Materials and Methods
This aspect contains materials and methods of the Mathematica pseudocodes of block Adams family. Block Adams family is divided into two parts; block Adams-Bashforth method and block Adams-Moulton method. Both pairs will be formulated using interpolation and collocation technique via a well coded multinomial and executed under the platform of Mathematica kernel to produce the desired results. The unification of block Adams family is described as
\(y(x)=\sum_{i=0}^{1}\alpha_{i}y_{n-i}+h\left[\sum_{i=0}^{3}{\beta_{i}^{*}\left(u\right)f_{n-i+1}+}\right]\)(2)
\(y\left(x\right)=\sum_{i=2}^{1}\alpha_{i}y_{n-i}+h\left[\sum_{i=0}^{1}{\beta_{i}^{*}\left(u\right)f_{n+i+1}+}\right].\)(3)
Equations (2) and (3) forms the block Adams-Bashforth method and block Adams-Moulton method of the block Adams family. Citing that\(y_{n+i}\) represents the mathematical estimate to the analytic effects \(y(x_{n+i})\) i.e. \(y(x_{n+i})\approx y_{n+i}\),\(f\left(x_{n-i},y_{n-i}\right)\approx f_{n-i}\text{\ \ }\) and\(f\left(x_{n+i},y_{n+i}\right)\approx f_{n+i}\text{\ \ }\)owning\(i=0,\ 1,\ 2,\ 3\). To reach equations (2) and (3), the mathematical multinomial is composed in the format of Mathematica pseudocodes that are required to estimate the analytic effect \(y(x)\) on trenchant definite length of time, \(\left[x_{n}\right]\) via the interpolating subroutine of the class (4) that is clearly characterized as
. (4)
Transforming (4) will bring-forth the Mathematica pseudocodes which can be constituted under the platform of Mathematica Kernel as
\(y\left[x_{-}\right]=w\left[0\right]+w\left[1\right]\frac{\left(x-x\left[n\right]\right)}{h_{1}}+w\left[2\right]\frac{\left(s-x\left[n\right]\right)}{h^{2}}^{3}+w\left[3\right]\frac{\left(x-x\left[n\right]\right)}{h}^{3},\)(5)
where\(w\left[0\right],\ w\left[1\right],\ w[2]\)and \(w[3]\) are unknown parametric quantity anticipated to be resolve in a unique way. Take for granted that equations (5) equates with the analytic effect at an approximate time-interval\({[x}_{n}],\ {[x}_{n-j}]\) to suit the estimates of
\({y(x}_{n})\approx y_{n}\), \({y(x}_{n-j})\approx y_{n-j}\). (6)
Demanding that the mathematical function (6) meet the requirements for (1) at the points \(\backslash nx_{n+j},\ j=1,\ 2,\ 3\) to formulate the next estimate as
\(y^{\prime}(x_{n+j})\approx f_{n+j}],\) \(j=0,\ 1,\ 2,3.\) (7)
To achieve this formulation of the block family, (5) is encrypted into Mathematica pseudocodes to carry out the process of interpolation and collocation.
A combination of (6) and (7) will give-rise to quartet structure which constitute \(Ax=d\).
