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Open AccessArticle

An Algorithm Based on GSVD for Image Encryption

1
Department of Islamic Education, University of Kufa, 31001 Al-Najaf, Iraq
2
Department of Mathematics, University of Kufa, 31001 Al-Najaf, Iraq
3
Department of Computer Science, University of Kufa, 31001 Al-Najaf, Iraq
*
Authors to whom correspondence should be addressed.
Math. Comput. Appl. 2017, 22(2), 28; https://doi.org/10.3390/mca22020028
Received: 28 January 2017 / Revised: 10 March 2017 / Accepted: 17 March 2017 / Published: 28 March 2017

Abstract

This paper represents a new image encryption algorithm based on modifying generalized singular value decomposition (GSVD) by decomposing the plain-image into two segments using GSVD with an exchanged key-image to produce the cipher-image. The exchanged key-image is used as an encrypting and decrypting image. Mathematically, this procedure is represented by transforming the plain-image’s matrix into two different matrices and applying the GSVD with the exchanged key-image’s matrix to obtain the cipher-image’s matrix. The two encoded segments can be kept in several places or assigned to a group of authorized persons. No one can obtain the information of the image easily without the knowledge of the decrypting key. This proposed algorithm is represented as one of the digital image encryption techniques used to enhance the security of images that have been sent between recipients.
Keywords: encryption; singular value decomposition; generalized singular value decomposition; image encryption encryption; singular value decomposition; generalized singular value decomposition; image encryption

1. Introduction

In this study, our focus is on the numerical solution of the initial value problems (IVPs) of second-order differential equations; whose first derivative does not appear explicitly of the form :
y = f ( x , y ) , x [ x 0 , X ] , y ( x 0 ) = y 0 , y ( x 0 ) = y 0 ,
whose solutions have a noticeable oscillatory character, where y d and f : d d are sufficiently differentiable. Such problems frequently occur in several areas of applied sciences such as: theoretical physics, celestial mechanics, nuclear chemistry, nuclear physics, electronics, molecular dynamics and elsewhere. Many numerical methods have been developed for the numerical integration of Equation (1), among them are Runge–Kutta–Nyström (RKN) methods. Exponentially/trigonometrically-fitted RKN methods have been studied by Simos in [1],Van de Vyver in [2], Kalogiratou and Simos in [3] and Shiwei Liu et al. in [4]. Franco in [5] proposed a 5(3) pair of explicit adapted Runge–Kutta–Nyström methods for the numerical integration of perturbed oscillators, Van de Vyver in [6,7] proposed a Runge–Kutta–Nyström pair for the numerical integration of perturbed oscillators, a 5(3) pair of explicit RKN methods for oscillatory problems. Senu et al. in [8] proposed an embedded explicit RKN method for solving oscillatory problems. Recently, Tsitouras in [9] proposed fitted modifications of RKN pairs, Franco et al. in [10] proposed two new embedded pair of explicit Runge-Kutta methods adapted to the numerical solution of oscillatory problems, and Anastassi and Kosti in [11] proposed a 6(4) optimized embedded Runge–Kutta–Nyström pair for the numerical solution of periodic problems.
In this paper, we develop a new efficient embedded explicit trigonometrically-fitted RKN method based on the technique proposed by Simos in [12] for Runge–Kutta (RK) methods. The constructed method can exactly integrate the test equation y = w 2 y and the numerical results show the efficacy of the new method. The remaining part of this paper is arranged as follows: in Section 2, we give the basic theory of an explicit Runge–Kutta–Nyström pair, the basic definition of trigonometrically-fitted RKN method and the derivation of an explicit trigonometrically-fitted RKN method. Section 3 deals with the derivation of the proposed method. In Section 4, we analyze the algebraic order of the new pair from their local truncation errors and give the interval of absolute stability of the new pair. In Section 5, we present the numerical results. In Section 6, we give a brief discussion about the graphs and the last section deals with the conclusions.

