Previous numerical methods for solving definite integrals based on Haar wavelets was introduced. In this research we extend the previous works by approximate the solution of improper integral based on Haar wavelets functions. Error analysis of the approximation by Haar wavelets are given. Numerical examples are conducted to show the accuracy of the method.

The current study provides a numerical method that is derived in a backward difference formulation for ordinary differential equations. The proposed method employs a constant step size algorithm of order 12. The backward difference formulation serves as a competitive algorithm for solving ordinary differential equations. In the current study, the backward difference method is used to analyze the dynamics of capital stocks in terms of depreciation rate for the capital–labor ratio. Results provided in this study will validate the accuracy of the backward difference algorithm hence proving it as a viable alternative for analyzing economic problems in the form of ordinary differential equations.

The public key cryptosystem is fundamental in safeguard communication in cyberspace. This paper described a new cryptosystem analogous to El-Gamal encryption scheme, which utilizing the Lucas sequence and Elliptic Curve. Similar to Elliptic Curve Cryptography (ECC) and Rivest-Shamir-Adleman (RSA), the proposed cryptosystem requires a precise hard mathematical problem as the essential part of security strength. The chosen plaintext attack (CPA) was employed to investigate the security of this cryptosystem. The result shows that the system is vulnerable against the CPA when the sender decrypts a plaintext with modified public key, where the cryptanalyst able to break the security of the proposed cryptosystem by recovering the plaintext even without knowing the secret key from either the sender or receiver.

In this work, generalize solution based on linear Legendre multi-wavelets are proposed for single, double and triple integrals with variable limits. To obtain the numerical approximations for integrals, an algorithm with the properties of linear Legendre multi-wavelets are applied. The main benefits of this method are its simple applicable and efficient. Furthermore, the error analysis for single, double and triple integrals is worked out to show the efficiency of the method. Numerical examples for the integrals are conducted by using linear Legendre multi-wavelets in order to validate the error estimation.

In the previous research, a direct computational method based on linear Legendre multi-wavelets has been applied for solving definite integrals. However, the error analysis to show the convergence of the method has not been discussed. Therefore, error analysis of the approximation method is established in the Holder classes Hs[0, 1] to show the efficiency of the method. The connections of the module of difference smoothness of the function is also established. Finally, some numerical examples of the implementation the method for the functions from Holder classes are presented.

The current study proposes a numerical method which solves nonlinear Fredholm and Volterra integral of the second kind using a combination of a Newton–Kantorovich and Haar wavelet. Error analysis for the Holder classes was established to ensure convergence of the Haar wavelets. Numerical examples will illustrate the accuracy and simplicity of Newton–Kantorovich and Haar wavelets. Numerical results of the current method were then compared with previous well-established methods.

The security of LUC type cryptosystems was investigated. In this study, chosen message attack was employed to analyze the security of LUC, LUC3 and LUC4,6 cryptosystems. The cryptanalyst invades the system by obtaining a signature without the sender’s consent, and use it to break the system. Finding shows that LUC4,6 cryptosystem is more resilient against chosen message attack compare with LUC and LUC3 cryptosystems.

Elliptic Curve Cryptosystem based on second order Lucas sequenceis a cryptosystem using elliptic curves over finite fields as a mask and incorporate with second order of Lucas sequence. The security of the Elliptic Curve Cryptography cryptosystem depends on the discrete logarithms. In this cryptosystem, Lucas sequence is employed to compute the ciphertext or recover the plaintext. The Elliptic Curve Cryptosystem based on second order Lucas sequence is vulnerable when the bit of the decryption key, $d$ flips by using fault based attack.

LUC-type cryptosystems are asymmetric key cryptosystems based on the Lucas sequence that is extended from RSA. The security challenge is comparable to RSA, which is based on the intractability of factoring a large number. This paper analysed the security of LUC, LUC3, and LUC4,6 cryptosystems using a common modulus attack.For a common modulus attack to be successful, a message must be transmitted to two distinct receivers with the same modulus. The strengths and limitations of the LUC, LUC3, and LUC4,6 cryptosystems when subjected to a common modulus attack were discussed as well. The results reveal that the LUC4,6 cryptosystem provides greater security than the LUC and LUC3.

This research proposes a three-point block method for solving Duffing type higher order ordinary differential equations (ODEs) which is also commonly referred as the Duffing oscillator. The research conducted implements a variable order step size technique for approximating the exact solution for the Duffing Oscillator. The proposed algorithm will be tested against various Duffing oscillators and numerical approximation will be compared with current viable methods. The accuracy and efficiency of the proposed method will be illustrated in the numerical results.

The current study was conducted to establish a new numerical method for solving Duffing type differential equations. Duffing type differential equations are often linked to damping issues in physical systems, which can be found in control process problems. The proposed method is developed using a three-point block method in backward difference form, which offers an accurate approximation of Duffing type differential equations with less computational cost. Applying an Adam's like predictor-corrector formulation, the three point block method is programmed with a recursive relationship between explicit and implicit coefficients to reduce computational cost. By establishing this recursive relationship, we established a corrector algorithm in terms of the predictor. This eliminates any undesired redundancy in the calculation when obtaining the corrector. The proposed method allows a more efficient solution without any significant loss of accuracy. Four types of Duffing differential equations are selected to test the viability of the method. Numerical results will show efficiency of the three-point block method compared against conventional and more established methods. The outcome of this research is a new method for successfully solving Duffing type differential equation and other ordinary differential equations that are found in the field of science and engineering. An added advantage of the three-point block method is its adaptability to parallel programming.

The current research aims to provide a viable numerical method for solving difficult engineering and science problems which are in the form of higher order ordinary differential equations. The proposed method approximates these ordinary differential equations using Newton-Gregory backward difference polynomial in predictor–corrector mode. The predictor–corrector algorithm is then fitted with a variable order step size algorithm to reduce computational cost. The variable order stepsize algorithm allows the method to predetermine the preferred level of accuracy with the added advantage of less computational cost. The method is subsequently programmed with a two-point block formulation which can be altered for parallel programming. This research also discusses order and stepsize strategies of the variable order stepsize algorithm. Stability and convergence estimations of the method are also established. Numerical results obtained will validate the accuracy and efficiency of the method using various types of linear and nonlinear higher order ordinary differential equations