What is the difference between linear and non linear equations?
A linear equation is always a polynomial of degree 1 (for example x+2y+3=0). In the two dimensional case, they always form lines; in other dimensions, they might also form planes, points, or hyperplanes. Their "shape" is always perfectly straight, with no curves of any kind. This is why we call them linear equations.Every other equation is nonlinear. Higher degree polynomials are nonlinear. Trigonometric functions (like sin or cos) are nonlinear. Square roots are nonlinear. The main exception is if the nonlinear piece can evaluate to a constant--for example, sqrt(4)*x is linear because sqrt(4) is just 2, and 2x is linear.Linear equations have some useful properties, mostly in that they are very easy to manipulate and solve. Although they are quite limited in what they can represent, it is often useful to try and approximate complicated systems using linear equations so that they will be easier to think about and deal with.Nonlinear equations, for the most part, are much harder to solve and manipulate. Sometimes you need them--nature doesn't always work in straight lines, and nor do mathematicians--but generally speaking, you can only solve nonlinear equations if the systems are fairly small and simple. Solving a linear system with a million interacting variables is very doable with a computer, and most nonlinear solvers aren't going to get even close to that.
Is it possible to solve a nonlinear equation system (no approximation) using a direct method (non iterative)?
Is it possible to solve a nonlinear equation system (no approximation) using a direct method (non iterative)?Yes! The simplest example is the famous solution[1] to a quadratic equation:[math]\quad ax^2+bx+c=0\Rightarrow x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}[/math]Footnotes[1] Quadratic formula
What are application of numerical methods in engineering?
There’s so many uses for numerical methods, it is impossible to list them all. But essentially, we can cover first the basic math problems they can be used for, which are often:Computing integrals and derivativesSolving differential equationsBuilding models based on data, be it through interpolation, Least Square, or other methodsRoot finding and numerical optimizationEstimating the solution to a set of linear and nonlinear equationsComputational geometryThere’s other areas I haven’t listed, but that’s some of the common fundamental uses. With respect to real world problems, here are some examples where numerical methods are used:Development and computation of optimal control algorithmsDevelopment of high fidelity simulations to model viscous flow around a race car to see if the wing designs generate sufficient downforceMachine learning algorithms, like estimating optimal weights of parametric models using only subsets of the full dataset (like stochastic gradient descent)Photorealistic rendererDesign optimization based on simulation and multi-objective optimization formulationsGame EnginesFiltering of noisy data based on an approximately expected model of the dynamics (Kalman Filter, Particle Filter, etc.)There are many more uses for numerical methods out there, but this will hopefully show a range of areas to prove its uses are broad.
What level of mathematics is required to solve Navier-Stokes equation?
To solve it mathematically (i.e. analytically) and obtain a general solution would require a level that is so high it is unknown. Of course, there are many simplifications that can be done in order to obtain solutions for simple problems, and even some complex ones.The Navier-Stokes equations are so complex that there is a Millenium Prize related to the nature of the Navier-Stokes equations. This prize is still unclaimed.However, it is fairly common to solve them numerically even for complex problems. For example, the simulation below shows an approximation of one solution of the Navier-Stokes equations for a turbulent flow:LES Analysis of Flow Around a Cylinder
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Can anyone tell me The Role of Linear Algebra in Computer Science?
Linear algebra is the study of Linear equations, matrix, determinants, and vectors spaces. The results of linear algebra have found application in such diverse fields as optics, quantum mechanics, display addressing, electric circuits, cryptography, computer graphics, economics, linear programming, solution of systems of differential equations, etc. The manipulation of matrices and determinants plays a central role in all applications of linear algebra. Linear algebra has a great role in: Mathematics, Computer graphics, Data Compression, Network Flow etc Linear Algebra is one of the most important areas in mathematics, with numerous applications in an extremely wide spectrum of disciplines in Science & Engineering. The language of vectors and matrices is an elegant way to describe (among other things) the way in which an object may be rotated, shifted (translated), or made larger or smaller (scaled). Image (jpg), video (MPG) and compression algorithms make use of Fourier transform a linear transformation. In all cases, the compression makes use of the fact that in Fourier space information can be cut away without disturbing the main information. Computer graphics uses linear algebra like matrix algebra, change of coordinates, geometry and 3-dimensional calculus. Also we can scale an object we can translate an object or we can rotate an object .Via linear matrix we can draw the pixels. The ideas of linear algebra are used throughout computer graphics. In fact, any area that concerns itself with numerical representations of geometry often will collect together numbers such as x,y,z positions into mathematical objects called vectors. Vectors and a related mathematical object called a matrix are used all the time in graphics. Linear programming uses a system of inequalities called constraints to maximize profit functions and minimize cost functions. all such problem occurring in industry are solved by a computer using linear algebra. The importance of linear algebra for applications has raisin in direct proportion to the increase in computing power. With each new generation of hardware and software trigging a demand for even great capabilities. Computer science this intricately linked with linear algebra through the explosive growth of parallel processing and large scale computation.
What are the key takeaways of Navier-Stokes equations in fluid dynamics?
There are broadly 3 equation of motions in fluid dynamics-Reynold’s equation of motion.Navier-Stokes Equation.Euler’s Equation of motion.The Navier-Stokes Equation governs the motion of fluid.Basically this equation is a system of non linear partial differential equations for abstract vector fields of any size.Navier-Stokes Equation finds its extensive application in fluid flow modeling,it predicts the the velocity and pressure in the given geometry.For detailed knowledge i will recommend you this link:What Are the Navier-Stokes Equations?Thank you,Hope this will help you :)