2. Basic Theory

The general form of an explicit k-stage RKN method is given by:
y n + 1 = y n + h y n + h 2 i = 1 k b i f ( x n + c i h , Y i ) ,
y n + 1 = y n + h i = 1 k d i f ( x n + c i h , Y i ) ,
Y i = y n + c i h y n + h 2 j = 1 i 1 a i j f ( x n + c i h , Y j ) ,
where y n + 1 and y n + 1 represent the approximation of y ( x n + 1 ) and y ( x n + 1 ) , respectively, where x n + 1 = x n + h , n = 0 , 1 , . . . , or in Butcher Tableau as :
Mca 22 00028 i001
where A is a matrix ( a i j ) k × k , c = ( c 1 , c 2 , . . . , c k ) T , b = ( b 1 , b 2 , . . . , b k ) , d = ( d 1 , d 2 , . . . , d k ) .
An embedded p ( q ) pair of RKN methods is based on the method ( c , A , b , d ) of order p and the other RKN method ( c , A , b ^ , d ^ ) of order q( q < p ). The higher order produces the solution ( y n + 1 , y n + 1 ), while the lower order method produces the solution y ^ n + 1 and y ^ n + 1 , which is used for the estimation of local truncation error only. An embedded pair is characterized by the Butcher tableau below:
Mca 22 00028 i002
In this work, a variable step size algorithm using the embedded technique is used because it provides cheap local error estimation. Local error estimation at the point x n + 1 = x n + h is determined by δ n + 1 = y ^ n + 1 y n + 1 and δ n + 1 = y ^ n + 1 y n + 1 .
Let Est n + 1 = max ( δ n + 1 , δ n + 1 ) represent local error estimation to control the step size h. For the numerical integration of the oscillatory problems, we use the step-size control procedure given by Shiwei in [4]:
  • if E s t n + 1 < T o l / 100 , h n + 1 = 2 h n ,
  • if T o l / 100 E s t n + 1 < T o l , h n + 1 = h n ,
  • if E s t n + 1 T o l , h n + 1 = h n / 2 and repeat the step,
where T o l is the tolerance (requested local error). It should be noted that the N th order approximation y n is used as the initial value for the (n+1)th step, that is to say, the embedded pair is applied in local extrapolation mode or higher order mode.
Definition 1.
Runge–Kutta–Nyström method (2)–(4) is said to be trigonometrically-fitted if it integrates exactly the function e i w x and e i w x or equivalently sin ( w x ) and cos ( w x ) with w > 0 the principal frequency of the problem when applied to the test equation y = w 2 y ; Leading to a system of equations as derived below:
When an explicit Runge–Kutta–Nyström method (2)–(4) is applied to the test equation y = w 2 y , the method becomes:
y n + 1 = y n + h y n + h 2 i = 1 k b i ( w 2 Y i ) ,
y n + 1 = y n + h i = 1 k d i ( w 2 Y i ) ,
where
Y 1 = y n ,
Y i = y n + c i h y n + h 2 j = 1 i 1 a i j ( w 2 Y j ) , i = 2 , 3 , . . . , k .
Let y n = e I w x . By computing the value of y n + 1 , y n and y n + 1 and substituting in the Equations (5)–(8) and by using e I v = cos ( v ) + I sin ( v ) and comparing the real and imaginary part, we obtain the following system of equations:
cos ( v ) = 1 v 2 i = 1 k b i ( 1 v 2 j = 1 i 1 a i j Y j e I w x ) ,
sin ( v ) = v v 2 i = 1 k b i c i v ,
sin ( v ) = v i = 1 k d i ( 1 v 2 j = 1 i 1 a i j Y j e I w x ) ,
cos ( v ) = 1 v 2 i = 1 k d i c i ,
where v = w h .

