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Rearranging equation | of (50) in increasing powers ¢*”), we have two conditions The coefficients R;(f, m), defined by Expansion Coefficients of f(z)~” m Appendix A

Figure 23 Rearranging equation | of (50) in increasing powers ¢*”), we have two conditions The coefficients R;(f, m), defined by Expansion Coefficients of f(z)~” m Appendix A

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The Recursive Formula for Burmann Coefficients C,(/; 4, Zo) This section shows how to calculate the coefficients of the Biirmann series recursively, which is easier to handle than (3) and more efficient when translated to symbolic pro- grams. If we use (1) and (2) to find convergent series representations of solutions to differ- ential equations, it is important to simplify the algorithms necessary to determine the expansion coefficients. This section shows how to calculate the coefficients of the Biirmann series recursively,
The Recursive Formula for Burmann Coefficients C,(/; 4, Zo) This section shows how to calculate the coefficients of the Biirmann series recursively, which is easier to handle than (3) and more efficient when translated to symbolic pro- grams. If we use (1) and (2) to find convergent series representations of solutions to differ- ential equations, it is important to simplify the algorithms necessary to determine the expansion coefficients. This section shows how to calculate the coefficients of the Biirmann series recursively,
Here are the coefficients B, as they are calculated with this procedure in their explicit ana- lytic form, according to the definition given in (2). Series [Biirmann[#*5 &, Sinh, {z, 0, 25}], {z, 0, 25}]
Here are the coefficients B, as they are calculated with this procedure in their explicit ana- lytic form, according to the definition given in (2). Series [Biirmann[#*5 &, Sinh, {z, 0, 25}], {z, 0, 25}]
A useful application of the recursion (5)—(7) to the case of the expansion of the inverse function y(z) of f(z) around z = z follows immediately: we have y(f(z)) = z and it fol- lows that the inverse function is the Biirmann series of z in powers of f(z) — f(Z) (see 181). By writing A useful application of the recursion (5)—(7) to the case of the expansion of the inverse
A useful application of the recursion (5)—(7) to the case of the expansion of the inverse function y(z) of f(z) around z = z follows immediately: we have y(f(z)) = z and it fol- lows that the inverse function is the Biirmann series of z in powers of f(z) — f(Z) (see 181). By writing A useful application of the recursion (5)—(7) to the case of the expansion of the inverse
Using the results listed in (10) for the Biirmann coefficients, we arrive at once at the expan- sion around Zp: w[z_] := Birmann[Log[1+H#] &, 1/ (1+#) &, {z, 0, 11}]
Using the results listed in (10) for the Biirmann coefficients, we arrive at once at the expan- sion around Zp: w[z_] := Birmann[Log[1+H#] &, 1/ (1+#) &, {z, 0, 11}]
m The Generalized Form of Burmann Series Based on a Combinatorial Approach
m The Generalized Form of Burmann Series Based on a Combinatorial Approach
While the expansion is valid for complex-valued functions, the plot shows only the real part of f. The Explicit Expression for the Coefficients 8,,(/; ¢, zo) in Terms of the Coefficients R;,(¢*, m)
While the expansion is valid for complex-valued functions, the plot shows only the real part of f. The Explicit Expression for the Coefficients 8,,(/; ¢, zo) in Terms of the Coefficients R;,(¢*, m)
The function fbiir shows how to apply (23) and (24) to the expansion of f(z) = 2° in The formula for the Biirmann coefficient 6,(f; 4,7) in terms of the coefficients R,(¢*, m) is thus given by
The function fbiir shows how to apply (23) and (24) to the expansion of f(z) = 2° in The formula for the Biirmann coefficient 6,(f; 4,7) in terms of the coefficients R,(¢*, m) is thus given by
As an example, we calculate an expansion of arcsin[z] up to order 15 according to (14) ir Use this definition for numerical purposes, such as for plotting. = The Generalized Form of Burmann Series
As an example, we calculate an expansion of arcsin[z] up to order 15 according to (14) ir Use this definition for numerical purposes, such as for plotting. = The Generalized Form of Burmann Series
A function calculating the expansion (32) is given below. To show that this approach is superior to a common Taylor expansion in a plot, we calculate the power series in z up to order 10. The result of this calculation performed up to order nine in © (@, 0, z) is
A function calculating the expansion (32) is given below. To show that this approach is superior to a common Taylor expansion in a plot, we calculate the power series in z up to order 10. The result of this calculation performed up to order nine in © (@, 0, z) is
The plot shows that the series in (32) has only a small constant offset error for larger values of z, whereas the Taylor expansion dramatically deviates for smaller values of z, although it converges uniformly for all values of z. The series in (32) converges uniformly for all z and gives the exact value for the error function. The rearrangement of terms, leading to erfg(z), is thus justified. Even for the lowest order, we will find a result that shows no unbounded error, unlike the Taylor series. The order v+1 of first nonvanishing derivative of 6(z) at Zo is 2
The plot shows that the series in (32) has only a small constant offset error for larger values of z, whereas the Taylor expansion dramatically deviates for smaller values of z, although it converges uniformly for all values of z. The series in (32) converges uniformly for all z and gives the exact value for the error function. The rearrangement of terms, leading to erfg(z), is thus justified. Even for the lowest order, we will find a result that shows no unbounded error, unlike the Taylor series. The order v+1 of first nonvanishing derivative of 6(z) at Zo is 2
m Application of Burmann Series to Problems of Nonlinear Heat Transfer This section applies the concept of Biirmann series to get solutions of nonlinear differen- tial equations. After a short introduction, an example from the field of the diffusion type is presented.
