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DOI: Current Special Offers. No Current Special Offers. The procedure begins with solving nonlinear differential equation by DTM. The effectiveness of the procedure is verified using a heat transfer nonlinear equation. The simulation result shows the significance of the proposed technique.
Theorem 3. Example 1. From equation 7 , we obtain k! The approximate solutions of Example 2 by DTM for y1 t , y2 t and y3 t are given in Tables 3, 4 and 5. Also, we saw that, if the numerical solution of the given problems are compared with their analytical solutions, the DTM is very effective and results are quite close. Sciences, 2 30 : Abdel-Naby, M.
Ramadan and S. Mohamed, Spline approximation for solving system of first order delay differential equation. Studia Univ. Ozkol, Solution of differential-difference equations by using differential transform method, Appl.
Remark 5. Problems for second-order equations. Differential transformations 4. To do this, we differentiate the first equation in 26 with respect to t. The Cauchy problem 28 — 30 can be integrated numerically applying the standard numerical methods [10—15], without fear of blow-up solutions.
Remark 6. Systems of differential-algebraic equations 5 and 26 are particular cases of parametrically defined non-linear differential equations, which are considered in [22, 23]. Exact and numerical solutions Example 5. Exact solution of this problem is defined by the formula The exact solution of this problem is determined by the formulas Example 6.
The exact solution of the problem 32 is defined by the formula The exact solution of the problem 33 is given by the formulas Therefore to accel- erate this process in the system 33 is useful additionally to make the exponential- type substitution Non-local transformations 5. Let us consider some possible choices of the function g in the system It follows that the method based on the non-local transformation 34 is a generalization of the method described in Section 4. Remark 7. Remark 8.
Exact and numerical solutions Example 7. The numerical solutions are in a good agreement, but the speed of the process of approaching the asymptote with respect to x is different. For comparison, similar calculations were performed with the aid of Maple according to the method based on the hodograph transformation see Sec- tion 5.
To control the solution process, the calculations were performed also with the aid of the other two problem solving environments, Mathematica 11 and MATLAB a. Example 8. However, in comparison with the method applied in Example 7, in this case the rate of approximation of the parametric solution to the asymptote is less which is not important for application of the standard numerical methods for solving similar problems.
Remark 9. For applying non-local transformations is not necessary to compute the integrals 34 or Comparison of efficiency of various transformations for numerical integra- tion of second-order blow-up ODE problems In Table 4, a comparison of the efficiency of the numerical integration meth- ods, based on various nonlocal transformations of the form 34 is presented by using the example of the test blow-up problem for the second-order ODE The comparison is based on the number of grid points needed to make calcula- tions with the same maximum error approximately equal to 0.
In all cases, for the integration of the transformed problems the standard fourth-order Runge— Kutta method was used. It can be seen that the arc-length transformation is the least effective, since the use of this transformation is associated with a large number of grid points in particular, when using the last two transformations, you need about and times less of a number of grid points.
The hodograph transformation has an intermediate moderate efficiency. Brief conclusions We describe two new methods for numerical integration of non-linear Cauchy problems for ODEs of the first- and second-orders, which have blow-up solutions.
It is shown that: i the proposed method based on the general non-local transformation includes, as particular cases, the method based on the hodograph transformation, the method of the arc-length transformation, and the method based on the dif- ferential transformation; ii methods based on special exp-type and modified differential transformations are more efficient than the method based on the hodograph transformation, the method of the arc-length transformation, and the method based on the differential transformation.
It is important to note that the method described in Section 5. References [1] M. Stuart and M.
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