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**Numerical Solutions of Ordinary Differential Equations**

4/17/2017 Dr.G.Suresh University

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**Ordinary Differential Equations**

Equations which are composed of an unknown function and its derivatives are called differential equations. Differential equations play a fundamental role in engineering because many physical phenomena are best formulated mathematically in terms of their rate of change. v - dependent variable t - independent variable 4/17/2017 Dr.G.Suresh University

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**Types of Differential Equations**

When a function involves one independent variable, the equation is called an ordinary differential equation (ODE). A partial differential equation (PDE) involves two or more independent variables. 4/17/2017 Dr.G.Suresh University

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**Model Ordinary Differential Equations**

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**Why Numerical solutions of ODE ?**

Many differential equations cannot be solved analytically, in which case we have to satisfy ourselves with an approximation to the solution. 4/17/2017 Dr.G.Suresh University

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Initial value problem Initial value problem associated with first order differential equations is 4/17/2017 Dr.G.Suresh University

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**Methods of solving IVP Taylor series Picard successive approximation**

Euler Modified Euler Runge-Kutta (RK) 4/17/2017 Dr.G.Suresh University

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Taylor Series Method The Taylor series method is a straight forward adaptation of classic calculus to develop the solution as an infinite series. The method is not strictly a numerical method but it is used in conjunction with numerical schemes. 4/17/2017 Dr.G.Suresh University

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Taylor Series Method The Taylor series expansion of y(x) about x = x0 is 4/17/2017 Dr.G.Suresh University

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Problem#1 The differential equation y'(x) = x + y satisfying y(0)=1. Determine the Taylor series solution. Also compare with the exact solution. {Analytical solution: y(x)=2ex - x -1(Cauchy linear differential equation).} 4/17/2017 Dr.G.Suresh University

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**Solution First, we find derivatives of y' = x + y, 4/17/2017**

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**Solution The Taylor series expansion of y(x) about x = x0 is 4/17/2017**

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**Comparison of solutions**

The exact solution is Therefore, Taylor series solution and exact solution are equal. 4/17/2017 Dr.G.Suresh University

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**Picardâ€™s Method The initial value Problem**

is equivalent to the following integral equation 4/17/2017 Dr.G.Suresh University

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**Picardâ€™s Method The Picardâ€™s nth approximate solution is 4/17/2017**

Dr.G.Suresh University

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Problem#1 Using Picardâ€™s process of successive approximation, obtain a solution up to the fifth approximation of the equation such that y = 1 when x = 0. Check your answer by finding the exact solution. Ans. Exact solution y(x) = 2ex â€“ (x+1). 4/17/2017 Dr.G.Suresh University

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Application Problem For the free-falling bungee jumper with linear drag, compute the velocity at 10 seconds of a free-falling parachutist of mass 80kg and drag coefficient 10 kg/s at 10 seconds, with initial velocity 20 m/s, using (i) Taylor series method (ii) Picardâ€™s method. 4/17/2017 Dr.G.Suresh University

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Solution Let v(t) be the velocity and â€˜aâ€™ be the acceleration of the parachutist at any time t. Then the total force acting on the parachutist is equal to the sum of the downward force FD due to gravity and upward force FU due to air resistance. i.e. F = FD + FU FD = mg FU Î± v (linear drag) FU = - Cd v(t). 4/17/2017 Dr.G.Suresh University

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Solution By Newton second law of motion F = ma. ma = mg â€“ Cd v Therefore, Given that m = 80kg, Cd = 10 kg/s, and initial velocity v(0)=20 m/s. Now solve the above IVP using Taylor and Picardâ€™s method. 4/17/2017 Dr.G.Suresh University

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Eulerâ€™s Method 4/17/2017 Dr.G.Suresh University

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**Interpretation of Euler Method**

y2 y1 y0 x x x x 4/17/2017 Dr.G.Suresh University

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**Interpretation of Euler Method**

Slope=f(x0,y0) y1 y1=y0+hf(x0,y0) hf(x0,y0) y0 x x x x h 4/17/2017 Dr.G.Suresh University