m Application of Burmann Series to Problems of Nonlinear Heat Transfer This section applies the concept of Biirmann series to get solutions of nonlinear differen- tial equations. After a short introduction, an example from the field of the diffusion type is presented.
While the first boundary condition in (35) is regular (i.e. T (0, t) = Ts and 6(0) = a), the second one is given in terms of the asymptotic expression lim,,,. T(z, t) = T,. The equiv- alent value o of the derivative (d@/d1),-o has to be estimated, which is performed by calculating the time evolution of the total thermal energy of the semi-infinite half space [18]. The approximate value of this energy integral (in terms of an algebraic expression) is determined in appendix B, where we use the Biirmann series (10) to approximate the energy integral. The explicit expression is displayed in appendix B, equation (56), which describes the dependence of o on the parameters with a relative accuracy of 0.37%. alent value o of the derivative (d@/d1),-9 has to be estimated, which is performed by
While the first boundary condition in (35) is regular (i.e. T (0, t) = Ts and 6(0) = a), the second one is given in terms of the asymptotic expression lim,,,. T(z, t) = T,. The equiv- alent value o of the derivative (d@/d1),-o has to be estimated, which is performed by calculating the time evolution of the total thermal energy of the semi-infinite half space [18]. The approximate value of this energy integral (in terms of an algebraic expression) is determined in appendix B, where we use the Biirmann series (10) to approximate the energy integral. The explicit expression is displayed in appendix B, equation (56), which describes the dependence of o on the parameters with a relative accuracy of 0.37%. alent value o of the derivative (d@/d1),-9 has to be estimated, which is performed by
If we consider y in (40) and (41) as a free parameter, we have to investigate how its choice influences the convergence of the corresponding Biirmann series. We write, using (37) and omitting the index y for w, All integrands are representable by uniformly convergent series expansions in powers of w <1, and thus the argument is similar to the one used in proving the convergence of the system (39). Furthermore, we observe that the expansion of the exponential function exp(-2 (B/ yy f dO...) converges more quickly for higher values of y. Thus, as long as y can be chosen to guarantee the condition Convergence of the Enhanced Burmann Series
If we consider y in (40) and (41) as a free parameter, we have to investigate how its choice influences the convergence of the corresponding Biirmann series. We write, using (37) and omitting the index y for w, All integrands are representable by uniformly convergent series expansions in powers of w <1, and thus the argument is similar to the one used in proving the convergence of the system (39). Furthermore, we observe that the expansion of the exponential function exp(-2 (B/ yy f dO...) converges more quickly for higher values of y. Thus, as long as y can be chosen to guarantee the condition Convergence of the Enhanced Burmann Series
We display the explicit expression for the third-order approximation below: If there is a solution y,; to equation (43) simultaneously fulfilling condition (44), we get an
We display the explicit expression for the third-order approximation below: If there is a solution y,; to equation (43) simultaneously fulfilling condition (44), we get an
This defines the cubic approximation according to (45)-(48).
This defines the cubic approximation according to (45)-(48).
The plot shows the numerically exact temperature profiles (the colored curves) and the exact solution’s third-order approximation according to equation (48) (the dotted lines), in the range z = 0-0.25 mat t = 10, 30, 100, 300, 1000, 3000, 10000 s. The plot shows the numerically exact temperature profiles (the colored curves) and the
The plot shows the numerically exact temperature profiles (the colored curves) and the exact solution’s third-order approximation according to equation (48) (the dotted lines), in the range z = 0-0.25 mat t = 10, 30, 100, 300, 1000, 3000, 10000 s. The plot shows the numerically exact temperature profiles (the colored curves) and the
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