3. Derivation of the Proposed Method

In this section, we will construct a new efficient embedded explicit trigonometrically-fitted RKN method.
In this study, Embedded Runge–Kutta–Nyström 5(4)M pair (ERKN5(4)M) is used as given by Senu in [13]. The coefficients of the method are given in Table 1.
Solving the above system of Equations (9)–(12) using the coefficients of the lower order method (order 4) for b 2 , b 3 , d 2 , d 3 , we obtain the solution as given in Equation (13).
b 2 = 5 , 380 , 190 , 812 , 500 cos v v 14 , 441 , 564 , 812 , 500 v + 514 , 613 , 717 , 975 v 3 + 19 , 821 , 755 , 625 , 000 sin v 462 , 315 , 000 v 3 1273 v 2 + 9800 72 , 954 , 403 , 664 v 5 1 , 050 , 695 , 364 v 7 730 , 168 , 753 , 125 sin v v 2 + 281 , 087 , 520 , 000 v 3 cos v + 70 , 271 , 880 , 000 v 4 sin v 462 , 315 , 000 v 3 1273 v 2 + 9800 , b 3 = 18 , 586 , 312 , 500 cos v v + 18 , 586 , 312 , 500 v 3 , 588 , 043 , 375 v 3 37 , 172 , 625 , 000 sin v + 62 , 403 , 600 v 5 867 , 000 v 3 1273 v 2 + 9800 10 , 749 , 212 v 7 + 4 , 646 , 578 , 125 sin v v 2 867 , 000 v 3 1273 v 2 + 9800 , d 2 = 82 , 672 , 800 sin v v 111 , 333 v 6 3 , 675 , 956 v 4 + 62 , 415 , 080 v 2 11 , 219 , 880 v 2 cos v 7104 v 2 1273 v 2 + 9800 + 304 , 584 , 000 cos v 304 , 584 , 000 4 , 319 , 232 v 3 sin v + 1 , 079 , 808 v 4 cos v 7104 v 2 1273 v 2 + 9800 , d 3 = 245 ( 673 , 831 , 200 sin v v 2 , 687 , 303 v 6 + 15 , 600 , 900 v 4 264 , 497 , 800 v 2 + 168 , 457 , 800 v 2 cos v 7 , 700 , 928 v 2 1273 v 2 + 9800 1 , 347 , 662 , 400 cos v + 1 , 347 , 662 , 400 ) 7 , 700 , 928 v 2 1273 v 2 + 9800 .
The corresponding Taylor series expansion of the solution is given in Equation (14):
b 2 = 411 , 163 3 , 399 , 375 + 206 , 089 99 , 960 , 000 v 2 + 32 , 195 , 419 61 , 225 , 500 , 000 v 4 16 , 326 , 175 , 118 , 467 129 , 602 , 138 , 400 , 000 , 000 v 6 + 253 , 277 , 042 , 450 , 924 , 651 13 , 971 , 110 , 519 , 520 , 000 , 000 , 000 v 8 4 , 241 , 152 , 320 , 070 , 814 , 549 , 399 1 , 779 , 919 , 480 , 186 , 848 , 000 , 000 , 000 , 000 v 10 + 3 , 741 , 076 , 822 , 521 , 180 , 957 , 593 , 411 12 , 076 , 069 , 088 , 652 , 307 , 200 , 000 , 000 , 000 , 000 v 12 61 , 913 , 938 , 732 , 739 , 966 , 085 , 182 , 108 , 639 1 , 538 , 491 , 201 , 894 , 303 , 937 , 280 , 000 , 000 , 000 , 000 , 000 v 14 + . . . , b 3 = 6 25 206 , 089 99 , 960 , 000 v 2 2 , 667 , 305 , 573 3 , 122 , 500 , 500 , 000 v 4 + 390 , 570 , 720 , 145 , 763 2 , 203 , 236 , 352 , 800 , 000 , 000 v 6 5 , 709 , 954 , 829 , 720 , 650 , 539 237 , 508 , 878 , 831 , 840 , 000 , 000 , 000 v 8 + 284 , 375 , 014 , 038 , 527 , 074 , 809 , 733 90 , 775 , 893 , 489 , 529 , 248 , 000 , 000 , 000 , 000 v 10 98 , 745 , 865 , 792 , 553 , 965 , 577 , 692 , 757 242 , 619 , 206 , 235 , 650 , 899 , 200 , 000 , 000 , 000 , 000 v 12 + 1 , 382 , 746 , 883 , 790 , 630 , 905 , 277 , 400 , 426 , 271 26 , 154 , 350 , 432 , 203 , 166 , 933 , 760 , 000 , 000 , 000 , 000 , 000 v 14 + . . . , d 2 = 235 888 + 11 