In this chapter, we discuss various methods for solvingthe partial differential equations (PDEs) of ordertwo. Second-order PDEs play a fascinating role inapplied sciences describing a variety of physicalproblems such as transverse vibrations of a stringas well as a membrane, longitudinal vibrations in abar, sound waves, electromagnetic waves, electricsignals in cables, gravitational potential,electrostatic potential, magnetostatic potential,irrotational motion of a fluid, steady currents,steady flow of heat, surface waves on a fluid,conduction of heat in a bar, diffusion in isotropicsubstances, slowing down of neutrons in a matter,transmission line, nuclear reactors, etc.

We shall begin this chapter with two standard forms oflinear PDEs, whose solutions are relatively morenatural. In the next section, first we classify thesemilinear PDEs and then for each class, we reducethe equation to its canonical form in order toobtain the general solution. Thereafter, awell-known general method, namely, Monge’s method,has been discussed for solving quasilinear andnon-linear equations. We also describe the Fouriermethod to solve homogeneous linear PDEs associatedwith boundary conditions (BCs). Finally, we presentthe derivations and solutions of some well-knownfundamental PDEs of mathematical physics.

5.1 Linear PDEs: Standard Forms

As discussed in Chapter 2, the most general linear PDEof order two in two independent variables x and y is of the form

Here, the coefficients R,S, and T donot vanish simultaneously.

In this section, by particularising the coefficientssuitably in Eq. (5.1), we obtain some standardforms, which on integrating reduce easily to thefirst-order PDEs. Then, solving these equations forp or q, we obtain the generalsolution as desired. In the following lines, wediscuss two types of such equations.

Type I (Equations Reducible to First-Order LinearEquations): In this type, we adopt thefollowing forms:

Such forms can be rewritten, respectively, as

These are linear PDEs of order one, in which eitherp or q is the dependent variableand hence can be solved by the method discussed inSection 3.2.

Now, we adopt some examples of these forms.

Example 5.1.Solve xr + p = 9x2y2.

Introduction

Curve fitting refers to the process of generating a curve that fits to a set of data points. For example, consider a series of data points that consists of ‘n’ data points [(x1,y1),(x2,y2 )… (xn,yn)]. The variables yi are related to their respective xi through a function f(xi). In this configuration, the residual (Ri) of the ith data pair is defined as the difference between the theoretical value and the observed value (Eqn. 3.1).

The aim of curve fitting is to minimize Ri, i.e., to minimize the difference between expected values and the observed values. From Eqn. 3.1, it is evident that the value of Ri can be positive for some values of ‘i’ and it can be negative for others. Therefore, the objective of ‘least square’ in curve fitting is to minimize the sum of square of all the residuals.

In the following sections, the least square fitting of different types of data sets will be discussed. The curve fitting for a linear and a non-linear data set is described in Sections 3.2 and 3.3 respectively. This is followed by polynomial fitting in Section 3.4. The next section explains the fitting procedure using built-in Scilab function. Applications of these methods have been discussed in Section 3.6.

Fitting of Linear Data

Consider a data set having ‘n’ data points [(x1,y1),(x2,y2),… (xn,yn)]. The linear relation between xi and yi is given by Eqn. 3.2.

Eqn. 3.2 represents a straight line whose slope is ‘m’ and the intercept on y-axis is ‘c’. The residual (Ri) is given by Eqn. 3.3.

Eqn. 3.4 gives the sum of squares of all the residuals.

The objective of least square fitting method is to find the values of m and c, such that they minimize S. The minimum value of S can be determined by differentiating it w.r.t. the slope and the constant and then equating the differential to zero, i.e.

In Eqn, 3.5,

Simultaneous solution of Eqns. 3.7 and 3.9 will give,

Eqns. 3.7 and 3.9 can also be written in the matrix notation, i.e.

Eqn. 3.12 implies that

In Eqn. 3.14, the first element of the matrix C will give the slope of the best fit curve. The second element of this matrix will give the intercept on the y-axis.

In this chapter, we introduce a number of parametric statistical models that have been used to model network data. The social network literature has named them $p_1$, $p_2$, and $p^{*}$ models, the last of which has also been referred to as an ERGM (exponential random graph model).

Introduction

Plant breeding is an age-old practice of genetic improvement of plants so that they become more useful to humans. It involves combining selected parental plants to obtain the next generation with an improved genetic potential of disease-resistance, stress-tolerance or better yields. The process includes manual crosses or controlled pollination followed by an artificial selection of progeny. Plant breeding, for its enormous benefits to mankind is widely practiced. However, the process is labor intensive and it takes many years for the integration of the required gene and development of a desired progeny. Another major limitation of conventional breeding is that the gene transfer can be achieved only in genetically related species/genera. Even in conspecific plants, the incompatibility of crosses becomes a major barrier. In the last few decades, these limitations of plant breeding have largely been overcome by modern plant genetic engineering/transformation techniques which allow insertion of foreign genes from one organism into another organism. Introduction of foreign genes is relatively less time consuming and does not require recipient and donor organisms to be genetically related to each other. For example, Bt-cotton which was created by genetically altering the cotton genome using genes from the soil bacterium Bacillus thuringiensis (Pursell & Perlak 2004).

