Finite Difference Method and MATLAB - Assignment Example

Table of Contents 1.1). Finite Difference Method and MATLAB 2 For this we need to calculate the heat exchange area and the log mean temperature difference, which is calculated as using the formula 2 1.2. Finite Element Method and MATLAB 4 9 1.5. Legendre functions 9 References 12 1.1). Finite Difference Method and MATLAB To demonstrate the usage of this finite difference method, diffusion equation is applied. It should be noted that heat diffusion depends on three factors change of time, temperature in both ends and resistance of the wire. Thus the diffusion equation to be considered in this case is Where D =: The overall heat transfer coefficient of the equipment can also be calculated from the equation For this we need to calculate the heat exchange area and the log mean temperature difference, which is calculated as using the formula Where, ΔT is the temperature difference between the two streams at the inlet of the hot stream end and   at the exit end. This can be solved using Matlab with the following condition; assume that the length of diffusing wire is 2m and takes is 12 seconds Let w be length of wire t be time. The following code is used to solve in matlab There are also special cases whereby the finite difference method algorithm is only applicable to modeling if one wants the sampled input to also become periodic. 1.2. Finite Element Method and MATLAB The following equation for steady turbulent flow can be solved using Finite Element Method and MATLAB. The equation is Where g is the gravitational acceleration, H is the flow depth, S0 is the bed slope, f is the Darcy–Weisbach friction factor which can be evaluated by application of the Manning and the Darcy–Weisbach formulae in the form, n is Manning’s roughness coefficient, is the density of the water, Ud is the depth-averaged stream wise velocity, q is the longitudinal unit flow rate(= UH), s is the channel side slope (1 : s, vertical : horizontal) and y is the lateral direction. Then is the dimensionless eddy viscosity coefficient given by is the depth averaged eddy viscosity is the shear stress in the x-direction on the plane perpendicular to the y-direction.  is the linear scale factor to apportion the secondary flow terms based upon the reach Averaged sinuosity,  (defined as the ratio of the channel thalweg length to the valley length), is the secondary flow parameter for a straight compound channel arethe depth-averaged local mean velocities in the x and y directions, Cuv is the meandering coefficient. The model can be solved using the finite element method with the parameters, f, ,  and Cuv determined empirically. Due to the dynamic nature of the finite element method is able to find very many applications. It is common to use the finite element method while river modeling. But it is often very useful when dealing with structures that are repetitive in nature such as crystals. The ability of finite element method to identify periodically repetitive components makes it a favorable method for this purpose. Other situations and applications that utilize the finite element method include sampling, and correlation and convolution theory just to name a few. In order to better comprehend how the finite element method, it works in the computation of Discrete Fourier Transform as shown in the code above('DFT of G (Avg,3)) , it is important to first understand the nature of Discrete Fourier Transform and how it correlates to the finite element method. The finite element method and the Discrete Fourier Transform differ in the way that each represents data. This means that the finite element method deals with complex numbers while the Discrete Fourier Transform deals with real numbers only. Therefore, finite element method is actually based on the complex version of Discrete Fourier Transform which utilizes complex numbers in performing its computations. However, the finite element method can be used to calculate the real version of Discrete Fourier Transform and this is where finite element method serves its most common function. It is thus important to understand how both the real Discrete Fourier Transform and the complex Discrete Fourier Transform store their data in order to be able to better comprehend how data from real Discrete Fourier Transform can be transferred into and out of complex Discrete Fourier Transform format in the use of finite element method algorithm to calculate the complex Discrete Fourier Transform. 1.3. Integral Transformation Method In this function will be demonstrated using transfer function G(s) = whilst maintaining satisfactory relative stability. Matlab will used as G(s) = = = >> num= [0 0 0 35]; >> den= [1 80 700 0]; >> bode(num,den) System performance specification was based upon closed-loop time domain criteria of rise time and damping ratio. The compensators were designed using analogue frequency response design methods, based upon open-loop frequency response criteria of phase margin (PM) and gain crossover frequency (cg). The design method assumed second order relationships between open-loop frequency and closed-loop time response measures. These assumptions were invested to determine their reasonableness in the system. Having obtained the required compensators in analogue form, a digital equivalent was evaluated. The performance of the continuous-time and discrete-time controllers was compared G(s) = : 1 20 ; 10 2K : 20-2K : 2K a = =50-K b = =10K a=10 K=40 20-2K = =50-K 20-50 = -K+2K K= -30 Combining (1), and (3) =⇒ -30 < K < 40. The use of the Integral Transformation Method is what has enabled many electronic techniques to be possible and practical today. This is because, unlike the previous approaches, the F Integral Transformation Method only utilizes hundreds of times fewer lines of code. For instance, it is common to find Integral Transformation Method utilizing only a few dozen lines of coded as compared to the previous versions which used thousands. 1.4. Bessel functions We begin by Where 1.5. Legendre functions This can demonstrated using Black Scholes Model of option valuing. The model uses l uses theoretical approach to calculate options’ call price. Legendre functions is formed using five key option determinants which include risk free interest rates, stock price, and time to expiration, stock price and volatility. The equation for solution a model is formed below C = SN(d1) – Xе-rTN(d2) P = Xе-rTN(-d2) - SN(-d1) Where l C and P is the price of European call and put options and d1 = d2 = = d1- in the equations above S is Current stock price, X is Exercise or strike price of option, r is Annual risk –free interest rate, T is time to expiration in years, σ is Standard deviation or volatility of the relative price change of the underlying stock price and N(x) is The cumulative normal distribution function let take an example where consider a European call option with three months to expiry. The stock price is 69, 25, the strike price is 65, the risk-free interest rate is 5, 75% per year, and the volatility is 15% per year. Thus, S=69, 25, X=65, T=0, 25, r=0, 0575, σ = 0, 3. The first step is to determine d1 and d2 d1 = = 1,073 d2= d1- = 0,998 we have value for the cumulative standard normal distribution N(d) for various values of das N(1,073) ≈ 0,8588, and N(0,998) ≈ 0,8413. Then е-rT = е-0,0575x0, 25 = 0, 9857 the equation C = SN(d1) – Xе-rTN(d2) Call = 69, 25 (0, 8588) – 65 (0, 9857) (0, 8413) = 5,574 This function enables calculation larger amounts of options at fast speeds which is good for investors with large amount of data therefore utilize this model extensively. However, the model’s approach of calculating options at its expiration makes it weaker to Binomial model whose calculations are done on several internals. References Gonzalez, RC, Woods, RE and Eddins, SL, 2009, Digital Image Processing Using MatLab’, New York: Prentice Hall. Read More

