Uses of difference equations
Introduction Mathematical computations frequently are based on equations that allow us to compute the value of a function recursively from a given set of values. Such an equation is called a difference equation. Difference equations usually describe the evolution of certain phenomena over the course of time. The means of difference equations is to find a relationship between what is happening now and what will happen in the near future. ... First Order Linear Difference Equations Linear Difference Equations When modelling the real world, we first consider only the most basic assumptions. This leads to solving linear difference equations. Linear difference equation is where the graph of the function is a straight line through the origin. ... Let’s substitute into which gives into which gives into which gives into which gives A simple pattern can be seen and therefore a solution for stated for t = 0,1,2 A general solution to the equation is for t = 0,1,2… Application of Linear Difference Equations A Simple Savings Account Suppose we start a savings account of £A on January 1st 2000. ... The equation can be written in a more general form: The equation is a first order linear difference equation. It would be tedious and times consuming to calculate say from the first order liner difference equation, so a better solution has to be found. At At At From the above, we can deduce that the solution of the first order linear difference equation is where … Now do a problem when A0 = £1000 and I= 5 percent. ... 9 grams of the material therefore: The difference equation for this radioactive material is: The general solution is Using the general solution you can work out the half life of the substance, that is in how many years, ,will Conclusion: It would take 69 years to find 5 grams of radioactive material remaining. Affine Difference Equation Affine difference equation is where the graph of is a straight line but does not go through the origin. There are two cases: Case 1: Affine difference equations that are in the form: where r and b are two constants. The above first order affine difference equation can be solved using complementary and particular solutions. ... To find C we use an initial condition where Therefore The solution to the first order affine difference equation where is: Case 2: Affine difference equations that are in the form: where b is a constant. Let Therefore From this we can conclude that the general solution is: The solution to the first order difference equation where is: Applications of affine first order linear difference equations Financial Applications Sinking Fund Annuity Find Complementary Solution: Let Complementary Solution is Find Particular Solution: Let and , substitute into Particular Solution is: The general solution is Therefore the general solution is To find C we will use an initial condition , which is the first deposit into the account. The solution to the sinking fund annuity difference equation where is: Now do an example mortgage repayment difference equation where is: Now do an example Supply and Demand We can relate supply and demand to affine linear difference equations. ... Relating Supply and Demand to Difference Equations From the first assumption the supply next year depends on the price this year. ... From this affine linear difference equation, if the price this year is £2.