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**Interpretation of Euler Method**

y2 y2=y1+hf(x1,y1) Slope=f(x1,y1) hf(x1,y1) Slope=f(x0,y0) y1 y1=y0+hf(x0,y0) hf(x0,y0) y0 x x x x h h 4/17/2017 Dr.G.Suresh University

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**Interpretation of Euler Method**

4/17/2017 Dr.G.Suresh University

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Problem#1 Using Eulerâ€™s method, find an approximate value of y corresponding to x=1, given that yÊ¹ = x + y such that y=1 when x=0. Ans: Taking h = 0.1, then y(0.1) = 1.1, y(0.2) = 1.22, y(0.3) = 1.36 y(0.4) = 1.53, y(0.5) = 1.72, y(0.6) = 1.94 y(0.7) = 2.19, y(0.8) = 2.48, y(0.9) = 2.81 y(1.0) = 3.18. 4/17/2017 Dr.G.Suresh University

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Problem#2 A cup of a coffee originally has temperature of 68oC. The temperature of the ambient is 21oC and the thermal constant is 0.017oC/min. Determine the temperature of the coffee from t = 0 to 10 minutes insteps of 2 minutes. Sol. The differential equation is TÊ¹(t) = -k(T - Ta) = (T - 21) T(0) = 68 4/17/2017 Dr.G.Suresh University

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**Improvements of Eulerâ€™s method**

A fundamental source of error in Eulerâ€™s method is that the derivative at the beginning of the interval is assumed to apply across the entire interval. Two simple modifications are available to circumvent this shortcoming: Heunâ€™s (Modified Euler) Method Midpoint (Improved Polygon) Method 4/17/2017 Dr.G.Suresh University

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**Modified Eulerâ€™s Method**

To improve the estimate of the slope, determine two derivatives for the interval: (1) At the initial point (2) At the end point The two derivatives are then averaged to obtain an improved estimate of the slope for the entire interval. 4/17/2017 Dr.G.Suresh University

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Euler Modified method To improve the estimate of the slope, determine two derivatives for the interval: At the initial point At the end point The two derivatives are then averaged to obtain an improved estimate of the slope for the entire interval.

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**Heunâ€™s method (improved)**

Original Huenâ€™s: Note that the corrector can be iterated to improve the accuracy of yi+1 .

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Problem#1 Using Modified Eulerâ€™s method, find an approximate value of y when x= 0.3, given that yÊ¹ = x + y and y=1 when x=0. Sol. Compare the IVP with the general IVP yÊ¹ = f(x, y), y(x0) = x0, we have f(x, y) = x + y, x0 = 0 and y0 = 1. Taking the step size h = 0.1, then x0 = 0, x1 = 0.1, x2 = 0.2, x3 = 0.3. Now we find the corresponding value of y, using modified Eulerâ€™s method.

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**Solution Step-1. To find y1 = y(x1) = y(0.1)**

Predictor formula (Eulerâ€™ formula) y1(0) = y0 + h f(x0, y0) = f(0, 1) = 1+0.1(0+1) =1.1 Corrector formula y1(k) = y0 + (h/2) [f(x0, y0) + f(x1, y1(k-1))] y1(1) = y0 + (h/2) [f(x0, y0) + f(x1, y1(0))] = 1 + (0.1/2) [f(0,1) + f(0.1, 1.1)] = [ ] =1.11

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**Solution y1(2) = y0 + (h/2) [f(x0, y0) + f(x1, y1(1))]**

= 1 + (0.1/2) [f(0,1) + f(0.1, 1.11)] = [ ] = y1(3) = y0 + (h/2) [f(x0, y0) + f(x1, y1(2))] = 1 + (0.1/2) [f(0,1) + f(0.1, )] = [ ] Therefore, y1 =

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**Solution Step-2. To find y2 = y(x2) = y(0.2)**

