Hi akhter,
Please refer below sample.
HTML
<asp:ScriptManager ID="ScriptManager1" runat="server"></asp:ScriptManager>
<asp:UpdatePanel ID="UpdatePanel1" runat="server">
    <ContentTemplate>
        <asp:DropDownList ID="ddlpurchase" runat="server" AutoPostBack="true"
            CssClass="form-control js-example-placeholder-single" Height="85px"
            ToolTip="Select " Width="280px">
            <asp:ListItem>---Select---</asp:ListItem>
            <asp:ListItem>ASPSnippets</asp:ListItem>
            <asp:ListItem>AspForums</asp:ListItem>
        </asp:DropDownList>
        <br />
        <asp:Button ID="btnRefresh" runat="server" Text="Refresh" />
    </ContentTemplate>
</asp:UpdatePanel>
<script type="text/javascript" src="https://ajax.googleapis.com/ajax/libs/jquery/1.8.3/jquery.min.js"></script>
<script type="text/javascript" src="https://cdnjs.cloudflare.com/ajax/libs/select2/4.0.3/js/select2.min.js"></script>
<link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/select2/4.0.3/css/select2.min.css" />
<link href="http://ajax.aspnetcdn.com/ajax/jquery.ui/1.8.9/themes/start/jquery-ui.css"
    rel="stylesheet" type="text/css" />
<script type="text/javascript">
    $(function () {
        $(".js-example-placeholder-single").select2({
            placeholder: "Select",
            allowClear: true
        });
    });
            
    var prm = Sys.WebForms.PageRequestManager.getInstance();
    if (prm != null) {
        prm.add_endRequest(function (sender, e) {
            if (sender._postBackSettings.panelsToUpdate != null) {
                $(".js-example-placeholder-single").select2({
                    placeholder: "Select",
                    allowClear: true
                });
            }
        });
    };
</script>