5760 v 4 26 , 759 56 , 448 , 000 v 6 + 30 , 179 , 399 414 , 892 , 800 , 000 v 8 974 , 755 , 501 , 787 100 , 632 , 248 , 640 , 000 , 000 v 10 + 32 , 337 , 223 , 905 , 299 , 251 25 , 641 , 096 , 953 , 472 , 000 , 000 , 000 v 12 41 , 171 , 207 , 949 , 008 , 915 , 273 251 , 282 , 750 , 144 , 025 , 600 , 000 , 000 , 000 v 14 + . . . , d 3 = 300 , 125 962 , 616 11 5760 v 4 + 38 , 503 56 , 448 , 000 v 6 121161749 1 , 244 , 678 , 400 , 000 v 8 + 1 , 283 , 041 , 005 , 779 100 , 632 , 248 , 640 , 000 , 000 v 10 42 , 487 , 644 , 873 , 726 , 467 25 , 641 , 096 , 953 , 472 , 000 , 000 , 000 v 12 + 162 , 263 , 941 , 292 , 140 , 283 , 723 753 , 848 , 250 , 432 , 076 , 800 , 000 , 000 , 000 v 14 + . . . . .
As v 0 , the coefficients b 2 , b 3 , d 2 and d 3 of the lower order method reduces to the coefficients of the original method (lower order ). That is to say, b 2 ( 0 ) , b 3 ( 0 ) , d 2 ( 0 ) and d 3 ( 0 ) are identical to b 2 , b 3 , d 2 and d 3 of the lower order method in ERKN5(4)M.
In a similar way, solving the above system of Equations (9)–(12) using the coefficients of the higher order method (order 5) for b 1 , b 2 , d 1 , d 2 , we obtain the solution as given in Equation (15):
b ^ 1 = 25 , 581 , 600 cos v v + 25 , 581 , 600 v 51 , 163 , 200 sin v + 426360 v 5 4 , 574 , 700 v 3 14 , 003 v 7 + 6 , 395 , 400 sin v v 2 25 , 581 , 600 v 3 , b ^ 2 = 4 , 528 , 800 sin v 4 , 528 , 800 v 37 , 740 v 5 + 455175 v 3 + 592 v 7 2 , 264 , 400 v 3 , d ^ 1 = 3 , 511 , 200 sin v v 14 , 003 v 6 + 146 , 300 v 4 1 , 213 , 100 v 2 + 7 , 022 , 400 + 877 , 800 v 2 cos v 7 , 022 , 400 cos v 3 , 511 , 200 v 2 , d ^ 2 = 310 , 800 cos v 310 , 800 12 , 950 v 4 + 114 , 275 v 2 + 296 v 6 155 , 400 v 2 .
The corresponding Taylor series expansion of the solution is given in Equation (16):
b ^ 1 = 479 5016 233 428 , 400 v 4 + 11 362 , 880 v 6 37 79 , 833 , 600 v 8 + 1 222 , 393 , 600 v 10 79 2 , 615 , 348 , 736 , 000 v 12 + 53 355 , 687 , 428 , 096 , 000 v 14 + . . . , b ^ 2 = 235 1776 + 29 214 , 200 v 4 1 181 , 440 v 6 + 1 19 , 958 , 400 v 8 1 3 , 113 , 510 , 400 v 10 + 1 653 , 837 , 184 , 000 v 12 1 177 , 843 , 714 , 048 , 000 v 14 + . . . , d ^ 1 = 479 5016 11 12 , 600 v 4 + 1 5040 v 6 29 7 , 257 , 600 v 8 + 23 479 , 001 , 600 v 10 67 174 , 356 , 582 , 400 v 12 + 23 10 , 461 , 394 , 944 , 000 v 14 + . . . , d ^ 2 = 235 888 + 11 12 , 600 v 4 1 20 , 160 v 6 + 1 1 , 814 , 400 v 8 1 239 , 500 , 800 v 10 + 1 43 , 589 , 145 , 600 v 12 1 10 , 461 , 394 , 944 , 000 v 14 + . . . .
As v 0 , the coefficients b ^ 1 , b ^ 2 , d ^ 1 and d ^ 2 of the higher order method reduces to the coefficients of the original method (higher order ). That is to say, b ^ 1 ( 0 ) , b ^ 2 ( 0 ) , d ^ 1 ( 0 ) and d ^ 2 ( 0 ) are identical to b ^ 1 , b ^ 2 , d ^ 1 and d ^ 2 of the higher order method in ERKN5(4)M.
The above two solutions found resulted in the new Embedded explicit trigonometrically-fitted Runge–Kutta–Nyström 5(4) pair (EETFRKN5(4)).