Predictor formula (Eulerâ€™ formula) y2(0) = y1 + h f(x1, y1) = f(0.1, ) = ( ) = Corrector formula y2(k) = y1 + (h/2) [f(x1, y1) + f(x2, y2(k-1))] y2(1) = y1 + (h/2) [f(x1, y1) + f(x2, y2(0))] = (0.1/2) [f(0.1,1.1105) + f(0.2, )] = [ ] =

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**Solution y2(2) = y1 + (h/2) [f(x1, y1) + f(x2, y2(1))]**

= (0.1/2) [f(0.1,1.1105) + f(0.2, )] = [ ] = y2(3) = y1 + (h/2) [f(x1, y1) + f(x2, y2(2))] = (0.1/2) [f(0.1,1.1105) + f(0.2, )] = [ ] Therefore, y2 = y(0.2)= Similarly, y3 = y(0.3) =

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Application Problem A storage tank (Fig.) contains a liquid at depth y where y = 0 when the tank is half full. Liquid is withdrawn at a constant flow rate Q=500m3/h to meet demands. The contents are resupplied at a sinusoidal rate 3Q sin2(t). The surface area of the tank is 1200 m2. Determine the depth of the tank from t= 0 to t=6 in steps of 2 hours, using modified Eulerâ€™s method. 4/17/2017 Dr.G.Suresh University

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**Problem#1 Q =500 m3/h 3Q sin2(t) 4/17/2017**

Dr.G.Suresh University

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Solution Let y(t) be the depth of the tank at any time t and A be the surface area of the tank. The depth of the tank is changed when volume of the liquid in the tank changed. Therefore, 4/17/2017 Dr.G.Suresh University

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Solution Since A is constant, it implies that Given that Q=500m3/h, A =1200 m2 and y(0) = 0. Substituting these value, we have Now solving the above initial value problem, using modified Eulerâ€™s method. 4/17/2017 Dr.G.Suresh University

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Runge - Kutta Method 4/17/2017 Dr.G.Suresh University

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**Martin Wilhelm Kutta (1867-1944)**

Carl Runge ( ) Martin Wilhelm Kutta ( ) 4/17/2017 Dr.G.Suresh University

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Introduction Runge-Kutta methods are very popular because of their good efficiency; and are used in most computer programs for differential equations. These methods agree with Taylorâ€™s series solution up to the term in hr, where r differs method to method and is called the order of the method. Eulerâ€™s and modified Eulerâ€™s are called the first and second order Runge-Kutta methods. The fourth order R-K method is most commonly used and is often referred to as R-K method only. 4/17/2017 Dr.G.Suresh University

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Working Rule To find increment â€˜kâ€™ of y corresponding to an increment h of x by R-K method from the initial value problem yÂ´ = f(x, y) with y(x0) = y0 is as follows Calculate k1 = h f(x0, y0) k2 = h f(x0+0.5h, y0+0.5k1) k3 = h f(x0+0.5h, y0+0.5k2) k4 = h f(x0+h, y0+k3) 4/17/2017 Dr.G.Suresh University

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**Working Rule Finally compute k = 1/6 (k1 + 2k2 + 2k3 + k4)**

is the weighted mean of k1, k2, k3 and k4. Therefore the required approximate value is y1 = y(x1) = y(x0 + h) = y0 + k. 4/17/2017 Dr.G.Suresh University

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**Graphical Representation**

k2 k4 k3 k1 xi xi + h/2 xi + h 4/17/2017 Dr.G.Suresh University

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Problem#1 Using Runge-Kutta method of fourth order, solve yÊ¹ = (y2 â€“ x2)/(y2 + x2) with y(0)=1 at x = 0.2, 0.4. Sol : - Compare the given problem with the general first order initial value problem yÊ¹=f(x, y) satisfying y(x0)=y0, we have f(x, y) = (y2 â€“ x2)/(y2 + x2) x0 = 0, y0 =1. Here we have to find value of y at x=0.2, 0.4. 4/17/2017 Dr.G.Suresh University

