tuple of roots absolutely continuous int.

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Sunto Proviamo che per ogni polinomio di gradom≤3, i cui coefficienti dipendano con sufficiente regolarità da un parametro realet, si possono selezionarem radici (indipendenti) che sono assolutamente continue int.

     

An efficient algorithm for computing the roots of general quadratic,cubic and quartic equations

While the solution to deriving the roots of the general quadratic equation is adequately covered in a typical classroom environment, the same is not true for the general cubic and quartic equations. To the best of our knowledge, we do not see the roots of the general cubic or quartic equation discussed in any typical algebra textbook at the undergraduate level. In this paper, we propose an efficient algorithm in order to calculate the roots of the general quadratic, cubic and quartic equations. Examples are given to demonstrate the usefulness of this proposed algorithm.     

Amplitude equations for SPDEs with cubic nonlinearities

For a quite general class of stochastic partial differential equations with cubic nonlinearities, we derive rigorously amplitude equations describing the essential dynamics using the natural separation of timescales near a change of stability. Typical examples are the Swift–Hohenberg equation, the Ginzburg–Landau (or Allen–Cahn) equation and some model from surface growth. We discuss the impact of degenerate noise on the dominant behaviour, and see that additive noise has the potential to stabilize the dynamics of the dominant modes. Furthermore, we discuss higher order corrections to the amplitude equation.     

Asymptotic solution of some singularly perturbed Fredholm integral equations

A formal approach is described for obtaining the asymptotic solution to a class of singularly perturbed Fredholm integral equations. The approach is illustrated through application to some example problems which arise in heat transfer, diffraction theory, crack mechanics, Markov processes and low order eigenvalue problems.     

On Newton-type methods for multiple roots with cubic convergence

We introduce two families of Newton-type methods for multiple roots with cubic convergence. A further Newton-type method for multiple roots with cubic convergence is presented that is related to quadrature. We also provide numerical tests that show that these new methods are competitive to other known methods for multiple roots.     

Existence of the mild solution for some fractional differential equations with nonlocal conditions

We are concerned in this paper with the existence of mild solutions to the Cauchy Problem for the fractional differential equation with nonlocal conditions: D ^q x(t)=Ax(t)+t ⁿ f(t,x(t),Bx(t)), t∈[0,T], n∈ℤ⁺, x(0)+g(x)=x ₀, where 0<q<1, A is the infinitesimal generator of a C ₀-semigroup of bounded linear operators on a Banach space X.     

Khayyam with Cabri: experiences of pre-service mathematics teachers with Khayyam's solution of cubic equations in dynamic geometry environment

The study reported in this article deals with the observed actionsof Turkish pre-service mathematics teachers in dynamic geometryenvironment (DGE) as they were learning Khayyam's method forsolving cubic equations formed as x³ + ax = b. Having learnedthe method, modelled it in DGE and verified the correctnessof the solution, students generated their own methods for solvingdifferent types of cubic equations such as x³ + ax² = b andx³ + a = bx in the light of Khayyam's method. With the presentedteaching experiment, students realized that Khayyam's mathematicsis different from theirs. We consider that this gave them anopportunity to have an insight about the cultural and socialaspects of mathematics. In addition, the teaching experimentshowed that dynamic geometry software is an excellent tool fordoing mathematics because of their dynamic nature and accurateconstructions. And, it can be easily concluded that the historyof mathematics is useful resource for enriching mathematicslearning environment.     

On the analytical solution of some families of transcendental equations

The problem of finding the exact analytical closed-form solution of some families of transcendental equations is studied, in some detail, by the Special Trans Function Theory (STFT). The mathematical genesis of the analytical closed-form solution is presented, and the structure of the theoretical derivation, proofs and numerical results confirm the validity and base principle of the STFT. Undoubtedly, the proposed analytical approach implies the qualitative improvement of the conventional analytical and numerical methods.