4. Algebraic Order and Error Analysis

In this section, we will find the local truncation error of the new methods and verify their algebraic order. We first find the Taylor series expansion of the actual solution y ( x n + h ) , the first derivative of the actual solution y ( x n + h ) , the approximate solution y n + 1 , and the first derivative of the approximate solution y n + 1 . The local truncation error (LTE) of y and its first derivative y is given as:
L T E = y n + 1 y ( x n + h ) , L T E d e r = y n + 1 y ( x n + h ) .
The L T E and L T E d e r of the lower order method (order 4) are:
L T E = 206 , 089 h 5 2 , 623 , 950 , 000 ( 3 y f y y y + 6 w 2 f x + 6 f y f x + ( y ) 3 f y y y + 3 y f x y + 6 ( f y ) 2 y + 3 y f y x x + 3 ( y ) 2 f x y y + 6 w 2 f y y + f x x x ) + O ( h 6 ) , L T E d e r = h 5 120 ( 4 ( y ) 3 f x y y y + ( y ) 4 f y y y y + 6 ( y ) 2 f x x y y + 3 ( y ) 2 f y y + 4 f x y f x + y f y x x + ( f y ) 2 y + f y f x x + 4 y f x x x y + f x x x x + 6 ( y ) 2 f y y y y + 5 ( y ) 2 f y y f y + 12 y f x y y y + 6 f y y f x y + 4 y f y y f x ) + O ( h 6 ) .
From Equation ( 18 ) , we can see that the algebraic order of the lower order method is 4 because all of the coefficients up to h 4 vanished. Similarly, the L T E and L T E d e r of the higher order method (order 5) are :
L T E = 479 h 6 10 , 281 , 600 ( 6 ( y ) 2 f y y y y + 1416 479 ( y ) 2 f y y f y + 12 y f x y y y + 5616 479 f y y f x y 2784 479 y f y y f x 4200 479 w 4 y + 6 ( y ) 2 f x x y y 2784 479 f x y f x + 4 y f x x x y + ( y ) 4 f y y y y + 4200 479 ( f y ) 2 y + 4 ( y ) 3 f x y y y + 3 ( y ) 2 f y y + 6 y f y x x + 4200 479 f y f x x + f x x x x ) + O ( h 7 ) , L T E d e r = h 6 720 ( 10 y f x x x y + 5 y f x x x x y + ( f y ) 3 y + 10 f y x x f x + ( y ) 5 f y y y y y + 10 y ( f x y ) 2 + 10 ( y ) 2 f x x x y y + 5 ( y ) 4 f x y y y y + ( f y ) 2 f x + 5 f x x f x y + f y f x x x + 18 y f y y f y y + f x x x x x + 5 ( y ) 3 ( f y y ) 2 + 10 ( y ) 3 f x x y y y + 15 ( y ) 2 f x y y + 10 ( y ) 2 f y y y f x + 11 ( y ) 3 f y y y f y + 15 y f y y y ( y ) 2 + 15 ( y ) 2 f y y f x y + 30 y f x x y y y + 8 f y y f x y + 10 y f y y f x + 13 f y y f y x x + 5 y f y y f x x + 30 ( y ) 2 f x y y y y + 20 y f x y y f x + 10 ( y ) 3 f y y y y y + 23 ( y ) 2 f y f x y y ) + O ( h 7 ) .
From Equation ( 19 ) , the higher order method has order 5 because all of the coeffients up to h 5 vanished.

Absolute Stability Analysis of the New Embedded Pair

The linear stability of the RKN method (2)–(4) is derived by applying it to the test equation y = w 2 y . Setting H = ( w h ) 2 , the numerical solution satisfies the following:
Z n + 1 = P ( H ) Z n ,
where Z n + 1 = y n + 1 h y n + 1 , Z n = y n h y n , P ( H ) = 1 + H b T N 1 e w h ( 1 + H b T N 1 c ) w h d T N 1 e 1 + H d T N 1 c , N = I H A , A = a 11 a 12 a 13 a 14 a 21 a 22 a 23 a 24 a 31 a 32 a 33 a 34 a 41 a 42 a 43 a 44 , I = 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 , b = [ b 1 , b 2 , b 3 , b 4 ] T , d = [ d 1 , d 2 , d 3 , d 4 ] T , e = [ 1 , 1 , 1 , 1 ] T , c = [ c 1 , c 2 , c 3 , c 4 ] T . It is assumed that P ( H ) has complex conjugate eigenvalues for sufficiently small values of v [14]. With this assumption, a periodic numerical solution is obtained whose characteristic depends on the eigenvalues of P ( H ) , which is called the stability matrix and its characteristic equation can be written as
λ 2 t r ( P ( H ) ) λ + d e t ( P ( H ) ) = 0 .
Definition 2.
An interval ( H b , 0 ) of the RKN method (2)–(4) is said to be absolutely stable if for all H ( H b , 0 ) , | λ 1 , 2 | < 1 , where λ 1 , 2 are the roots of P ( H ) .
Hence, we obtain the approximate interval of absolute stability of the higher order method (order 5) of our new embedded pair as ( 36.99 , 0 ) and the lower order method (order 4) has no interval of absolute stability.