\(matrixa=\par \begin{Bmatrix}\left\{1,0,0,0\right\},\\ \left\{0,1,-2,3\right\},\\ \left\{0,1,-4,12\right\},\\ \left\{0,1,-6,27\right\},\\ \end{Bmatrix};\)
\(d=\left\{y\left[n\right],\ f\left[n-1\right],f\left[n-2\right],f[n-3]\right\};\)
\(\left\{b,c,l,q\right\}=Inverse\left[\text{matrixa}\right]\text{.d}\), (8)
\(matrixa=\par \begin{Bmatrix}\left\{1,-2,4,-8\right\},\\ \left\{0,1,2,3\right\},\\ \left\{0,1,4,12\right\},\\ \left\{0,1,6,27\right\}\\ \end{Bmatrix};\)
\(d=\left\{y\left[n-2\right],\ f\left[n+1\right],f\left[n+2\right],f[n+3]\right\};\)
\(\left\{b,c,l,q\right\}=Inverse\left[\text{matrixa}\right]\text{.d}\). (9)
This moves on to evaluate at certain fixed time-interval of\(x_{n+j},\ j=1,\ 2,\ 3\) and generates the encrypted continuous Mathematica pseudocodes for block Adams-Bashforth method of the form
\(y\left[x_{-}\right]=\left(1\right)y\left[n\right]+\left(\frac{{3\left(x-x\left[n\right]\right)}^{1}}{h}+\frac{5}{4}\frac{\left(x-x\left[n\right]\right)^{2}}{4h^{2}}+\frac{\left(x-x\left[n\right]\right)^{3}}{6h^{3}}\right)\text{\ f}\left[n-1\right]h+\left(-3\frac{\left(x-x\left[n\right]\right)^{1}}{h}-2\frac{\left(x-x\left[n\right]\right)^{2}}{4h^{2}}-\frac{1}{3}\frac{\left(x-x\left[n\right]\right)^{3}}{6h^{3}}\right)\text{\ f}\left[n-2\right]h+\left(\frac{\left(x-x\left[n\right]\right)^{1}}{h}+\frac{3}{4}\frac{\left(x-x\left[n\right]\right)^{2}}{4h^{2}}+\frac{\left(x-x\left[n\right]\right)^{3}}{6h^{3}}\right)\text{\ f}\left[n-3\right]h\), (10)
while the continuous Mathematica pseudocodes for block Adams-Moulton method is seen as
\(y\left[x_{-}\right]=\left(1\right)y\left[n-2\right]+\left(\frac{37}{3}+\frac{3\left(x-x\left[n\right]\right)^{1}}{h}-\frac{{5\left(x-x\left[n\right]\right)}^{2}}{4h^{2}}+\frac{\left(x-x\left[n\right]\right)^{3}}{6h^{3}}\right)\text{\ f}\left[n+1\right]h+\left(-\frac{50}{3}-\frac{3\left(x-x\left[n\right]\right)^{1}}{h}+\frac{{2\left(x-x\left[n\right]\right)}^{2}}{h^{2}}-\frac{\left(x-x\left[n\right]\right)^{3}}{3h^{3}}\right)\text{\ f}\left[n+2\right]+\left(\frac{19}{3}+\frac{\left(x-x\left[n\right]\right)^{1}}{h}-\frac{{3\left(x-x\left[n\right]\right)}^{2}}{4h^{2}}+\frac{\left(x-x\left[n\right]\right)^{3}}{6h^{3}}\right)\text{\ f}\left[n+3\right]\). (11)
Equations (10) and (11) are called the Mathematica pseudocodes of block Adams family designed under the platform of Mathematical Kernel. This turns out to produce the block Adams-Bashforth method of order three expressed below:
\(y\left[n+1\right]=y\left[n\right]+h\left(\frac{53}{12}f\left[n-1\right]-\frac{16}{3}f\left[n-2\right]+\frac{23}{12}f\left[n-3\right]\right)\),
\(y\left[n+2\right]=y\left[n\right]+h\left(\frac{37}{3}f\left[n-1\right]-\frac{50}{3}f\left[n-2\right]+\frac{19}{3}f[n-3]\right)\), (12)
\(y\left[n+3\right]=y\left[n\right]+h\left(\frac{99}{4}f\left[n-1\right]-36f\left[n-2\right]+\frac{57}{4}f[n-3]\right).\)
Similar manner, the block Adams-Moulton method of order three is given below:
\(y\left[n+1\right]=y\left[n-2\right]+h\left(\frac{57}{4}f\left[n+1\right]-18f\left[n+2\right]+\frac{27}{4}f[n+3]\right)\),
\(y\left[n+2\right]=y\left[n-2\right]+h\left(\frac{44}{3}f\left[n+1\right]-\frac{52}{3}f\left[n+2\right]+\frac{20}{3}f[n+3]\right)\), (13)
\(y\left[n+3\right]=y\left[n-2\right]+h\left(\frac{175}{12}f\left[n+1\right]-\frac{50}{3}f\left[n+2\right]+\frac{85}{12}f[n+3]\right)\).
Equations (12) and (13) represent the block Adams family of order three. Study [1, 12-14, 19-23, 28] for more info.