The technique of developing transgenic plants has become an integral part of crop improvement programs across the world (Eapen 2011). Although, transgenic approach for crop improvement has several advantages over conventional breeding, it suffers from some drawbacks as well. The transformation techniques used are expensive, genotype-dependent and involve time taking procedures such as in vitro culture and regeneration of explants. Therefore, alternate methods of quick and easy transformation are being developed. One such approach which offers several advantages is transformation of germ cells of plants instead of somatic cells/tissues. Germ cells of plants include sperm cells in the pollen grain (male germ cell) and egg cell in the female gametophyte (female germ cell). The process of introduction of desired genes into germ cells is known as Germline Transformation. This method of transformation has the potential to produce genetically modified plants within less time as it bypasses some tedious steps of in vitro regeneration. The genetically transformed germ cells can also be used directly for fertilization to recover transgenics without tissue culture and the procedure is known as in planta transformation.

This is a common scene on many city streets and may be one that is familiar to you. When you look at the man with the sign, do you feel empathy? Anger? Frustration? Why do you think he is in this situation? What would need to happen for him to be economically secure? What do you feel when you look at the woman? How do you think she came to be financially secure? If you walked past the man in this photo, would you stop to talk with him or give him money if you had any to spare? Would you support paying higher taxes if it assisted low-income people in your community? Would you vote for a politician who supported fining people who ask for money in public places?

Introduction

Zygote is a unique cell from which the life cycle of angiosperms begins. It is a product of fertilization between the sperm and the egg cell. Zygote undergoes organized divisions and cell-specifications to give rise to an embryo, a young sporophyte. The process of development of an embryo from a zygote is known as embryogenesis. A particular pattern, form, and polarity exists during the development of embryo which is the outcome of several cellular, molecular and genetic mechanisms. These programmed changes enable the embryo to form the future sporophyte as a unique entity. The present chapter deals with the embryogenesis in angiosperms, encompassing embryo patterning, genetics and physiology involved. Embryos can also develop from somatic cells under in vitro conditions and resulting embryos are known as somatic embryos and the process as somatic embryogenesis. In this chapter, the term embryogenesis will be used for only zygotic embryogenesis.

Structure of the Embryo

Typical mature embryos in both monocots and dicots are similar in the basic design as they share similar embryogeny, at least up to a particular stage (i.e., octant, as will be discussed in Section 10.4); after which ontogenetic differences appear between them. A typical dicot embryo comprises of an embryonal axis with two cotyledons attached to it laterally. The part of embryonal axis above the level of cotyledons is known as the epicotyl; which terminates into the embryonic shoot (also called plumule). The part of embryonal axis below the level of cotyledons is known as the hypocotyl; which gives rise to an embryonic root (also called radicle) (Fig. 10.1 A). A typical monocot embryo differs from a dicot embryo in having only one cotyledon attached to the embryonal axis (Fig. 10.1 B).

The embryos in cereals like Zea mays, Triticum sp., Oryza sp. need special mention because of additional embryonic organs associated with them namely, scutellum, coleoptile and coleorhiza (Fig. 10.1 C). Scutellum is thought to be a seed leaf or a single massive cotyledon in cereals, which fully covers the embryo. Lateral to the scutellum, a short embryonal axis is attached which is divided into an epicotyl (above the level of scutellum) and a hypocotyl (below the level of scutellum). The epicotyl comprises of several young leaves covered by a sheath called coleoptile.

Programs need to be correct. Programs also need to be fast. In order to write efficient programs, one surely must know how to evaluate efficiency. One might take recourse to our prior understanding of efficiency in the sequential context and compare observed parallel performance to observed sequential performance. Or, we can define parallel efficiency independent of sequential performance. We may yet draw inspiration from the way efficiency is evaluated in a sequential context. Into that scheme, we would need to incorporate the impact of an increasing number of processors deployed to solve the given problem.

Efficiency has two metrics. The first is in an abstract setting, for example, the asymptotic analysis of the underlying algorithm. The second is concrete – how well does the algorithm's implementation behave in practice on the available hardware and on data sizes of interest. Both are important.

There is no substitute for measuring the performance of the real implementation on real data. On the other hand, developing and testing iteratively on large parallel systems is prohibitively expensive. Most development occurs on a small scale: using only a few processors, p, on small input of size n. The extrapolation of these tests to a much larger scale is deceptively hard, and we often must resort to simplified models and analysis tools.

Asymptotic analysis on simple models is sometimes criticized because it oversimplifies several complex dynamics (like cache behavior, out-of-order execution on multiple execution engines, instruction dependencies, etc.) and conceals constant multipliers. Nonetheless, with large input sizes that are common in parallel applications, asymptotic measures do have value. They can be computed somewhat easily, in a standardized setting and without requiring iterations on large supercomputers. And, concealing constants is a choice to some degree. Useful constants can and should be retained. Nonetheless, the abstract part of our analysis will employ the big-O notation to describe the number of steps an algorithm takes. It is a function of the input size n and the number of processors p.

Asymptotic notation or not, the time t(n, p) to solve a problem in parallel is a function of n and p. For this purpose, we will generally count in p the number of sequential processors – they complete their program instructions in sequence.