5. Numerical Results

In order to show the efficiency of the new method, we use the following RK and RKN pairs for the numerical comparison:
  • EETFRKN5(4)M: The new embedded explicit trigonometrically-fitted Runge–Kutta–Nyström pair (EETFRKN5(4)) derived in this paper,
  • ERKN6(4)6ER: A 6(4) optimized embedded Runge–Kutta–Nyström pair derived by Anastassi and Kosti in [11],
  • ERKN4(3): The embedded Runge–Kutta–Nyström method obtained by Van de Vyver in [6],
  • ARKN5(3)S: A 5(3) pair of explicit adapted Runge–Kutta–Nyström method derived by Franco in [5],
  • DOP5(4): A 5(4) Dormand and Prince embedded Runge–Kutta method given by Butcher in [15],
and by considering the following problems.
Problem 1.
(Almost Periodic Orbit Problem) Senu et al. in [8]
y 1 = y 1 + 0.001 cos ( x ) , y 1 ( 0 ) = 1 , y 1 ( 0 ) = 0 , y 2 = y 2 + 0.001 sin ( x ) , y 2 ( 0 ) = 0 , y 2 ( 0 ) = 0.9995 , x e n d = 100 .
The exact solution is
y 1 ( x ) = cos ( x ) + 0.0005 x cos ( x ) ,
y 2 ( x ) = sin ( x ) 0.0005 x sin ( x ) .
Problem 2.
(Nonlinear System) Fang et al. in [16]
y 1 = 4 x 2 y 1 2 y 2 ( y 1 2 + y 2 2 ) 1 / 2 , y 1 ( 0 ) = 1 , y 1 ( 0 ) = 0 , y 2 = 4 x 2 y 2 + 2 y 1 ( y 1 2 + y 2 2 ) 1 / 2 , y 2 ( 0 ) = 0 , y 2 ( 0 ) = 0 , x e n d = 10 .
The exact solution is
y 1 ( x ) = cos ( x 2 ) , y 2 ( x ) = sin ( x 2 ) .
Problem 3.
(Nonhomogeneous System) Senu et al. in [8]
y 1 = v 2 y 1 ( x ) + v 2 f ( x ) + f ( x ) , y 1 ( 0 ) = a + f ( 0 ) , y 1 ( 0 ) = f ( 0 ) , y 2 = v 2 y 2 ( x ) + v 2 f ( x ) + f ( x ) , y 2 ( 0 ) = f ( 0 ) , y 2 ( 0 ) = a v + f ( 0 ) , x e n d = 100 .
The exact solution is
y 1 ( x ) = a cos ( v x ) + f ( x ) , y 2 ( x ) = a sin ( v x ) + f ( x ) ,
where v = 1.0 , a = 0.1 are parameters and f ( x ) = e 10 x .
Problem 4.
(Almost Periodic Problem) Van de Vyver in [6]
y 1 = y 1 + ϵ cos ( Ψ x ) , y 1 ( 0 ) = 1 , y 1 ( 0 ) = 0 , y 2 = y 2 + ϵ sin ( Ψ x ) , y 2 ( 0 ) = 0 , y 2 ( 0 ) = 1 .
The exact solution is
y 1 ( x ) = ( 1 ϵ Ψ 2 ) ( 1 Ψ 2 ) cos ( x ) + ϵ ( 1 Ψ 2 ) cos ( Ψ x ) ,
y 2 ( x ) = ( 1 ϵ Ψ Ψ 2 ) ( 1 Ψ 2 ) sin ( x ) + ϵ ( 1 Ψ 2 ) sin ( Ψ x ) ,
where ϵ = 0.001 , Ψ = 0.1 and x e n d = 100 .
The numerical results are shown in Table 2, Table 3, Table 4 and Table 5.
We further display the performance of these methods graphically in Figure 1, Figure 2, Figure 3 and Figure 4. The tolerances used for Problem 1 are: Tol = 10 2 i , i = 1 , 2 , 3 , 5 , and that of Problems 2–4 are: Tol = 10 2 i , i = 1 , 2 , 3 , 4 , 5 .

6. Discussion

For all of the problems tested and from Figure 1, Figure 2, Figure 3 and Figure 4, we can deduce that our new method has a lower number of function evaluations per step, which signifies that our new method has less computational costs than the other existing methods, and is therefore more suitable for solving second order ordinary differential equations with oscillatory solutions than the other existing methods in the literature.

7. Conclusions

In this study, we have derived a new efficient 5(4) embedded pair of explicit trigonometrically-fitted Runge–Kutta–Nyström methods for the solution of oscillatory initial value problems. The numerical results obtained indicate that the function evaluations per step of the new method are less when compared with the other existing embedded pairs. Hence, the new method has less computational costs than the other existing methods, and, therefore, the efficiency of the new method is higher than the other existing methods.

Acknowledgments

The authors are very grateful to the reviewers of this manuscript for their constructive comments and the Institute of Mathematical Research (INSPEM), Department of Mathematics, Universiti Putra Malaysia for the support and assistance given during the research work.

