aspg-3-4156-1791943240 20261014020040 American Scientific Publishing Group admin@americaspg.com American Scientific Publishing Group Fusion: Practice and Applications FPA 2770-0070 2692-4048 2026 21 1 Ahmed Bakheet Mustansiriyah University, College of Science, Department of Mathematics, Baghdad, Iraq Ali Abdulhussein Mustansiriyah University, College of Science, Department of Mathematics, Baghdad, Iraq Laheeb Muhsen Noman Mustansiriyah University, College of Science, Department of Mathematics, Baghdad, Iraq Normal false false false EN-US X-NONE AR-SA DefSemiHidden="false" DefQFormat="false" DefPriority="99" LatentStyleCount="371"> UnhideWhenUsed="true" QFormat="true" Name="heading 2"/> UnhideWhenUsed="true" QFormat="true" Name="heading 3"/> UnhideWhenUsed="true" QFormat="true" Name="heading 4"/> UnhideWhenUsed="true" QFormat="true" Name="heading 5"/> UnhideWhenUsed="true" QFormat="true" Name="heading 6"/> UnhideWhenUsed="true" QFormat="true" Name="heading 7"/> UnhideWhenUsed="true" 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7 Colorful Accent 2"/> Name="Grid Table 1 Light Accent 3"/> Name="Grid Table 6 Colorful Accent 3"/> Name="Grid Table 7 Colorful Accent 3"/> Name="Grid Table 1 Light Accent 4"/> Name="Grid Table 6 Colorful Accent 4"/> Name="Grid Table 7 Colorful Accent 4"/> Name="Grid Table 1 Light Accent 5"/> Name="Grid Table 6 Colorful Accent 5"/> Name="Grid Table 7 Colorful Accent 5"/> Name="Grid Table 1 Light Accent 6"/> Name="Grid Table 6 Colorful Accent 6"/> Name="Grid Table 7 Colorful Accent 6"/> Name="List Table 1 Light Accent 1"/> Name="List Table 6 Colorful Accent 1"/> Name="List Table 7 Colorful Accent 1"/> Name="List Table 1 Light Accent 2"/> Name="List Table 6 Colorful Accent 2"/> Name="List Table 7 Colorful Accent 2"/> Name="List Table 1 Light Accent 3"/> Name="List Table 6 Colorful Accent 3"/> Name="List Table 7 Colorful Accent 3"/> Name="List Table 1 Light Accent 4"/> Name="List Table 6 Colorful Accent 4"/> Name="List Table 7 Colorful Accent 4"/> Name="List Table 1 Light Accent 5"/> Name="List Table 6 Colorful Accent 5"/> Name="List Table 7 Colorful Accent 5"/> Name="List Table 1 Light Accent 6"/> Name="List Table 6 Colorful Accent 6"/> Name="List Table 7 Colorful Accent 6"/> /* Style Definitions */ table.MsoNormalTable {mso-style-name:"Table Normal"; mso-tstyle-rowband-size:0; mso-tstyle-colband-size:0; mso-style-noshow:yes; mso-style-priority:99; mso-style-parent:""; mso-padding-alt:0in 5.4pt 0in 5.4pt; mso-para-margin-top:0in; mso-para-margin-right:0in; mso-para-margin-bottom:8.0pt; mso-para-margin-left:0in; line-height:107%; mso-pagination:widow-orphan; font-size:11.0pt; font-family:"Calibri",sans-serif; mso-ascii-font-family:Calibri; mso-ascii-theme-font:minor-latin; mso-hansi-font-family:Calibri; mso-hansi-theme-font:minor-latin; mso-bidi-font-family:Arial; mso-bidi-theme-font:minor-bidi;} The main goal of this paper is to obtain a special form of asymptotic solutions to the scalar version of the two body problem whenever the two bodies collide on the real line at the collision time. It has been shown that the desired asymptotic solution maintains certain properties when t approaches the collision time. However, it is not easy to Handel such a mission without the employment of successive approximations technique. The successive approximations technique has been modified and adjusted to serve as the main tool in the process of obtaining such solution. Moreover, it has been shown that the series of successive approximations converges absolutely and uniformly to a continuous function that approaches to 0 when t attains the collision time in a certain interval. The problem of one dimensional collision between the two bodies has been solved asymptotically at the collision time. 2026 307 335 4156 https://creativecommons.org/licenses/by/4.0/ 10.54216/FPA.210122 https://americaspg.com/journal/3/article/4156 https://americaspg.com/storage/71760717489.pdf