In this study, we considered time-fractional nonlinear telegraph equations (TFNLTEs). For solving the TFNLTEs, we deployed a method known as the Multistep Modified Reduced Differential Transform Method (MMRDTM). Prior to the multistep technique, the nonlinear term in TFNLTEs is replaced with corresponding Adomian polynomials. It can be observed that the MMRDTM is much simpler and more straightforward. On top of that, it works exceptionally where the obtained solutions are more accurately approximated over time. To demonstrate the performance of the MMRDTM in terms of its capabilities and accuracies, we provided two numerical examples of solving TFNLTEs by using MMRDTM and Modified Reduced Differential Transform Method (MRDTM). By comparing the absolute errors of the obtained solutions by both methods, we demonstrated that the solutions provided by the MMRDTM much closer to the exact solutions compared to the corresponding solutions yielded by the MRDTM. This justified that the MMRDTM provides highly accurate and precise solutions for the TFNLTEs.

Chaos-based image encryption system is an encryption method that uses chaotic systems in encrypting digital images for the purpose of enhancing security. Chaos theory exhibits some distinctive characteristics, which include butterfly-like patterns, unpredictable behavior, and sensitive dependence to initial conditions. . In the past few decades, image encryption based on a single chaotic map has been a common technique in encrypting images. However, there are still unauthorized interceptors who illegally access and obtain the private information of the image. With that being the case, an encryption method that applies more than one chaotic system will contribute to creating a more complex relationship between the original and encrypted images, making it challenging for unauthorized individuals to decipher or extract the original content of the image without the appropriate decryption key. In this study, two chaotic maps, 3D Lorenz Map and 3D Chen system, are applied to generate random cryptographic keys for encrypting color images using permutation and diffusion mechanisms. The proposed algorithm that employs two chaotic maps is proven to be an effective system that achieves excellent results in security.

The Multistep Modified Reduced Differential Transform Method (MMRDTM) is proposed in this paper. It is implemented to solve the forced nonlinear Korteweg-de Vries (fnKdV) equations. There are several advantages of the proposed method. Using the suggested approach, a high-speed converging series can be approximated analytically. Furthermore, the number of calculated terms is reduced significantly. The nonlinear term in fnKdV equations is substituted with corresponding Adomian polynomials before applying the multi-step technique. As a result, we provided simpler and more effective ways to solve fnKdV equations. On top of that, the solutions can be approximated more accurately over a longer period of time. To show the MMRDTM's capability and accuracy, we consider several of the fnKdV examples to illustrate the proposed method’s potential in analytical approximation. Then, the features of the solutions are represented in tabular and graphical forms. In conclusion, the proposed method delivers highly accurate and precise solutions for these types of equations.

We may secure the content of text, audio, image and video during their transmission from one party to another party via an open channel such as the internet by using cryptograph. Logistic-Sine System (LSS) is a combination on two 1D chaotic maps which are Logistic Map and Sine Map. By applying the LSS into cryptography, the image encryption and decryption can be performed. This study is focusing on the performance test of the image encryption and decryption processes by using the LSS. For comparison purpose, we compare the performance of the encryption and decryption by using two different chaotic systems, which are the LSS and Logistic-Tent System (LTS). The result shows that system with LSS is less efficient than LTS in term of encryption time but both systems have similar efficiency in term of decryption time.

In this study, sixth-order boundary value problems for linear and nonlinear differential equations have been solved by using Differential Transformation Method (DTM). The numerical solutions are given in several examples. For each example, the solution given by DTM is compared with the exact solution. Absolute relative error (ARE) for each iteration can be computed. Therefore, the maximum absolute relative error (MARE) of the DTM can be obtained. To show that the solution given by the DTM has higher level of accuracy, the absolute relative error of the DTM has been compared with the other methods such as Adomian decomposition method with Green’s function, modified decomposition method (MDM), homotopy perturbation method (HPM), Variational Iteration Method (VIM) and Quintic B-Spline Collocation Method. Comparison graphs are given at the end of this paper. The obtained result shows that the proposed method is able to provide better approximation in term of accuracy.

This study proposed a new version of Multistep Modified Reduced Differential Transform Method (MMRDTM) which is applicable for obtaining semi-analytical approximation on Non-linear Telegraph Equations (NLTEs) with source term. Through the experimental approach, we unveiled various advantages of the proposed method, especially its ability to analytically approximate high-speed converging series. Moreover, a significant reduction in the number of calculated terms can also be observed. Before deploying the multistep technique, the substitution of non-linear term in NLTEs with the corresponding Adomian polynomial takes place. Consequently, a simpler technique for approximating NLTEs with source term was discovered. Besides that, the solutions can be estimated more precisely over a longer time. To justify our findings, solutions of NLTEs with source term are obtained using the MMRDTM. The accuracy of the proposed method for solving these equations was tested by comparing the absolute errors between the solutions obtained by the proposed method and the Modified Reduced Differential Transform Method (MRDTM) with the exact solution. The obtained solutions evidenced that the proposed MMRDTM can deliver highly precise approximations for the NLTEs with source term.