Author Contributions

M.A.D. and N.S. conceived and designed the experiments; M.A.D. performed the experiments; M.A.D. and N.S. analyzed the data; F.I. Supervised; M.A.D. wrote the paper.

Conflicts of Interest

The authors declare no conflict of interest.

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Figure 1. Efficiency curves for Problem 1.
Figure 1. Efficiency curves for Problem 1.
Mca 22 00028 g001
Figure 2. Efficiency curves for Problem 2.
Figure 2. Efficiency curves for Problem 2.
Mca 22 00028 g002
Figure 3. Efficiency curves for Problem 3.
Figure 3. Efficiency curves for Problem 3.
Mca 22 00028 g003
Figure 4. Efficiency curves for Problem 4.
Figure 4. Efficiency curves for Problem 4.
Mca 22 00028 g004
Table 1. The Embedded Runge–Kutta–Nyström 5(4)M (ERKN5(4)M) Method in [13].
Table 1. The Embedded Runge–Kutta–Nyström 5(4)M (ERKN5(4)M) Method in [13].
0
1 2 1 8
19 70 2907 343000 1216 42875
44 51 6624772 128538819 6273905 54121608 210498365 1028310552
479 5016 235 1776 145775 641744 309519 6873416
479 5016 235 888 300125 962616 2255067 6873416
184883 2021250 411163 3399375 6 25 593028 12464375
479 5016 235 888 300125 962616 2255067 6873416
Table 2. Numerical results for Problem 1.
Table 2. Numerical results for Problem 1.
TOLMETHODSTEPFCNFSTEPMAXETIME (s)
10 2 EETFRKN5(4)M6325201.589499(-4)0.312
ARKN5(3)S24296809.829659(-1)0.812
ERKN6(4)6ER215129007.375736(-1)1.063
ERKN4(3)578231512.175738(-1)1.031
DOP5(4)12172607.747317(-3)0.531
10 4 EETFRKN5(4)M13252803.336685(-6)0.359
ARKN5(3)S1044417916.406274(-2)1.547
ERKN6(4)6ER825495505.685758(-2)1.531
ERKN4(3)290111,61028.650517(-3)2.000
DOP5(4)262157201.273632(-4)1.531
10 6 EETFRKN5(4)M283113207.329552(-8)1.230
ARKN5(3)S449117,97023.460856(-3)3.044
ERKN6(4)6ER318019,09023.817832(-3)2.904
ERKN4(3)29,131116,53648.569318(-5)14.371
DOP5(4)561336602.521377(-6)2.531
10 10 EETFRKN5(4)M654261601.089911(-9)1.250
ARKN5(3)S83,362333,46041.003239(-5)41.557
ERKN6(4)6ER94,782568,71754.291853(-6)41.977
ERKN4(3)2,940,66811,762,69688.077835(-9)1347.362
DOP5(4)521031,26003.630829(-11)4.531
EETFRKN5(4)M: The new embedded explicit trigonometrically-fitted Runge–Kutta–Nyström pair derived in this paper; ARKN5(3)S: A 5(3) pair of explicit Adapted Runge–Kutta–Nyström method derived by Franco in [5]; ERKN4(3): The embedded Runge–Kutta–Nyström method obtained by Van de Vyver in [6]; ERKN6(4)6ER: A 6(4) optimized embedded Runge–Kutta–Nyström pair derived by Anastassi and Kosti in [11]; DOP5(4): A 5(4) embedded Runge–Kutta method given by Butcher in [15]; TOL: The tolerance used; STEP: Execution steps; FCN: Function evaluation; FSTEP: Are the failed steps; MAXE: Maximum global error; TIME (s): Execution time in seconds.
Table 3. Numerical results for Problem 2.
Table 3. Numerical results for Problem 2.
TOLMETHODSTEPFCNFSTEPMAXETIME (s)
10 2 EETFRKN5(4)M7732041.868952(-2)0.500
ARKN5(3)S391157957.585297(-2)1.297
ERKN6(4)6ER412250262.409650(-1)1.685
ERKN4(3)1391558574.620804(-2)1.984
DOP5(4)14890841.003445(-1)0.703
10 4 EETFRKN5(4)M20382441.883175(-4)0.844