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Solution Taking step size h = 0.2. Step(1) To find y(0.2) k1 = h f(x0, y0) = 0.2 f(0, 1) = 0.2 (12-02)/(12+02) = 0.2 k2 = h f(x0+0.5h, y0+0.5k1) = 0.2 f(0+0.5(0.2), (0.2)) = 0.2 f(0.1, 1.1) = 4/17/2017 Dr.G.Suresh University

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Solution k3 = h f(x0+0.5h, y0+0.5k2) = 0.2 f(0+0.5(0.2), 1+0.5( )) = 0.2 f(0.1, ) = k4 = h f(x0+h, y0+k3) = 0.2 f(0+0.2, ) = 0.2 f(0.2, ) = 4/17/2017 Dr.G.Suresh University

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Solution k = 1/6 (k1 + 2k2 + 2k3 + k4) = 1/6 (0.2 +2( )+2(0.1967) ) = Hence y1 = y0 + k y(0.2) = = Step(2) To find y(0.4) k1 = h f(x1, y1) = 0.2 f(0.2, ) = 4/17/2017 Dr.G.Suresh University

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solution k2 = h f(x1+0.5h, y1+0.5k1) = 0.2 f( (0.2), (0.1891)) = 0.2 f(0.3, ) = k3 = h f(x1+0.5h, y1+0.5k2) = 0.2 f( (0.2), (0.1795)) = 0.2 f(0.3, ) = 4/17/2017 Dr.G.Suresh University

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Solution k4 = h f(x1+h, y1+k3) = 0.2 f( , ) = 0.2 f(0.4, ) = k = 1/6 (k1 + 2k2 + 2k3 + k4) = 1/6 ( (0.1795)+2(0.1793) ) = Hence y2 = y1 + k y(0.4) = = 4/17/2017 Dr.G.Suresh University

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Problem#2 Apply Runge-Kutta method to find an approximate value of y for x=0.2 in steps of 0.1, if yÊ¹ = x + y2 given that y=1 where x = 0. Sol : - Compare the given problem with the general first order initial value problem yÊ¹=f(x, y) satisfying y(x0)=y0, we have f(x, y) = x + y2 , x0 = 0, y0 =1. Here we have to find value of y at x = 0.2. 4/17/2017 Dr.G.Suresh University

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Solution Taking step size h = 0.1. Step(1) To find y(0.1) k1 = h f(x0, y0) = 0.1 f(0, 1) = 0.1 (0 + 12) = 0.1 k2 = h f(x0+0.5h, y0+0.5k1) = 0.1 f(0+0.5(0.1), (0.1)) = 0.2 f(0.05, 1.05) = 4/17/2017 Dr.G.Suresh University

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Solution k3 = h f(x0+0.5h, y0+0.5k2) = 0.1 f(0+0.5(0.1), 1+0.5(0.1152)) = 0.1 f(0.05, ) = k4 = h f(x0+h, y0+k3) = 0.1 f(0+0.1, ) = 0.1 f(0.1, ) = 4/17/2017 Dr.G.Suresh University

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Solution k = 1/6 (k1 + 2k2 + 2k3 + k4) = 1/6 (0.1 +2(0.1152)+2( ) ) = â‰ˆ Hence y1 = y0 + k y(0.1) = = Step(2) To find y(0.2) k1 = h f(x1, y1) = 0.1 f(0.1, ) = 4/17/2017 Dr.G.Suresh University

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solution k2 = h f(x1+0.5h, y1+0.5k1) = 0.1 f( (0.1), (0.1347)) = 0.2 f(0.15, ) = k3 = h f(x1+0.5h, y1+0.5k2) = 0.1 f( (0.1), (0.1551)) = 0.1 f(0.15, 1.194) = 4/17/2017 Dr.G.Suresh University

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Solution k4 = h f(x1+h, y1+k3) = 0.1 f( , ) = 0.1 f(0.2, ) = k = 1/6 (k1 + 2k2 + 2k3 + k4) = 1/6 ( (0.1551)+2(0.1576) ) = Hence y2 = y1 + k y(0.2) = = 4/17/2017 Dr.G.Suresh University

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