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Here's a dip into one of the more complex branches of mathematics, Analysis  the area of mathematics that deals largely with change and rates of change. I'll warn readers in advance that this is not my area of expertise, so what is presented here is my simpleminded understanding of some background information. IntroductionDifferential equations contain information about the rate of change of an dependent variable with respect to some independent variable. To solve the equation, we integrate (sum) these little changes to get a final answer. Sometimes there are analytical solutions to these types of problems, often there are not. In these cases computers come to the rescue with numerical solutions. RungeKuttaIf the equation meets certain conditions then the classical 4th order RungeKutta method is a good choice for computer solution because it is both fast and accurate. The "4th order" part refers to the fact that the algorithm takes a weighted average of 4 estimates of the derivative for each calculated point which reduces total error in proportion to the 4th power of the chosen step size. One of the conditions for RungeKutta is "Ordinary", only one independent variable, and the abbreviation ODE is frequently used as shorthand reference for Ordinary Differential Equation. We'll do that from now on. Many real world phenomena can be described by ODEs that meet the conditions required for RungeKutta including our current interest, time/location problems. I'll assume that we're in the time/location domain for the rest of this discussion. Initial ConditionsODEs do not completely specify where an object will be at some point in time unless we know where it was and how fast it was moving at some other point in time. Thus we also need to specify Initial Conditions when using RungeKutta. (A variation for future implementation uses Boundary Values instead  we know where it object was at two points in time, but not it's velocity.) 2nd OrderThe usual notation for derivatives uses ' marks, so for variable x, x' is the first derivative and x'' is the second derivative. RungeKutta requires that ODEs be linear, that is contain first derivatives only. Fortunately it can handle systems with multiple equations and multiple dependent variables and it is easy to split an equation that contains a 2nd derivative into two equations that contain only first derivatives by assigning a new variable (and equation) x2=x'. Now any prior occurrence of x'' becomes x2'. In our second order cases, the called procedure will take care of estimating the first derivative given a second derivative, as well as estimating the variable value from the 1st derivative. It will be up to us to tell it what 2nd derivative value to use at any given point in time, T, given T, X, and X' values. Systems of EquationsSometimes there are "coupled" systems where several objects are affect each other, planets in the solar system, or springs that connect things together for example. Described below is a second RungeKutta procedure that can solve these systems if we provide an array of initial conditions, one for each element on the system, and an array of functions to do the necessary 2nd derivative calculations at any point in time. Enough background, let's move on to  The ImplementationUnit URungeKutta4 defines a type Float (currently double) that applies to all of the floating point variables used. There are two procedures presented here: RungeKuttaICRungeKuttaIC solves a single Second Order ODE with Initial Conditions. The calling sequence is
where Lower Limit and Upper Limit are type Float and give the range of the independent variable (0 to 2.5 seconds for example) InitialValue and InitialDeriv are the Float values of the independent variable when the dependent variable is at LowerLimit. ReturnInterval is the Float independent variable specifying the independent variable increment for result points to be passed back to the caller. CalcInterval is the Float independent variable increment between calculation points Note: ReturnInterval should be a multiple of CalcInterval. CalcInterval influences the accuracy of the result so we may, for example calculate results every hundredth of a second (CalcInterval) but only return result to the caller for each tenth of a second (ResultInterval). Error: return error codes
UserFunc is the name of a caller supplied function with calling sequence : Function UserFunc(T, X, XPrime:Float):Float; which must return a second derivative value for the (T, X, Xprime) values received. T is the current independent variable value, X the current dependent variable value, and XPrime the current 1st derivative value for X. This function is defined with the "of object" condition, so your function should be defined within an object, usually your TForm1 main form object. CallBackFunc is the name of a caller supplied function which receives the T, X, and XPrime values every ReturnInterval increments of the independent variable. Calling sequence is Function CallBackFunc(T, X, XPrime:Float): boolean; Parameters are the same as described above  return True to continue the calculation, return False to stop the RungeKutta procedure before the independent variable reaches Upperlimit. RungeKutta2ndOrderIC_SystemSolves a system of Second Order ODEs with Initial Conditions. The calling sequence is RungeKutta2ndOrderIC_System( LowerLimit, UpperLimit, InitialValues, ReturnInterval, CalcInterval, Error, NumEquations, UserFuncs, Test2CallBackFunc); where Lower Limit and Upper Limit are type Float and give the range of the independent variable (0 to 2.5 seconds for example) InitialValues is defined as type TnVector; an array of [1..10] of TnData records containing the initial conditions for each dependent variable. TnData defined as : TNData = record ReturnInterval is the Float independent variable increment between result points to be passed back to the caller. CalcInterval is the Float independent variable increment between calculation points Note: ReturnInterval should be a multiple of CalcInterval. Error: return error codes
NumEquations is the number of equations in the system. UserFuncs is type TFuncVect, an array [1..10] of functions to be called while calculating points The calling function for each function is : Function UserFuncN(v:TNVector):Float; which must return a second derivative value for the Nth dependent variable. V is an array [0..10] of TnData records as described above. Function type is defined with the "of object" condition, so your functions should be defined within an object, usually your TForm1 main form object. Note that variables are numbered 1 through 10; V[0].X is used to contain the current value of the independent variable. CallBackFunc is the name of a caller supplied function which receives an array of Tndata records containing the T, X, and XPrime values each dependent variable every ReturnInterval increments of the independent variable. Calling sequence is Function Test2CallBackFunc(v:TNVector):boolean; Parameters are the same as described above  return True to continue the calculation, return False to stop the RungeKutta procedure before the independent variable reaches Upperlimit. Running/Exploring the ProgramWhew! Examining sample code will probably be an easier way to learn how to use these routines. The code below contains two test cases from Borland's Numerical Methods Toolbox, a simple "springmass" system and "two pendulums coupled by a weak spring". Also a Simple Pendulum I coded myself just to make sure understood the ODE part. I added some simple animation "just for fun".

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