ARKN5(3)S1768709063.050204(-3)2.250
ERKN6(4)6ER211912,74971.047787(-2)2.188
ERKN4(3)627825,13682.250399(-3)5.234
DOP5(4)369223448.921876(-4)1.020
10 6 EETFRKN5(4)M507204041.389967(-6)1.609
ARKN5(3)S800132,02571.295624(-4)5.636
ERKN6(4)6ER974458,50484.798031(-4)6.125
ERKN4(3)30,571122,31198.727528(-5)15.296
DOP5(4)915551047.372093(-6)2.703
10 8 EETFRKN5(4)M1208484441.809555(-8)1.844
ARKN5(3)S36,949147,82396.030980(-6)19.549
ERKN6(4)6ER42,606255,68192.359393(-5)20.213
ERKN4(3)300,521120,214108.837019(-7)134.762
DOP5(4)220213,23247.347484(-8)3.124
10 10 EETFRKN5(4)M296811,88751.891552(-10)3.584
ARKN5(3)S180,503722,042103.607369(-7)86.196
ERKN6(4)6ER202,6031,215,673111.132031(-6)86.814
ERKN4(3)3,017,35412,069,452121.646494(-8)1620.306
DOP5(4)537832,29351.071751(-9)5.223
Table 4. Numerical results for Problem 3.
Table 4. Numerical results for Problem 3.
TOLMETHODSTEPFCNFSTEPMAXETIME (s)
10 2 EETFRKN5(4)M6526312.665158(-3)0.250
ARKN5(3)S13354131.336913(-1)0.328
ERKN6(4)6ER219132426.973318(-2)0.422
ERKN4(3)588236132.107025(-2)0.719
DOP5(4)12777221.107309(+1)0.344
10 4 EETFRKN5(4)M13855517.897400(-5)0.359
ARKN5(3)S551221332.752336(-2)0.828
ERKN6(4)6ER429258932.187558(-2)0.609
ERKN4(3)300512,03558.461092(-4)1.703
DOP5(4)529318421.1000002(+1)1.030
10 6 EETFRKN5(4)M16466225.757119(-7)0.422
ARKN5(3)S452318,137159.530356(-4)2.188
ERKN6(4)6ER168810,15351.495030(-3)1.047
ERKN4(3)15,70962,85773.359187(-5)6.748
DOP5(4)1143686821.099999(+1)2.344
10 8 EETFRKN5(4)M355142628.919276(-9)0.625
ARKN5(3)S994339,79681.652088(-4)4.453
ERKN6(4)6ER655039,33061.003134(-4)3.069
ERKN4(3)85,151340,63191.337546(-6)36.152
DOP5(4)248314,91331.100000(+1)3.031
10 10 EETFRKN5(4)M774310221.223295(-10)0.781
ARKN5(3)S43,042172,18664.372055(-6)17.310
ERKN6(4)6ER48,932293,63281.680967(-6)20.141
ERKN4(3)855,3493,421,429111.314822(-8)356.970
DOP5(4)537932,28931.100000(+1)4.344
Table 5. Numerical results for Problem 4.
Table 5. Numerical results for Problem 4.
TOLMETHODSTEPFCNFSTEPMAXETIME (s)
10 2 EETFRKN5(4)M12248805.791290(-4)0.424
ARKN5(3)S24296802.111440(-1)0.453
ERKN6(4)6ER215129007.335990(-1)0.453
ERKN4(3)578231512.168852(-1)0.844
DOP5(4)12172607.766793(-3)0.429
10 4 EETFRKN5(4)M262104801.294696(-5)0.578
ARKN5(3)S1044417911.141697(-2)1.109
ERKN6(4)6ER825495515.673723(-2)0.781
ERKN4(3)290111,61028.642825(-3)1.547
DOP5(4)262157201.275701(-4)1.422
10 6 EETFRKN5(4)M562224802.671335(-7)0.656
ARKN5(3)S449117,97026.156293(-4)2.219
ERKN6(4)6ER318019,09023.814429(-3)1.656
ERKN4(3)29,131116,53648.562316(-5)12.500
DOP5(4)561336602.521051(-6)2.422
10 8 EETFRKN5(4)M1211484405.680715(-9)0.859
ARKN5(3)S1934777,39733.316106(-5)8.094
ERKN6(4)6ER24,548147,30846.394316(-5)10.047
ERKN4(3)292,6761,170,72268.477710(-7)123.098
DOP5(4)1210726005.343219(-8)3.422
10 10 EETFRKN5(4)M26221048801.301748(-10)1.500
ARKN5(3)S83,362333,46041.785978(-6)34.359
ERKN6(4)6ER94,782568,71754.2883486(-6)37.406
ERKN4(3)2,940,66811,762,69688.074738(-9)1164.595
DOP5(4)521031,26003.620826(-11)4.422
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