Other situations and applications that utilize the finite element method include sampling, and correlation and convolution theory just to name a few. In order to better comprehend how the finite element method, it works in the computation of Discrete Fourier Transform as shown in the code above('DFT of G (Avg,3)) , it is important to first understand the nature of Discrete Fourier Transform and how it correlates to the finite element method. The finite element method and the Discrete Fourier Transform differ in the way that each represents data.

This means that the finite element method deals with complex numbers while the Discrete Fourier Transform deals with real numbers only. Therefore, finite element method is actually based on the complex version of Discrete Fourier Transform which utilizes complex numbers in performing its computations. However, the finite element method can be used to calculate the real version of Discrete Fourier Transform and this is where finite element method serves its most common function. It is thus important to understand how both the real Discrete Fourier Transform and the complex Discrete Fourier Transform store their data in order to be able to better comprehend how data from real Discrete Fourier Transform can be transferred into and out of complex Discrete Fourier Transform format in the use of finite element method algorithm to calculate the complex Discrete Fourier Transform. 1.3.

Integral Transformation Method In this function will be demonstrated using transfer function G(s) = whilst maintaining satisfactory relative stability. Matlab will used as G(s) = = = >> num= [0 0 0 35]; >> den= [1 80 700 0]; >> bode(num,den) System performance specification was based upon closed-loop time domain criteria of rise time and damping ratio. The compensators were designed using analogue frequency response design methods, based upon open-loop frequency response criteria of phase margin (PM) and gain crossover frequency (cg).

The design method assumed second order relationships between open-loop frequency and closed-loop time response measures. These assumptions were invested to determine their reasonableness in the system. Having obtained the required compensators in analogue form, a digital equivalent was evaluated. The performance of the continuous-time and discrete-time controllers was compared G(s) = : 1 20 ; 10 2K : 20-2K : 2K a = =50-K b = =10K a=10 K=40 20-2K = =50-K 20-50 = -K+2K K= -30 Combining (1), and (3) =⇒ -30 < K < 40.

The use of the Integral Transformation Method is what has enabled many electronic techniques to be possible and practical today. This is because, unlike the previous approaches, the F Integral Transformation Method only utilizes hundreds of times fewer lines of code. For instance, it is common to find Integral Transformation Method utilizing only a few dozen lines of coded as compared to the previous versions which used thousands. 1.4. Bessel functions We begin by Where 1.5. Legendre functions This can demonstrated using Black Scholes Model of option valuing.

The model uses l uses theoretical approach to calculate options’ call price. Legendre functions is formed using five key option determinants which include risk free interest rates, stock price, and time to expiration, stock price and volatility. The equation for solution a model is formed below C = SN(d1) – Xе-rTN(d2) P = Xе-rTN(-d2) - SN(-d1) Where l C and P is the price of European call and put options and d1 = d2 = = d1- in the equations above S is Current stock price, X is Exercise or strike price of option, r is Annual risk –free interest rate, T is time to expiration in years, σ is Standard deviation or volatility of the relative price change of the underlying stock price and N(x) is The cumulative normal distribution function let take an example where consider a European call option with three months to expiry.

The stock price is 69, 25, the strike price is 65, the risk-free interest rate is 5, 75% per year, and the volatility is 15% per year.

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