Step Methods to Solve Differential Equations" has been approved by the Board of Examiners for the partial fulfillment of the requirements for the deeree of 'Master of Philosophy in the Department of Mathematics, Khulna University of Engineering &' technology. To get the solution of particular problem is it is necessary to determine the values of the arbitrary constants.If all the conditions are prescribed at the initial point of the domain, the conditions are called initial conditions and the equation together with the initial conditions is called the initial value problem (I.V.P.).
So usually the exact value cannot be calculated, but it is forced to estimate an approximate value of the dependent variable, even if the desired degree of accuracy can be achieved. The integer p is called the order of the method.. which is neglected is the relative discretization or local truncation error. From the terminology itself, it is clear that the value of the dependent variable is kiiown at one point, then we will be able to calculate the value of it at the next point with the known step size in case of single step method.
If in order to calculate the value of the dependent variable at a time with known step size, it is only necessary to know the value of the dependent variable at the very first point, then the method is called a single-step method. In section 2.1, a brief description and related ideas for the single-step method will be presented.
SINGLE STEP METIJOI)
Many one-step methods have been developed, but only the Runge-Kutta limit is considered here. The proposed five new formulas, S two 5th and 6th order formulas and one 7th order formula, will be presented in Section 2.3. It is also necessary to ensure that formula (2.1.2) is insensitive to small changes in local errors.
A necessary and sufficient condition for the convergence of a regular one-step method of order p is consistency.
RUNGE-KUTTA METHODS
In the modified Euler method, the slope of the solution curve is approximated by the slopes of the curve at the end points of each stichinterval in the calculation of the solution. The natural generalization of this concept is to calculate the slope by taking a weighted average of the slopes obtained at multiple points in each subinterval. However, the implementation of the scheme differs from the modified Euler method so that the developed algorithm is clear in nature.
From (2.2.4) we can interpret the increment function as the linear combination of the slopes at x, and at several other points between x, and x, 1. To define the parameters c's, a's and W'S in the above equation, y, in the diagram, is expanded in terms of step length h and the resulting equation is then compared with Taylor series expansion of the solution of the differential equation to a certain number of terms, for example p. It is interesting to note that for any i >4 the maximum possible order p of the RK method is always less than i.
Before proceeding, we would like to point out the advantages and disadvantages of the RK method, which are listed below. We note that no choice of parameter c7 will cause the leading term 7 to disappear for all.
Proposed Formulas
In the previous section, the derivation of the second-order RK method is presented, and it can be seen that the number of obtained equations is smaller than the number of entered constants. The derivation of each of them is not presented here, but the derivation of the 6th order formula is presented in Appendix-A. In the following, we will list the coefficients and weights of the proposed forms.
To demonstrate the use of the proposed formulas, we have chosen a non-linear first-order differential equation = y + xy 2. We will try to find the value of the dependent variable atx0.1 and atx0.2 i.e. I) and (O.2) will be estimated. This form is chosen as it can be reduced to a linear one and as such its exact solution can be easily obtained.
Multi Step Methods
Multi Step Methods
- Explicit Multi Step Methods
- Implicit Multi Step Methods
If is independent of y 1 then the general multi-step is called an explicit, open or predictor method: otherwise an implicit, closed or corrector method. The special cases of the linear multistep method (3.1.5) are used to solve the initial value problem (3.1 .1). An alternative form of formula (3.1.11) can be obtained if the differences V"f, are expressed in terms of the function values j;,.
It is clear from (3.1.9) that with k calculated values we obtain explicit multistep methods of order k, since the truncation error is of the form ch, where c is independent of/i. In the preceding subsection we expressed y,1 in terms of previously calculated ordinates and slopcs. The Newton backward difference formula that interpolates at these k-1 I points in terms of = (x - x,1 ) / h is given by.
It is obvious from (3.1.15) that the implicit multistep methods are of an order higher than the corresponding explicit multistep methods with the same number of previously calculated ordinates and slopes.
Adams - Bashforth formula
To derive the relations for the Adams-Bashlorth method, we write the differential equation ClY = f(x,y).
Adams- Moulton formula
Demonstrations
In this chapter, comparisons will be made between the proposed formula and the existing formula of the same order. For this purpose, a selected problem will be solved by all the proposed (except 7th order) and known same order RK methods and the result obtained by them will be compared by calculating the percentage error. For 2nd interval we have, . which again corresponds to the exact value to four decimal places.
Conclusion
APPENDIX-A
